LMFDB - Embedded newform 150.4.c.a.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,4,Mod(49,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,-8,0,-12,0,0,-18,0,-120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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:	no (minimal twist has level 30)
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.a.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} +32.0000i q^{7} +8.00000i q^{8} -9.00000 q^{9} -60.0000 q^{11} +12.0000i q^{12} +34.0000i q^{13} +64.0000 q^{14} +16.0000 q^{16} +42.0000i q^{17} +18.0000i q^{18} +76.0000 q^{19} +96.0000 q^{21} +120.000i q^{22} +24.0000 q^{24} +68.0000 q^{26} +27.0000i q^{27} -128.000i q^{28} -6.00000 q^{29} -232.000 q^{31} -32.0000i q^{32} +180.000i q^{33} +84.0000 q^{34} +36.0000 q^{36} +134.000i q^{37} -152.000i q^{38} +102.000 q^{39} +234.000 q^{41} -192.000i q^{42} +412.000i q^{43} +240.000 q^{44} -360.000i q^{47} -48.0000i q^{48} -681.000 q^{49} +126.000 q^{51} -136.000i q^{52} -222.000i q^{53} +54.0000 q^{54} -256.000 q^{56} -228.000i q^{57} +12.0000i q^{58} -660.000 q^{59} -490.000 q^{61} +464.000i q^{62} -288.000i q^{63} -64.0000 q^{64} +360.000 q^{66} +812.000i q^{67} -168.000i q^{68} +120.000 q^{71} -72.0000i q^{72} -746.000i q^{73} +268.000 q^{74} -304.000 q^{76} -1920.00i q^{77} -204.000i q^{78} -152.000 q^{79} +81.0000 q^{81} -468.000i q^{82} +804.000i q^{83} -384.000 q^{84} +824.000 q^{86} +18.0000i q^{87} -480.000i q^{88} +678.000 q^{89} -1088.00 q^{91} +696.000i q^{93} -720.000 q^{94} -96.0000 q^{96} +194.000i q^{97} +1362.00i q^{98} +540.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} - 120 q^{11} + 128 q^{14} + 32 q^{16} + 152 q^{19} + 192 q^{21} + 48 q^{24} + 136 q^{26} - 12 q^{29} - 464 q^{31} + 168 q^{34} + 72 q^{36} + 204 q^{39} + 468 q^{41} + 480 q^{44}+ \cdots + 1080 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 - 120 * q^11 + 128 * q^14 + 32 * q^16 + 152 * q^19 + 192 * q^21 + 48 * q^24 + 136 * q^26 - 12 * q^29 - 464 * q^31 + 168 * q^34 + 72 * q^36 + 204 * q^39 + 468 * q^41 + 480 * q^44 - 1362 * q^49 + 252 * q^51 + 108 * q^54 - 512 * q^56 - 1320 * q^59 - 980 * q^61 - 128 * q^64 + 720 * q^66 + 240 * q^71 + 536 * q^74 - 608 * q^76 - 304 * q^79 + 162 * q^81 - 768 * q^84 + 1648 * q^86 + 1356 * q^89 - 2176 * q^91 - 1440 * q^94 - 192 * q^96 + 1080 * 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
$7$	32.0000i	1.72784i	0.503631	+	0.863919i	$0.331997\pi$
$7$	32.0000i	1.72784i	−0.503631	+	0.863919i	$0.668003\pi$
$8$	8.00000i	0.353553i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−60.0000	−1.64461	−0.822304	−	0.569049i	$-0.807311\pi$
$11$	−60.0000	−1.64461	−0.822304	+	0.569049i	$0.807311\pi$
$12$	12.0000i	0.288675i
$13$	34.0000i	0.725377i	0.931910	+	0.362689i	$0.118141\pi$
$13$	34.0000i	0.725377i	−0.931910	+	0.362689i	$0.881859\pi$
$14$	64.0000	1.22177
$15$	0	0
$16$	16.0000	0.250000
$17$	42.0000i	0.599206i	0.954064	+	0.299603i	$0.0968542\pi$
$17$	42.0000i	0.599206i	−0.954064	+	0.299603i	$0.903146\pi$
$18$	18.0000i	0.235702i
$19$	76.0000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	76.0000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	96.0000	0.997567
$22$	120.000i	1.16291i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	68.0000	0.512919
$27$	27.0000i	0.192450i
$28$	− 128.000i	− 0.863919i
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	−232.000	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$31$	−232.000	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$32$	− 32.0000i	− 0.176777i
$33$	180.000i	0.949514i
$34$	84.0000	0.423702
$35$	0	0
$36$	36.0000	0.166667
$37$	134.000i	0.595391i	0.954661	+	0.297695i	$0.0962180\pi$
$37$	134.000i	0.595391i	−0.954661	+	0.297695i	$0.903782\pi$
$38$	− 152.000i	− 0.648886i
$39$	102.000	0.418797
$40$	0	0
$41$	234.000	0.891333	0.445667	−	0.895199i	$-0.352967\pi$
$41$	234.000	0.891333	0.445667	+	0.895199i	$0.352967\pi$
$42$	− 192.000i	− 0.705387i
$43$	412.000i	1.46115i	0.682833	+	0.730575i	$0.260748\pi$
$43$	412.000i	1.46115i	−0.682833	+	0.730575i	$0.739252\pi$
$44$	240.000	0.822304
$45$	0	0
$46$	0	0
$47$	− 360.000i	− 1.11726i	−0.829416	−	0.558632i	$-0.811326\pi$
$47$	− 360.000i	− 1.11726i	0.829416	−	0.558632i	$-0.188674\pi$
$48$	− 48.0000i	− 0.144338i
$49$	−681.000	−1.98542
$50$	0	0
$51$	126.000	0.345952
$52$	− 136.000i	− 0.362689i
$53$	− 222.000i	− 0.575359i	−0.957727	−	0.287680i	$-0.907116\pi$
$53$	− 222.000i	− 0.575359i	0.957727	−	0.287680i	$-0.0928838\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	−256.000	−0.610883
$57$	− 228.000i	− 0.529813i
$58$	12.0000i	0.0271668i
$59$	−660.000	−1.45635	−0.728175	−	0.685391i	$-0.759631\pi$
$59$	−660.000	−1.45635	−0.728175	+	0.685391i	$0.759631\pi$
$60$	0	0
$61$	−490.000	−1.02849	−0.514246	−	0.857642i	$-0.671928\pi$
$61$	−490.000	−1.02849	−0.514246	+	0.857642i	$0.671928\pi$
$62$	464.000i	0.950453i
$63$	− 288.000i	− 0.575946i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	360.000	0.671408
$67$	812.000i	1.48062i	0.672265	+	0.740310i	$0.265321\pi$
$67$	812.000i	1.48062i	−0.672265	+	0.740310i	$0.734679\pi$
$68$	− 168.000i	− 0.299603i
$69$	0	0
$70$	0	0
$71$	120.000	0.200583	0.100291	−	0.994958i	$-0.468022\pi$
$71$	120.000	0.200583	0.100291	+	0.994958i	$0.468022\pi$
$72$	− 72.0000i	− 0.117851i
$73$	− 746.000i	− 1.19606i	−0.801472	−	0.598032i	$-0.795949\pi$
$73$	− 746.000i	− 1.19606i	0.801472	−	0.598032i	$-0.204051\pi$
$74$	268.000	0.421005
$75$	0	0
$76$	−304.000	−0.458831
$77$	− 1920.00i	− 2.84161i
$78$	− 204.000i	− 0.296134i
$79$	−152.000	−0.216473	−0.108236	−	0.994125i	$-0.534520\pi$
$79$	−152.000	−0.216473	−0.108236	+	0.994125i	$0.534520\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 468.000i	− 0.630268i
$83$	804.000i	1.06326i	0.846977	+	0.531629i	$0.178420\pi$
$83$	804.000i	1.06326i	−0.846977	+	0.531629i	$0.821580\pi$
$84$	−384.000	−0.498784
$85$	0	0
$86$	824.000	1.03319
$87$	18.0000i	0.0221816i
$88$	− 480.000i	− 0.581456i
$89$	678.000	0.807504	0.403752	−	0.914868i	$-0.367706\pi$
$89$	678.000	0.807504	0.403752	+	0.914868i	$0.367706\pi$
$90$	0	0
$91$	−1088.00	−1.25333
$92$	0	0
$93$	696.000i	0.776041i
$94$	−720.000	−0.790025
$95$	0	0
$96$	−96.0000	−0.102062
$97$	194.000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	194.000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	1362.00i	1.40391i
$99$	540.000	0.548202
$100$	0	0
$101$	798.000	0.786178	0.393089	−	0.919500i	$-0.371406\pi$
$101$	798.000	0.786178	0.393089	+	0.919500i	$0.371406\pi$
$102$	− 252.000i	− 0.244625i
$103$	− 1088.00i	− 1.04081i	−0.853918	−	0.520407i	$-0.825780\pi$
$103$	− 1088.00i	− 1.04081i	0.853918	−	0.520407i	$-0.174220\pi$
$104$	−272.000	−0.256460
$105$	0	0
$106$	−444.000	−0.406840
$107$	1716.00i	1.55039i	0.631721	+	0.775196i	$0.282349\pi$
$107$	1716.00i	1.55039i	−0.631721	+	0.775196i	$0.717651\pi$
$108$	− 108.000i	− 0.0962250i
$109$	970.000	0.852378	0.426189	−	0.904634i	$-0.359856\pi$
$109$	970.000	0.852378	0.426189	+	0.904634i	$0.359856\pi$
$110$	0	0
$111$	402.000	0.343749
$112$	512.000i	0.431959i
$113$	− 426.000i	− 0.354643i	−0.984153	−	0.177322i	$-0.943257\pi$
$113$	− 426.000i	− 0.354643i	0.984153	−	0.177322i	$-0.0567433\pi$
$114$	−456.000	−0.374634
$115$	0	0
$116$	24.0000	0.0192099
$117$	− 306.000i	− 0.241792i
$118$	1320.00i	1.02980i
$119$	−1344.00	−1.03533
$120$	0	0
$121$	2269.00	1.70473
$122$	980.000i	0.727254i
$123$	− 702.000i	− 0.514611i
$124$	928.000	0.672071
$125$	0	0
$126$	−576.000	−0.407255
$127$	200.000i	0.139741i	0.997556	+	0.0698706i	$0.0222586\pi$
$127$	200.000i	0.139741i	−0.997556	+	0.0698706i	$0.977741\pi$
$128$	128.000i	0.0883883i
$129$	1236.00	0.843595
$130$	0	0
$131$	60.0000	0.0400170	0.0200085	−	0.999800i	$-0.493631\pi$
$131$	60.0000	0.0400170	0.0200085	+	0.999800i	$0.493631\pi$
$132$	− 720.000i	− 0.474757i
$133$	2432.00i	1.58557i
$134$	1624.00	1.04696
$135$	0	0
$136$	−336.000	−0.211851
$137$	642.000i	0.400363i	0.979759	+	0.200182i	$0.0641532\pi$
$137$	642.000i	0.400363i	−0.979759	+	0.200182i	$0.935847\pi$
$138$	0	0
$139$	2836.00	1.73055	0.865275	−	0.501298i	$-0.167144\pi$
$139$	2836.00	1.73055	0.865275	+	0.501298i	$0.167144\pi$
$140$	0	0
$141$	−1080.00	−0.645053
$142$	− 240.000i	− 0.141833i
$143$	− 2040.00i	− 1.19296i
$144$	−144.000	−0.0833333
$145$	0	0
$146$	−1492.00	−0.845745
$147$	2043.00i	1.14628i
$148$	− 536.000i	− 0.297695i
$149$	1554.00	0.854420	0.427210	−	0.904152i	$-0.359496\pi$
$149$	1554.00	0.854420	0.427210	+	0.904152i	$0.359496\pi$
$150$	0	0
$151$	−2272.00	−1.22446	−0.612228	−	0.790682i	$-0.709726\pi$
$151$	−2272.00	−1.22446	−0.612228	+	0.790682i	$0.709726\pi$
$152$	608.000i	0.324443i
$153$	− 378.000i	− 0.199735i
$154$	−3840.00	−2.00932
$155$	0	0
$156$	−408.000	−0.209398
$157$	1694.00i	0.861120i	0.902562	+	0.430560i	$0.141684\pi$
$157$	1694.00i	0.861120i	−0.902562	+	0.430560i	$0.858316\pi$
$158$	304.000i	0.153069i
$159$	−666.000	−0.332184
$160$	0	0
$161$	0	0
$162$	− 162.000i	− 0.0785674i
$163$	52.0000i	0.0249874i	0.999922	+	0.0124937i	$0.00397698\pi$
$163$	52.0000i	0.0249874i	−0.999922	+	0.0124937i	$0.996023\pi$
$164$	−936.000	−0.445667
$165$	0	0
$166$	1608.00	0.751837
$167$	− 1200.00i	− 0.556041i	−0.960575	−	0.278020i	$-0.910322\pi$
$167$	− 1200.00i	− 0.556041i	0.960575	−	0.278020i	$-0.0896783\pi$
$168$	768.000i	0.352693i
$169$	1041.00	0.473828
$170$	0	0
$171$	−684.000	−0.305888
$172$	− 1648.00i	− 0.730575i
$173$	− 54.0000i	− 0.0237315i	−0.999930	−	0.0118657i	$-0.996223\pi$
$173$	− 54.0000i	− 0.0237315i	0.999930	−	0.0118657i	$-0.00377707\pi$
$174$	36.0000	0.0156848
$175$	0	0
$176$	−960.000	−0.411152
$177$	1980.00i	0.840824i
$178$	− 1356.00i	− 0.570992i
$179$	−876.000	−0.365784	−0.182892	−	0.983133i	$-0.558546\pi$
$179$	−876.000	−0.365784	−0.182892	+	0.983133i	$0.558546\pi$
$180$	0	0
$181$	3854.00	1.58268	0.791341	−	0.611375i	$-0.209383\pi$
$181$	3854.00	1.58268	0.791341	+	0.611375i	$0.209383\pi$
$182$	2176.00i	0.886241i
$183$	1470.00i	0.593801i
$184$	0	0
$185$	0	0
$186$	1392.00	0.548744
$187$	− 2520.00i	− 0.985458i
$188$	1440.00i	0.558632i
$189$	−864.000	−0.332522
$190$	0	0
$191$	−2784.00	−1.05468	−0.527338	−	0.849656i	$-0.676810\pi$
$191$	−2784.00	−1.05468	−0.527338	+	0.849656i	$0.676810\pi$
$192$	192.000i	0.0721688i
$193$	− 914.000i	− 0.340887i	−0.985367	−	0.170443i	$-0.945480\pi$
$193$	− 914.000i	− 0.340887i	0.985367	−	0.170443i	$-0.0545200\pi$
$194$	388.000	0.143592
$195$	0	0
$196$	2724.00	0.992711
$197$	− 5202.00i	− 1.88136i	−0.339300	−	0.940678i	$-0.610190\pi$
$197$	− 5202.00i	− 1.88136i	0.339300	−	0.940678i	$-0.389810\pi$
$198$	− 1080.00i	− 0.387638i
$199$	−3152.00	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$199$	−3152.00	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$200$	0	0
$201$	2436.00	0.854837
$202$	− 1596.00i	− 0.555912i
$203$	− 192.000i	− 0.0663830i
$204$	−504.000	−0.172976
$205$	0	0
$206$	−2176.00	−0.735967
$207$	0	0
$208$	544.000i	0.181344i
$209$	−4560.00	−1.50920
$210$	0	0
$211$	740.000	0.241439	0.120720	−	0.992687i	$-0.461480\pi$
$211$	740.000	0.241439	0.120720	+	0.992687i	$0.461480\pi$
$212$	888.000i	0.287680i
$213$	− 360.000i	− 0.115807i
$214$	3432.00	1.09629
$215$	0	0
$216$	−216.000	−0.0680414
$217$	− 7424.00i	− 2.32246i
$218$	− 1940.00i	− 0.602722i
$219$	−2238.00	−0.690548
$220$	0	0
$221$	−1428.00	−0.434650
$222$	− 804.000i	− 0.243067i
$223$	520.000i	0.156151i	0.996947	+	0.0780757i	$0.0248776\pi$
$223$	520.000i	0.156151i	−0.996947	+	0.0780757i	$0.975122\pi$
$224$	1024.00	0.305441
$225$	0	0
$226$	−852.000	−0.250771
$227$	396.000i	0.115786i	0.998323	+	0.0578930i	$0.0184382\pi$
$227$	396.000i	0.115786i	−0.998323	+	0.0578930i	$0.981562\pi$
$228$	912.000i	0.264906i
$229$	1330.00	0.383794	0.191897	−	0.981415i	$-0.438536\pi$
$229$	1330.00	0.383794	0.191897	+	0.981415i	$0.438536\pi$
$230$	0	0
$231$	−5760.00	−1.64061
$232$	− 48.0000i	− 0.0135834i
$233$	− 4866.00i	− 1.36816i	−0.729405	−	0.684082i	$-0.760203\pi$
$233$	− 4866.00i	− 1.36816i	0.729405	−	0.684082i	$-0.239797\pi$
$234$	−612.000	−0.170973
$235$	0	0
$236$	2640.00	0.728175
$237$	456.000i	0.124981i
$238$	2688.00i	0.732089i
$239$	1824.00	0.493660	0.246830	−	0.969059i	$-0.420611\pi$
$239$	1824.00	0.493660	0.246830	+	0.969059i	$0.420611\pi$
$240$	0	0
$241$	6482.00	1.73254	0.866270	−	0.499575i	$-0.166511\pi$
$241$	6482.00	1.73254	0.866270	+	0.499575i	$0.166511\pi$
$242$	− 4538.00i	− 1.20543i
$243$	− 243.000i	− 0.0641500i
$244$	1960.00	0.514246
$245$	0	0
$246$	−1404.00	−0.363885
$247$	2584.00i	0.665652i
$248$	− 1856.00i	− 0.475226i
$249$	2412.00	0.613873
$250$	0	0
$251$	1476.00	0.371172	0.185586	−	0.982628i	$-0.440582\pi$
$251$	1476.00	0.371172	0.185586	+	0.982628i	$0.440582\pi$
$252$	1152.00i	0.287973i
$253$	0	0
$254$	400.000	0.0988119
$255$	0	0
$256$	256.000	0.0625000
$257$	4314.00i	1.04708i	0.852001	+	0.523541i	$0.175389\pi$
$257$	4314.00i	1.04708i	−0.852001	+	0.523541i	$0.824611\pi$
$258$	− 2472.00i	− 0.596512i
$259$	−4288.00	−1.02874
$260$	0	0
$261$	54.0000	0.0128066
$262$	− 120.000i	− 0.0282963i
$263$	5280.00i	1.23794i	0.785414	+	0.618971i	$0.212450\pi$
$263$	5280.00i	1.23794i	−0.785414	+	0.618971i	$0.787550\pi$
$264$	−1440.00	−0.335704
$265$	0	0
$266$	4864.00	1.12117
$267$	− 2034.00i	− 0.466213i
$268$	− 3248.00i	− 0.740310i
$269$	−5526.00	−1.25251	−0.626257	−	0.779617i	$-0.715414\pi$
$269$	−5526.00	−1.25251	−0.626257	+	0.779617i	$0.715414\pi$
$270$	0	0
$271$	2024.00	0.453687	0.226844	−	0.973931i	$-0.427159\pi$
$271$	2024.00	0.453687	0.226844	+	0.973931i	$0.427159\pi$
$272$	672.000i	0.149801i
$273$	3264.00i	0.723613i
$274$	1284.00	0.283100
$275$	0	0
$276$	0	0
$277$	2054.00i	0.445534i	0.974872	+	0.222767i	$0.0715089\pi$
$277$	2054.00i	0.445534i	−0.974872	+	0.222767i	$0.928491\pi$
$278$	− 5672.00i	− 1.22368i
$279$	2088.00	0.448048
$280$	0	0
$281$	−7302.00	−1.55018	−0.775090	−	0.631850i	$-0.782296\pi$
$281$	−7302.00	−1.55018	−0.775090	+	0.631850i	$0.782296\pi$
$282$	2160.00i	0.456121i
$283$	3724.00i	0.782222i	0.920344	+	0.391111i	$0.127909\pi$
$283$	3724.00i	0.782222i	−0.920344	+	0.391111i	$0.872091\pi$
$284$	−480.000	−0.100291
$285$	0	0
$286$	−4080.00	−0.843551
$287$	7488.00i	1.54008i
$288$	288.000i	0.0589256i
$289$	3149.00	0.640953
$290$	0	0
$291$	582.000	0.117242
$292$	2984.00i	0.598032i
$293$	7218.00i	1.43918i	0.694399	+	0.719591i	$0.255670\pi$
$293$	7218.00i	1.43918i	−0.694399	+	0.719591i	$0.744330\pi$
$294$	4086.00	0.810545
$295$	0	0
$296$	−1072.00	−0.210502
$297$	− 1620.00i	− 0.316505i
$298$	− 3108.00i	− 0.604166i
$299$	0	0
$300$	0	0
$301$	−13184.0	−2.52463
$302$	4544.00i	0.865821i
$303$	− 2394.00i	− 0.453900i
$304$	1216.00	0.229416
$305$	0	0
$306$	−756.000	−0.141234
$307$	2540.00i	0.472200i	0.971729	+	0.236100i	$0.0758693\pi$
$307$	2540.00i	0.472200i	−0.971729	+	0.236100i	$0.924131\pi$
$308$	7680.00i	1.42081i
$309$	−3264.00	−0.600914
$310$	0	0
$311$	1560.00	0.284436	0.142218	−	0.989835i	$-0.454577\pi$
$311$	1560.00	0.284436	0.142218	+	0.989835i	$0.454577\pi$
$312$	816.000i	0.148067i
$313$	934.000i	0.168667i	0.996438	+	0.0843335i	$0.0268761\pi$
$313$	934.000i	0.168667i	−0.996438	+	0.0843335i	$0.973124\pi$
$314$	3388.00	0.608904
$315$	0	0
$316$	608.000	0.108236
$317$	− 1674.00i	− 0.296597i	−0.988943	−	0.148298i	$-0.952620\pi$
$317$	− 1674.00i	− 0.296597i	0.988943	−	0.148298i	$-0.0473796\pi$
$318$	1332.00i	0.234889i
$319$	360.000	0.0631854
$320$	0	0
$321$	5148.00	0.895119
$322$	0	0
$323$	3192.00i	0.549869i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	104.000	0.0176688
$327$	− 2910.00i	− 0.492120i
$328$	1872.00i	0.315134i
$329$	11520.0	1.93045
$330$	0	0
$331$	−3988.00	−0.662237	−0.331118	−	0.943589i	$-0.607426\pi$
$331$	−3988.00	−0.662237	−0.331118	+	0.943589i	$0.607426\pi$
$332$	− 3216.00i	− 0.531629i
$333$	− 1206.00i	− 0.198464i
$334$	−2400.00	−0.393180
$335$	0	0
$336$	1536.00	0.249392
$337$	2.00000i	0 0.000323285i	1.00000		0.000161642i	$5.14524e-5\pi$
$337$	2.00000i	0 0.000323285i	−1.00000		0.000161642i	$0.999949\pi$
$338$	− 2082.00i	− 0.335047i
$339$	−1278.00	−0.204753
$340$	0	0
$341$	13920.0	2.21059
$342$	1368.00i	0.216295i
$343$	− 10816.0i	− 1.70265i
$344$	−3296.00	−0.516594
$345$	0	0
$346$	−108.000	−0.0167807
$347$	1764.00i	0.272901i	0.990647	+	0.136450i	$0.0435694\pi$
$347$	1764.00i	0.272901i	−0.990647	+	0.136450i	$0.956431\pi$
$348$	− 72.0000i	− 0.0110908i
$349$	−4310.00	−0.661057	−0.330529	−	0.943796i	$-0.607227\pi$
$349$	−4310.00	−0.661057	−0.330529	+	0.943796i	$0.607227\pi$
$350$	0	0
$351$	−918.000	−0.139599
$352$	1920.00i	0.290728i
$353$	− 138.000i	− 0.0208074i	−0.999946	−	0.0104037i	$-0.996688\pi$
$353$	− 138.000i	− 0.0208074i	0.999946	−	0.0104037i	$-0.00331165\pi$
$354$	3960.00	0.594553
$355$	0	0
$356$	−2712.00	−0.403752
$357$	4032.00i	0.597748i
$358$	1752.00i	0.258648i
$359$	11976.0	1.76064	0.880319	−	0.474382i	$-0.157328\pi$
$359$	11976.0	1.76064	0.880319	+	0.474382i	$0.157328\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	− 7708.00i	− 1.11913i
$363$	− 6807.00i	− 0.984228i
$364$	4352.00	0.626667
$365$	0	0
$366$	2940.00	0.419880
$367$	9704.00i	1.38023i	0.723699	+	0.690115i	$0.242440\pi$
$367$	9704.00i	1.38023i	−0.723699	+	0.690115i	$0.757560\pi$
$368$	0	0
$369$	−2106.00	−0.297111
$370$	0	0
$371$	7104.00	0.994128
$372$	− 2784.00i	− 0.388021i
$373$	8122.00i	1.12746i	0.825960	+	0.563728i	$0.190633\pi$
$373$	8122.00i	1.12746i	−0.825960	+	0.563728i	$0.809367\pi$
$374$	−5040.00	−0.696824
$375$	0	0
$376$	2880.00	0.395012
$377$	− 204.000i	− 0.0278688i
$378$	1728.00i	0.235129i
$379$	−3404.00	−0.461350	−0.230675	−	0.973031i	$-0.574093\pi$
$379$	−3404.00	−0.461350	−0.230675	+	0.973031i	$0.574093\pi$
$380$	0	0
$381$	600.000	0.0806796
$382$	5568.00i	0.745769i
$383$	2520.00i	0.336204i	0.985770	+	0.168102i	$0.0537637\pi$
$383$	2520.00i	0.336204i	−0.985770	+	0.168102i	$0.946236\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−1828.00	−0.241043
$387$	− 3708.00i	− 0.487050i
$388$	− 776.000i	− 0.101535i
$389$	−1566.00	−0.204111	−0.102056	−	0.994779i	$-0.532542\pi$
$389$	−1566.00	−0.204111	−0.102056	+	0.994779i	$0.532542\pi$
$390$	0	0
$391$	0	0
$392$	− 5448.00i	− 0.701953i
$393$	− 180.000i	− 0.0231038i
$394$	−10404.0	−1.33032
$395$	0	0
$396$	−2160.00	−0.274101
$397$	− 4354.00i	− 0.550431i	−0.961383	−	0.275215i	$-0.911251\pi$
$397$	− 4354.00i	− 0.550431i	0.961383	−	0.275215i	$-0.0887492\pi$
$398$	6304.00i	0.793947i
$399$	7296.00	0.915431
$400$	0	0
$401$	−8046.00	−1.00199	−0.500995	−	0.865450i	$-0.667033\pi$
$401$	−8046.00	−1.00199	−0.500995	+	0.865450i	$0.667033\pi$
$402$	− 4872.00i	− 0.604461i
$403$	− 7888.00i	− 0.975011i
$404$	−3192.00	−0.393089
$405$	0	0
$406$	−384.000	−0.0469399
$407$	− 8040.00i	− 0.979184i
$408$	1008.00i	0.122312i
$409$	2806.00	0.339237	0.169618	−	0.985510i	$-0.445747\pi$
$409$	2806.00	0.339237	0.169618	+	0.985510i	$0.445747\pi$
$410$	0	0
$411$	1926.00	0.231150
$412$	4352.00i	0.520407i
$413$	− 21120.0i	− 2.51634i
$414$	0	0
$415$	0	0
$416$	1088.00	0.128230
$417$	− 8508.00i	− 0.999133i
$418$	9120.00i	1.06716i
$419$	−11580.0	−1.35017	−0.675084	−	0.737741i	$-0.735892\pi$
$419$	−11580.0	−1.35017	−0.675084	+	0.737741i	$0.735892\pi$
$420$	0	0
$421$	−370.000	−0.0428330	−0.0214165	−	0.999771i	$-0.506818\pi$
$421$	−370.000	−0.0428330	−0.0214165	+	0.999771i	$0.506818\pi$
$422$	− 1480.00i	− 0.170723i
$423$	3240.00i	0.372421i
$424$	1776.00	0.203420
$425$	0	0
$426$	−720.000	−0.0818876
$427$	− 15680.0i	− 1.77707i
$428$	− 6864.00i	− 0.775196i
$429$	−6120.00	−0.688756
$430$	0	0
$431$	5040.00	0.563267	0.281634	−	0.959522i	$-0.409124\pi$
$431$	5040.00	0.563267	0.281634	+	0.959522i	$0.409124\pi$
$432$	432.000i	0.0481125i
$433$	3742.00i	0.415310i	0.978202	+	0.207655i	$0.0665831\pi$
$433$	3742.00i	0.415310i	−0.978202	+	0.207655i	$0.933417\pi$
$434$	−14848.0	−1.64223
$435$	0	0
$436$	−3880.00	−0.426189
$437$	0	0
$438$	4476.00i	0.488291i
$439$	6208.00	0.674924	0.337462	−	0.941339i	$-0.390432\pi$
$439$	6208.00	0.674924	0.337462	+	0.941339i	$0.390432\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	2856.00i	0.307344i
$443$	15564.0i	1.66923i	0.550835	+	0.834614i	$0.314309\pi$
$443$	15564.0i	1.66923i	−0.550835	+	0.834614i	$0.685691\pi$
$444$	−1608.00	−0.171875
$445$	0	0
$446$	1040.00	0.110416
$447$	− 4662.00i	− 0.493300i
$448$	− 2048.00i	− 0.215980i
$449$	15774.0	1.65795	0.828977	−	0.559283i	$-0.188924\pi$
$449$	15774.0	1.65795	0.828977	+	0.559283i	$0.188924\pi$
$450$	0	0
$451$	−14040.0	−1.46589
$452$	1704.00i	0.177322i
$453$	6816.00i	0.706940i
$454$	792.000	0.0818731
$455$	0	0
$456$	1824.00	0.187317
$457$	9722.00i	0.995133i	0.867426	+	0.497567i	$0.165773\pi$
$457$	9722.00i	0.995133i	−0.867426	+	0.497567i	$0.834227\pi$
$458$	− 2660.00i	− 0.271383i
$459$	−1134.00	−0.115317
$460$	0	0
$461$	−10890.0	−1.10021	−0.550106	−	0.835095i	$-0.685413\pi$
$461$	−10890.0	−1.10021	−0.550106	+	0.835095i	$0.685413\pi$
$462$	11520.0i	1.16008i
$463$	− 15128.0i	− 1.51848i	−0.650809	−	0.759242i	$-0.725570\pi$
$463$	− 15128.0i	− 1.51848i	0.650809	−	0.759242i	$-0.274430\pi$
$464$	−96.0000	−0.00960493
$465$	0	0
$466$	−9732.00	−0.967438
$467$	10668.0i	1.05708i	0.848909	+	0.528540i	$0.177260\pi$
$467$	10668.0i	1.05708i	−0.848909	+	0.528540i	$0.822740\pi$
$468$	1224.00i	0.120896i
$469$	−25984.0	−2.55827
$470$	0	0
$471$	5082.00	0.497168
$472$	− 5280.00i	− 0.514898i
$473$	− 24720.0i	− 2.40302i
$474$	912.000	0.0883746
$475$	0	0
$476$	5376.00	0.517665
$477$	1998.00i	0.191786i
$478$	− 3648.00i	− 0.349070i
$479$	−15264.0	−1.45601	−0.728006	−	0.685571i	$-0.759553\pi$
$479$	−15264.0	−1.45601	−0.728006	+	0.685571i	$0.759553\pi$
$480$	0	0
$481$	−4556.00	−0.431883
$482$	− 12964.0i	− 1.22509i
$483$	0	0
$484$	−9076.00	−0.852367
$485$	0	0
$486$	−486.000	−0.0453609
$487$	− 5776.00i	− 0.537445i	−0.963218	−	0.268722i	$-0.913399\pi$
$487$	− 5776.00i	− 0.537445i	0.963218	−	0.268722i	$-0.0866014\pi$
$488$	− 3920.00i	− 0.363627i
$489$	156.000	0.0144265
$490$	0	0
$491$	14244.0	1.30921	0.654606	−	0.755971i	$-0.272835\pi$
$491$	14244.0	1.30921	0.654606	+	0.755971i	$0.272835\pi$
$492$	2808.00i	0.257306i
$493$	− 252.000i	− 0.0230213i
$494$	5168.00	0.470687
$495$	0	0
$496$	−3712.00	−0.336036
$497$	3840.00i	0.346575i
$498$	− 4824.00i	− 0.434074i
$499$	17116.0	1.53551	0.767753	−	0.640746i	$-0.221375\pi$
$499$	17116.0	1.53551	0.767753	+	0.640746i	$0.221375\pi$
$500$	0	0
$501$	−3600.00	−0.321030
$502$	− 2952.00i	− 0.262459i
$503$	16848.0i	1.49347i	0.665122	+	0.746735i	$0.268380\pi$
$503$	16848.0i	1.49347i	−0.665122	+	0.746735i	$0.731620\pi$
$504$	2304.00	0.203628
$505$	0	0
$506$	0	0
$507$	− 3123.00i	− 0.273565i
$508$	− 800.000i	− 0.0698706i
$509$	3834.00	0.333868	0.166934	−	0.985968i	$-0.446613\pi$
$509$	3834.00	0.333868	0.166934	+	0.985968i	$0.446613\pi$
$510$	0	0
$511$	23872.0	2.06660
$512$	− 512.000i	− 0.0441942i
$513$	2052.00i	0.176604i
$514$	8628.00	0.740398
$515$	0	0
$516$	−4944.00	−0.421797
$517$	21600.0i	1.83746i
$518$	8576.00i	0.727428i
$519$	−162.000	−0.0137014
$520$	0	0
$521$	−18822.0	−1.58274	−0.791369	−	0.611338i	$-0.790631\pi$
$521$	−18822.0	−1.58274	−0.791369	+	0.611338i	$0.790631\pi$
$522$	− 108.000i	− 0.00905562i
$523$	15340.0i	1.28255i	0.767313	+	0.641273i	$0.221593\pi$
$523$	15340.0i	1.28255i	−0.767313	+	0.641273i	$0.778407\pi$
$524$	−240.000	−0.0200085
$525$	0	0
$526$	10560.0	0.875357
$527$	− 9744.00i	− 0.805418i
$528$	2880.00i	0.237379i
$529$	12167.0	1.00000
$530$	0	0
$531$	5940.00	0.485450
$532$	− 9728.00i	− 0.792786i
$533$	7956.00i	0.646553i
$534$	−4068.00	−0.329662
$535$	0	0
$536$	−6496.00	−0.523478
$537$	2628.00i	0.211185i
$538$	11052.0i	0.885661i
$539$	40860.0	3.26524
$540$	0	0
$541$	18950.0	1.50596	0.752980	−	0.658044i	$-0.228616\pi$
$541$	18950.0	1.50596	0.752980	+	0.658044i	$0.228616\pi$
$542$	− 4048.00i	− 0.320805i
$543$	− 11562.0i	− 0.913762i
$544$	1344.00	0.105926
$545$	0	0
$546$	6528.00	0.511671
$547$	− 10036.0i	− 0.784476i	−0.919864	−	0.392238i	$-0.871701\pi$
$547$	− 10036.0i	− 0.784476i	0.919864	−	0.392238i	$-0.128299\pi$
$548$	− 2568.00i	− 0.200182i
$549$	4410.00	0.342831
$550$	0	0
$551$	−456.000	−0.0352564
$552$	0	0
$553$	− 4864.00i	− 0.374030i
$554$	4108.00	0.315040
$555$	0	0
$556$	−11344.0	−0.865275
$557$	10326.0i	0.785506i	0.919644	+	0.392753i	$0.128477\pi$
$557$	10326.0i	0.785506i	−0.919644	+	0.392753i	$0.871523\pi$
$558$	− 4176.00i	− 0.316818i
$559$	−14008.0	−1.05988
$560$	0	0
$561$	−7560.00	−0.568954
$562$	14604.0i	1.09614i
$563$	− 4524.00i	− 0.338657i	−0.985560	−	0.169328i	$-0.945840\pi$
$563$	− 4524.00i	− 0.338657i	0.985560	−	0.169328i	$-0.0541599\pi$
$564$	4320.00	0.322526
$565$	0	0
$566$	7448.00	0.553114
$567$	2592.00i	0.191982i
$568$	960.000i	0.0709167i
$569$	−16362.0	−1.20550	−0.602751	−	0.797929i	$-0.705929\pi$
$569$	−16362.0	−1.20550	−0.602751	+	0.797929i	$0.705929\pi$
$570$	0	0
$571$	6620.00	0.485181	0.242591	−	0.970129i	$-0.422003\pi$
$571$	6620.00	0.485181	0.242591	+	0.970129i	$0.422003\pi$
$572$	8160.00i	0.596480i
$573$	8352.00i	0.608918i
$574$	14976.0	1.08900
$575$	0	0
$576$	576.000	0.0416667
$577$	8834.00i	0.637373i	0.947860	+	0.318687i	$0.103242\pi$
$577$	8834.00i	0.637373i	−0.947860	+	0.318687i	$0.896758\pi$
$578$	− 6298.00i	− 0.453222i
$579$	−2742.00	−0.196811
$580$	0	0
$581$	−25728.0	−1.83714
$582$	− 1164.00i	− 0.0829027i
$583$	13320.0i	0.946240i
$584$	5968.00	0.422873
$585$	0	0
$586$	14436.0	1.01765
$587$	3636.00i	0.255662i	0.991796	+	0.127831i	$0.0408016\pi$
$587$	3636.00i	0.255662i	−0.991796	+	0.127831i	$0.959198\pi$
$588$	− 8172.00i	− 0.573142i
$589$	−17632.0	−1.23347
$590$	0	0
$591$	−15606.0	−1.08620
$592$	2144.00i	0.148848i
$593$	− 6570.00i	− 0.454971i	−0.973782	−	0.227485i	$-0.926950\pi$
$593$	− 6570.00i	− 0.454971i	0.973782	−	0.227485i	$-0.0730504\pi$
$594$	−3240.00	−0.223803
$595$	0	0
$596$	−6216.00	−0.427210
$597$	9456.00i	0.648255i
$598$	0	0
$599$	−16584.0	−1.13123	−0.565613	−	0.824671i	$-0.691360\pi$
$599$	−16584.0	−1.13123	−0.565613	+	0.824671i	$0.691360\pi$
$600$	0	0
$601$	−502.000	−0.0340716	−0.0170358	−	0.999855i	$-0.505423\pi$
$601$	−502.000	−0.0340716	−0.0170358	+	0.999855i	$0.505423\pi$
$602$	26368.0i	1.78518i
$603$	− 7308.00i	− 0.493540i
$604$	9088.00	0.612228
$605$	0	0
$606$	−4788.00	−0.320956
$607$	− 18568.0i	− 1.24160i	−0.783969	−	0.620801i	$-0.786808\pi$
$607$	− 18568.0i	− 1.24160i	0.783969	−	0.620801i	$-0.213192\pi$
$608$	− 2432.00i	− 0.162221i
$609$	−576.000	−0.0383263
$610$	0	0
$611$	12240.0	0.810438
$612$	1512.00i	0.0998676i
$613$	13114.0i	0.864061i	0.901859	+	0.432031i	$0.142203\pi$
$613$	13114.0i	0.864061i	−0.901859	+	0.432031i	$0.857797\pi$
$614$	5080.00	0.333896
$615$	0	0
$616$	15360.0	1.00466
$617$	5250.00i	0.342556i	0.985223	+	0.171278i	$0.0547896\pi$
$617$	5250.00i	0.342556i	−0.985223	+	0.171278i	$0.945210\pi$
$618$	6528.00i	0.424910i
$619$	10804.0	0.701534	0.350767	−	0.936463i	$-0.385921\pi$
$619$	10804.0	0.701534	0.350767	+	0.936463i	$0.385921\pi$
$620$	0	0
$621$	0	0
$622$	− 3120.00i	− 0.201126i
$623$	21696.0i	1.39524i
$624$	1632.00	0.104699
$625$	0	0
$626$	1868.00	0.119266
$627$	13680.0i	0.871334i
$628$	− 6776.00i	− 0.430560i
$629$	−5628.00	−0.356762
$630$	0	0
$631$	−27088.0	−1.70896	−0.854482	−	0.519481i	$-0.826125\pi$
$631$	−27088.0	−1.70896	−0.854482	+	0.519481i	$0.826125\pi$
$632$	− 1216.00i	− 0.0765346i
$633$	− 2220.00i	− 0.139395i
$634$	−3348.00	−0.209726
$635$	0	0
$636$	2664.00	0.166092
$637$	− 23154.0i	− 1.44018i
$638$	− 720.000i	− 0.0446788i
$639$	−1080.00	−0.0668609
$640$	0	0
$641$	18930.0	1.16644	0.583222	−	0.812313i	$-0.301792\pi$
$641$	18930.0	1.16644	0.583222	+	0.812313i	$0.301792\pi$
$642$	− 10296.0i	− 0.632945i
$643$	− 20108.0i	− 1.23325i	−0.787256	−	0.616627i	$-0.788499\pi$
$643$	− 20108.0i	− 1.23325i	0.787256	−	0.616627i	$-0.211501\pi$
$644$	0	0
$645$	0	0
$646$	6384.00	0.388816
$647$	− 7152.00i	− 0.434581i	−0.976107	−	0.217291i	$-0.930278\pi$
$647$	− 7152.00i	− 0.434581i	0.976107	−	0.217291i	$-0.0697219\pi$
$648$	648.000i	0.0392837i
$649$	39600.0	2.39512
$650$	0	0
$651$	−22272.0	−1.34087
$652$	− 208.000i	− 0.0124937i
$653$	31626.0i	1.89528i	0.319333	+	0.947642i	$0.396541\pi$
$653$	31626.0i	1.89528i	−0.319333	+	0.947642i	$0.603459\pi$
$654$	−5820.00	−0.347982
$655$	0	0
$656$	3744.00	0.222833
$657$	6714.00i	0.398688i
$658$	− 23040.0i	− 1.36503i
$659$	−28092.0	−1.66056	−0.830280	−	0.557347i	$-0.811819\pi$
$659$	−28092.0	−1.66056	−0.830280	+	0.557347i	$0.811819\pi$
$660$	0	0
$661$	−13186.0	−0.775909	−0.387955	−	0.921678i	$-0.626818\pi$
$661$	−13186.0	−0.775909	−0.387955	+	0.921678i	$0.626818\pi$
$662$	7976.00i	0.468272i
$663$	4284.00i	0.250945i
$664$	−6432.00	−0.375919
$665$	0	0
$666$	−2412.00	−0.140335
$667$	0	0
$668$	4800.00i	0.278020i
$669$	1560.00	0.0901541
$670$	0	0
$671$	29400.0	1.69147
$672$	− 3072.00i	− 0.176347i
$673$	− 5138.00i	− 0.294287i	−0.989115	−	0.147144i	$-0.952992\pi$
$673$	− 5138.00i	− 0.294287i	0.989115	−	0.147144i	$-0.0470080\pi$
$674$	4.00000	0.000228597 0
$675$	0	0
$676$	−4164.00	−0.236914
$677$	6078.00i	0.345047i	0.985005	+	0.172523i	$0.0551920\pi$
$677$	6078.00i	0.345047i	−0.985005	+	0.172523i	$0.944808\pi$
$678$	2556.00i	0.144783i
$679$	−6208.00	−0.350871
$680$	0	0
$681$	1188.00	0.0668491
$682$	− 27840.0i	− 1.56312i
$683$	− 32244.0i	− 1.80642i	−0.429203	−	0.903208i	$-0.641205\pi$
$683$	− 32244.0i	− 1.80642i	0.429203	−	0.903208i	$-0.358795\pi$
$684$	2736.00	0.152944
$685$	0	0
$686$	−21632.0	−1.20396
$687$	− 3990.00i	− 0.221584i
$688$	6592.00i	0.365287i
$689$	7548.00	0.417353
$690$	0	0
$691$	4484.00	0.246859	0.123429	−	0.992353i	$-0.460611\pi$
$691$	4484.00	0.246859	0.123429	+	0.992353i	$0.460611\pi$
$692$	216.000i	0.0118657i
$693$	17280.0i	0.947205i
$694$	3528.00	0.192970
$695$	0	0
$696$	−144.000	−0.00784239
$697$	9828.00i	0.534092i
$698$	8620.00i	0.467438i
$699$	−14598.0	−0.789910
$700$	0	0
$701$	−30426.0	−1.63934	−0.819668	−	0.572839i	$-0.805842\pi$
$701$	−30426.0	−1.63934	−0.819668	+	0.572839i	$0.805842\pi$
$702$	1836.00i	0.0987113i
$703$	10184.0i	0.546368i
$704$	3840.00	0.205576
$705$	0	0
$706$	−276.000	−0.0147130
$707$	25536.0i	1.35839i
$708$	− 7920.00i	− 0.420412i
$709$	−13262.0	−0.702489	−0.351245	−	0.936284i	$-0.614241\pi$
$709$	−13262.0	−0.702489	−0.351245	+	0.936284i	$0.614241\pi$
$710$	0	0
$711$	1368.00	0.0721575
$712$	5424.00i	0.285496i
$713$	0	0
$714$	8064.00	0.422672
$715$	0	0
$716$	3504.00	0.182892
$717$	− 5472.00i	− 0.285015i
$718$	− 23952.0i	− 1.24496i
$719$	−13920.0	−0.722014	−0.361007	−	0.932563i	$-0.617567\pi$
$719$	−13920.0	−0.722014	−0.361007	+	0.932563i	$0.617567\pi$
$720$	0	0
$721$	34816.0	1.79836
$722$	2166.00i	0.111648i
$723$	− 19446.0i	− 1.00028i
$724$	−15416.0	−0.791341
$725$	0	0
$726$	−13614.0	−0.695954
$727$	− 9376.00i	− 0.478317i	−0.970981	−	0.239159i	$-0.923128\pi$
$727$	− 9376.00i	− 0.478317i	0.970981	−	0.239159i	$-0.0768716\pi$
$728$	− 8704.00i	− 0.443120i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−17304.0	−0.875529
$732$	− 5880.00i	− 0.296900i
$733$	− 6014.00i	− 0.303045i	−0.988454	−	0.151523i	$-0.951582\pi$
$733$	− 6014.00i	− 0.303045i	0.988454	−	0.151523i	$-0.0484176\pi$
$734$	19408.0	0.975971
$735$	0	0
$736$	0	0
$737$	− 48720.0i	− 2.43504i
$738$	4212.00i	0.210089i
$739$	7468.00	0.371739	0.185869	−	0.982574i	$-0.440490\pi$
$739$	7468.00	0.371739	0.185869	+	0.982574i	$0.440490\pi$
$740$	0	0
$741$	7752.00	0.384314
$742$	− 14208.0i	− 0.702954i
$743$	− 31248.0i	− 1.54290i	−0.636287	−	0.771452i	$-0.719531\pi$
$743$	− 31248.0i	− 1.54290i	0.636287	−	0.771452i	$-0.280469\pi$
$744$	−5568.00	−0.274372
$745$	0	0
$746$	16244.0	0.797232
$747$	− 7236.00i	− 0.354420i
$748$	10080.0i	0.492729i
$749$	−54912.0	−2.67883
$750$	0	0
$751$	32840.0	1.59567	0.797835	−	0.602875i	$-0.205978\pi$
$751$	32840.0	1.59567	0.797835	+	0.602875i	$0.205978\pi$
$752$	− 5760.00i	− 0.279316i
$753$	− 4428.00i	− 0.214297i
$754$	−408.000	−0.0197062
$755$	0	0
$756$	3456.00	0.166261
$757$	− 19066.0i	− 0.915410i	−0.889104	−	0.457705i	$-0.848672\pi$
$757$	− 19066.0i	− 0.915410i	0.889104	−	0.457705i	$-0.151328\pi$
$758$	6808.00i	0.326224i
$759$	0	0
$760$	0	0
$761$	6858.00	0.326678	0.163339	−	0.986570i	$-0.447773\pi$
$761$	6858.00	0.326678	0.163339	+	0.986570i	$0.447773\pi$
$762$	− 1200.00i	− 0.0570491i
$763$	31040.0i	1.47277i
$764$	11136.0	0.527338
$765$	0	0
$766$	5040.00	0.237732
$767$	− 22440.0i	− 1.05640i
$768$	− 768.000i	− 0.0360844i
$769$	−22178.0	−1.04000	−0.519999	−	0.854167i	$-0.674068\pi$
$769$	−22178.0	−1.04000	−0.519999	+	0.854167i	$0.674068\pi$
$770$	0	0
$771$	12942.0	0.604533
$772$	3656.00i	0.170443i
$773$	− 14286.0i	− 0.664724i	−0.943152	−	0.332362i	$-0.892154\pi$
$773$	− 14286.0i	− 0.664724i	0.943152	−	0.332362i	$-0.107846\pi$
$774$	−7416.00	−0.344396
$775$	0	0
$776$	−1552.00	−0.0717958
$777$	12864.0i	0.593943i
$778$	3132.00i	0.144329i
$779$	17784.0	0.817943
$780$	0	0
$781$	−7200.00	−0.329880
$782$	0	0
$783$	− 162.000i	− 0.00739388i
$784$	−10896.0	−0.496356
$785$	0	0
$786$	−360.000	−0.0163369
$787$	− 18868.0i	− 0.854602i	−0.904109	−	0.427301i	$-0.859465\pi$
$787$	− 18868.0i	− 0.854602i	0.904109	−	0.427301i	$-0.140535\pi$
$788$	20808.0i	0.940678i
$789$	15840.0	0.714726
$790$	0	0
$791$	13632.0	0.612766
$792$	4320.00i	0.193819i
$793$	− 16660.0i	− 0.746045i
$794$	−8708.00	−0.389213
$795$	0	0
$796$	12608.0	0.561405
$797$	− 21690.0i	− 0.963989i	−0.876174	−	0.481994i	$-0.839913\pi$
$797$	− 21690.0i	− 0.963989i	0.876174	−	0.481994i	$-0.160087\pi$
$798$	− 14592.0i	− 0.647307i
$799$	15120.0	0.669471
$800$	0	0
$801$	−6102.00	−0.269168
$802$	16092.0i	0.708514i
$803$	44760.0i	1.96706i
$804$	−9744.00	−0.427418
$805$	0	0
$806$	−15776.0	−0.689437
$807$	16578.0i	0.723139i
$808$	6384.00i	0.277956i
$809$	24726.0	1.07456	0.537281	−	0.843404i	$-0.319452\pi$
$809$	24726.0	1.07456	0.537281	+	0.843404i	$0.319452\pi$
$810$	0	0
$811$	−2644.00	−0.114480	−0.0572401	−	0.998360i	$-0.518230\pi$
$811$	−2644.00	−0.114480	−0.0572401	+	0.998360i	$0.518230\pi$
$812$	768.000i	0.0331915i
$813$	− 6072.00i	− 0.261936i
$814$	−16080.0	−0.692388
$815$	0	0
$816$	2016.00	0.0864879
$817$	31312.0i	1.34084i
$818$	− 5612.00i	− 0.239877i
$819$	9792.00	0.417778
$820$	0	0
$821$	−37842.0	−1.60864	−0.804321	−	0.594195i	$-0.797471\pi$
$821$	−37842.0	−1.60864	−0.804321	+	0.594195i	$0.797471\pi$
$822$	− 3852.00i	− 0.163448i
$823$	880.000i	0.0372720i	0.999826	+	0.0186360i	$0.00593237\pi$
$823$	880.000i	0.0372720i	−0.999826	+	0.0186360i	$0.994068\pi$
$824$	8704.00	0.367983
$825$	0	0
$826$	−42240.0	−1.77932
$827$	− 12876.0i	− 0.541406i	−0.962663	−	0.270703i	$-0.912744\pi$
$827$	− 12876.0i	− 0.541406i	0.962663	−	0.270703i	$-0.0872561\pi$
$828$	0	0
$829$	25498.0	1.06825	0.534127	−	0.845404i	$-0.320641\pi$
$829$	25498.0	1.06825	0.534127	+	0.845404i	$0.320641\pi$
$830$	0	0
$831$	6162.00	0.257229
$832$	− 2176.00i	− 0.0906721i
$833$	− 28602.0i	− 1.18968i
$834$	−17016.0	−0.706494
$835$	0	0
$836$	18240.0	0.754598
$837$	− 6264.00i	− 0.258680i
$838$	23160.0i	0.954712i
$839$	40584.0	1.66998	0.834991	−	0.550263i	$-0.185473\pi$
$839$	40584.0	1.66998	0.834991	+	0.550263i	$0.185473\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	740.000i	0.0302875i
$843$	21906.0i	0.894997i
$844$	−2960.00	−0.120720
$845$	0	0
$846$	6480.00	0.263342
$847$	72608.0i	2.94550i
$848$	− 3552.00i	− 0.143840i
$849$	11172.0	0.451616
$850$	0	0
$851$	0	0
$852$	1440.00i	0.0579033i
$853$	25738.0i	1.03312i	0.856251	+	0.516561i	$0.172788\pi$
$853$	25738.0i	1.03312i	−0.856251	+	0.516561i	$0.827212\pi$
$854$	−31360.0	−1.25658
$855$	0	0
$856$	−13728.0	−0.548146
$857$	13314.0i	0.530686i	0.964154	+	0.265343i	$0.0854851\pi$
$857$	13314.0i	0.530686i	−0.964154	+	0.265343i	$0.914515\pi$
$858$	12240.0i	0.487024i
$859$	−24524.0	−0.974096	−0.487048	−	0.873375i	$-0.661926\pi$
$859$	−24524.0	−0.974096	−0.487048	+	0.873375i	$0.661926\pi$
$860$	0	0
$861$	22464.0	0.889165
$862$	− 10080.0i	− 0.398290i
$863$	− 5592.00i	− 0.220572i	−0.993900	−	0.110286i	$-0.964823\pi$
$863$	− 5592.00i	− 0.220572i	0.993900	−	0.110286i	$-0.0351767\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	7484.00	0.293668
$867$	− 9447.00i	− 0.370054i
$868$	29696.0i	1.16123i
$869$	9120.00	0.356012
$870$	0	0
$871$	−27608.0	−1.07401
$872$	7760.00i	0.301361i
$873$	− 1746.00i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	8952.00	0.345274
$877$	− 14386.0i	− 0.553912i	−0.960883	−	0.276956i	$-0.910674\pi$
$877$	− 14386.0i	− 0.553912i	0.960883	−	0.276956i	$-0.0893256\pi$
$878$	− 12416.0i	− 0.477243i
$879$	21654.0	0.830912
$880$	0	0
$881$	47106.0	1.80141	0.900705	−	0.434432i	$-0.143051\pi$
$881$	47106.0	1.80141	0.900705	+	0.434432i	$0.143051\pi$
$882$	− 12258.0i	− 0.467969i
$883$	− 51548.0i	− 1.96458i	−0.187354	−	0.982292i	$-0.559991\pi$
$883$	− 51548.0i	− 1.96458i	0.187354	−	0.982292i	$-0.440009\pi$
$884$	5712.00	0.217325
$885$	0	0
$886$	31128.0	1.18032
$887$	34080.0i	1.29007i	0.764152	+	0.645036i	$0.223158\pi$
$887$	34080.0i	1.29007i	−0.764152	+	0.645036i	$0.776842\pi$
$888$	3216.00i	0.121534i
$889$	−6400.00	−0.241450
$890$	0	0
$891$	−4860.00	−0.182734
$892$	− 2080.00i	− 0.0780757i
$893$	− 27360.0i	− 1.02527i
$894$	−9324.00	−0.348816
$895$	0	0
$896$	−4096.00	−0.152721
$897$	0	0
$898$	− 31548.0i	− 1.17235i
$899$	1392.00	0.0516416
$900$	0	0
$901$	9324.00	0.344759
$902$	28080.0i	1.03654i
$903$	39552.0i	1.45759i
$904$	3408.00	0.125385
$905$	0	0
$906$	13632.0	0.499882
$907$	25748.0i	0.942611i	0.881970	+	0.471306i	$0.156217\pi$
$907$	25748.0i	0.942611i	−0.881970	+	0.471306i	$0.843783\pi$
$908$	− 1584.00i	− 0.0578930i
$909$	−7182.00	−0.262059
$910$	0	0
$911$	−24768.0	−0.900769	−0.450384	−	0.892835i	$-0.648713\pi$
$911$	−24768.0	−0.900769	−0.450384	+	0.892835i	$0.648713\pi$
$912$	− 3648.00i	− 0.132453i
$913$	− 48240.0i	− 1.74864i
$914$	19444.0	0.703666
$915$	0	0
$916$	−5320.00	−0.191897
$917$	1920.00i	0.0691428i
$918$	2268.00i	0.0815416i
$919$	31264.0	1.12220	0.561101	−	0.827747i	$-0.310378\pi$
$919$	31264.0	1.12220	0.561101	+	0.827747i	$0.310378\pi$
$920$	0	0
$921$	7620.00	0.272625
$922$	21780.0i	0.777968i
$923$	4080.00i	0.145498i
$924$	23040.0	0.820303
$925$	0	0
$926$	−30256.0	−1.07373
$927$	9792.00i	0.346938i
$928$	192.000i	0.00679171i
$929$	6174.00	0.218043	0.109022	−	0.994039i	$-0.465228\pi$
$929$	6174.00	0.218043	0.109022	+	0.994039i	$0.465228\pi$
$930$	0	0
$931$	−51756.0	−1.82195
$932$	19464.0i	0.684082i
$933$	− 4680.00i	− 0.164219i
$934$	21336.0	0.747468
$935$	0	0
$936$	2448.00	0.0854865
$937$	28922.0i	1.00837i	0.863596	+	0.504184i	$0.168207\pi$
$937$	28922.0i	1.00837i	−0.863596	+	0.504184i	$0.831793\pi$
$938$	51968.0i	1.80897i
$939$	2802.00	0.0973800
$940$	0	0
$941$	29238.0	1.01289	0.506446	−	0.862272i	$-0.330959\pi$
$941$	29238.0	1.01289	0.506446	+	0.862272i	$0.330959\pi$
$942$	− 10164.0i	− 0.351551i
$943$	0	0
$944$	−10560.0	−0.364088
$945$	0	0
$946$	−49440.0	−1.69919
$947$	− 2868.00i	− 0.0984134i	−0.998789	−	0.0492067i	$-0.984331\pi$
$947$	− 2868.00i	− 0.0984134i	0.998789	−	0.0492067i	$-0.0156693\pi$
$948$	− 1824.00i	− 0.0624903i
$949$	25364.0	0.867598
$950$	0	0
$951$	−5022.00	−0.171240
$952$	− 10752.0i	− 0.366044i
$953$	− 24018.0i	− 0.816390i	−0.912895	−	0.408195i	$-0.866158\pi$
$953$	− 24018.0i	− 0.816390i	0.912895	−	0.408195i	$-0.133842\pi$
$954$	3996.00	0.135613
$955$	0	0
$956$	−7296.00	−0.246830
$957$	− 1080.00i	− 0.0364801i
$958$	30528.0i	1.02956i
$959$	−20544.0	−0.691763
$960$	0	0
$961$	24033.0	0.806720
$962$	9112.00i	0.305387i
$963$	− 15444.0i	− 0.516797i
$964$	−25928.0	−0.866270
$965$	0	0
$966$	0	0
$967$	25712.0i	0.855059i	0.904001	+	0.427530i	$0.140616\pi$
$967$	25712.0i	0.855059i	−0.904001	+	0.427530i	$0.859384\pi$
$968$	18152.0i	0.602714i
$969$	9576.00	0.317467
$970$	0	0
$971$	−12396.0	−0.409688	−0.204844	−	0.978795i	$-0.565669\pi$
$971$	−12396.0	−0.409688	−0.204844	+	0.978795i	$0.565669\pi$
$972$	972.000i	0.0320750i
$973$	90752.0i	2.99011i
$974$	−11552.0	−0.380031
$975$	0	0
$976$	−7840.00	−0.257123
$977$	− 46614.0i	− 1.52642i	−0.646150	−	0.763211i	$-0.723622\pi$
$977$	− 46614.0i	− 1.52642i	0.646150	−	0.763211i	$-0.276378\pi$
$978$	− 312.000i	− 0.0102011i
$979$	−40680.0	−1.32803
$980$	0	0
$981$	−8730.00	−0.284126
$982$	− 28488.0i	− 0.925752i
$983$	672.000i	0.0218041i	0.999941	+	0.0109021i	$0.00347031\pi$
$983$	672.000i	0.0218041i	−0.999941	+	0.0109021i	$0.996530\pi$
$984$	5616.00	0.181943
$985$	0	0
$986$	−504.000	−0.0162785
$987$	− 34560.0i	− 1.11455i
$988$	− 10336.0i	− 0.332826i
$989$	0	0
$990$	0	0
$991$	−38776.0	−1.24295	−0.621473	−	0.783435i	$-0.713466\pi$
$991$	−38776.0	−1.24295	−0.621473	+	0.783435i	$0.713466\pi$
$992$	7424.00i	0.237613i
$993$	11964.0i	0.382342i
$994$	7680.00	0.245065
$995$	0	0
$996$	−9648.00	−0.306936
$997$	30422.0i	0.966374i	0.875517	+	0.483187i	$0.160521\pi$
$997$	30422.0i	0.966374i	−0.875517	+	0.483187i	$0.839479\pi$
$998$	− 34232.0i	− 1.08577i
$999$	−3618.00	−0.114583

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.4.c.a.49.1		2
3.2	odd	2		450.4.c.k.199.2		2
4.3	odd	2		1200.4.f.u.49.2		2
5.2	odd	4		150.4.a.e.1.1		1
5.3	odd	4		30.4.a.a.1.1	✓	1
5.4	even	2	inner	150.4.c.a.49.2		2
15.2	even	4		450.4.a.b.1.1		1
15.8	even	4		90.4.a.d.1.1		1
15.14	odd	2		450.4.c.k.199.1		2
20.3	even	4		240.4.a.c.1.1		1
20.7	even	4		1200.4.a.bk.1.1		1
20.19	odd	2		1200.4.f.u.49.1		2
35.13	even	4		1470.4.a.a.1.1		1
40.3	even	4		960.4.a.s.1.1		1
40.13	odd	4		960.4.a.j.1.1		1
45.13	odd	12		810.4.e.m.541.1		2
45.23	even	12		810.4.e.e.541.1		2
45.38	even	12		810.4.e.e.271.1		2
45.43	odd	12		810.4.e.m.271.1		2
60.23	odd	4		720.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.a.1.1	✓	1	5.3	odd	4
90.4.a.d.1.1		1	15.8	even	4
150.4.a.e.1.1		1	5.2	odd	4
150.4.c.a.49.1		2	1.1	even	1	trivial
150.4.c.a.49.2		2	5.4	even	2	inner
240.4.a.c.1.1		1	20.3	even	4
450.4.a.b.1.1		1	15.2	even	4
450.4.c.k.199.1		2	15.14	odd	2
450.4.c.k.199.2		2	3.2	odd	2
720.4.a.b.1.1		1	60.23	odd	4
810.4.e.e.271.1		2	45.38	even	12
810.4.e.e.541.1		2	45.23	even	12
810.4.e.m.271.1		2	45.43	odd	12
810.4.e.m.541.1		2	45.13	odd	12
960.4.a.j.1.1		1	40.13	odd	4
960.4.a.s.1.1		1	40.3	even	4
1200.4.a.bk.1.1		1	20.7	even	4
1200.4.f.u.49.1		2	20.19	odd	2
1200.4.f.u.49.2		2	4.3	odd	2
1470.4.a.a.1.1		1	35.13	even	4

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} +32.0000i q^{7} +8.00000i q^{8} -9.00000 q^{9} -60.0000 q^{11} +12.0000i q^{12} +34.0000i q^{13} +64.0000 q^{14} +16.0000 q^{16} +42.0000i q^{17} +18.0000i q^{18} +76.0000 q^{19} +96.0000 q^{21} +120.000i q^{22} +24.0000 q^{24} +68.0000 q^{26} +27.0000i q^{27} -128.000i q^{28} -6.00000 q^{29} -232.000 q^{31} -32.0000i q^{32} +180.000i q^{33} +84.0000 q^{34} +36.0000 q^{36} +134.000i q^{37} -152.000i q^{38} +102.000 q^{39} +234.000 q^{41} -192.000i q^{42} +412.000i q^{43} +240.000 q^{44} -360.000i q^{47} -48.0000i q^{48} -681.000 q^{49} +126.000 q^{51} -136.000i q^{52} -222.000i q^{53} +54.0000 q^{54} -256.000 q^{56} -228.000i q^{57} +12.0000i q^{58} -660.000 q^{59} -490.000 q^{61} +464.000i q^{62} -288.000i q^{63} -64.0000 q^{64} +360.000 q^{66} +812.000i q^{67} -168.000i q^{68} +120.000 q^{71} -72.0000i q^{72} -746.000i q^{73} +268.000 q^{74} -304.000 q^{76} -1920.00i q^{77} -204.000i q^{78} -152.000 q^{79} +81.0000 q^{81} -468.000i q^{82} +804.000i q^{83} -384.000 q^{84} +824.000 q^{86} +18.0000i q^{87} -480.000i q^{88} +678.000 q^{89} -1088.00 q^{91} +696.000i q^{93} -720.000 q^{94} -96.0000 q^{96} +194.000i q^{97} +1362.00i q^{98} +540.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} - 120 q^{11} + 128 q^{14} + 32 q^{16} + 152 q^{19} + 192 q^{21} + 48 q^{24} + 136 q^{26} - 12 q^{29} - 464 q^{31} + 168 q^{34} + 72 q^{36} + 204 q^{39} + 468 q^{41} + 480 q^{44}+ \cdots + 1080 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 - 120 * q^11 + 128 * q^14 + 32 * q^16 + 152 * q^19 + 192 * q^21 + 48 * q^24 + 136 * q^26 - 12 * q^29 - 464 * q^31 + 168 * q^34 + 72 * q^36 + 204 * q^39 + 468 * q^41 + 480 * q^44 - 1362 * q^49 + 252 * q^51 + 108 * q^54 - 512 * q^56 - 1320 * q^59 - 980 * q^61 - 128 * q^64 + 720 * q^66 + 240 * q^71 + 536 * q^74 - 608 * q^76 - 304 * q^79 + 162 * q^81 - 768 * q^84 + 1648 * q^86 + 1356 * q^89 - 2176 * q^91 - 1440 * q^94 - 192 * q^96 + 1080 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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