LMFDB - Embedded newform 1200.4.f.u.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (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:	$70.8022920069$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.u.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -32.0000i q^{7} -9.00000 q^{9} +60.0000 q^{11} +34.0000i q^{13} +42.0000i q^{17} -76.0000 q^{19} +96.0000 q^{21} -27.0000i q^{27} -6.00000 q^{29} +232.000 q^{31} +180.000i q^{33} +134.000i q^{37} -102.000 q^{39} +234.000 q^{41} -412.000i q^{43} +360.000i q^{47} -681.000 q^{49} -126.000 q^{51} -222.000i q^{53} -228.000i q^{57} +660.000 q^{59} -490.000 q^{61} +288.000i q^{63} -812.000i q^{67} -120.000 q^{71} -746.000i q^{73} -1920.00i q^{77} +152.000 q^{79} +81.0000 q^{81} -804.000i q^{83} -18.0000i q^{87} +678.000 q^{89} +1088.00 q^{91} +696.000i q^{93} +194.000i q^{97} -540.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 120 q^{11} - 152 q^{19} + 192 q^{21} - 12 q^{29} + 464 q^{31} - 204 q^{39} + 468 q^{41} - 1362 q^{49} - 252 q^{51} + 1320 q^{59} - 980 q^{61} - 240 q^{71} + 304 q^{79} + 162 q^{81} + 1356 q^{89}+ \cdots - 1080 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 120 * q^11 - 152 * q^19 + 192 * q^21 - 12 * q^29 + 464 * q^31 - 204 * q^39 + 468 * q^41 - 1362 * q^49 - 252 * q^51 + 1320 * q^59 - 980 * q^61 - 240 * q^71 + 304 * q^79 + 162 * q^81 + 1356 * q^89 + 2176 * q^91 - 1080 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 32.0000i	− 1.72784i	−0.503631	−	0.863919i	$-0.668003\pi$
$7$	− 32.0000i	− 1.72784i	0.503631	−	0.863919i	$-0.331997\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	60.0000	1.64461	0.822304	−	0.569049i	$-0.192689\pi$
$11$	60.0000	1.64461	0.822304	+	0.569049i	$0.192689\pi$
$12$	0	0
$13$	34.0000i	0.725377i	0.931910	+	0.362689i	$0.118141\pi$
$13$	34.0000i	0.725377i	−0.931910	+	0.362689i	$0.881859\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	42.0000i	0.599206i	0.954064	+	0.299603i	$0.0968542\pi$
$17$	42.0000i	0.599206i	−0.954064	+	0.299603i	$0.903146\pi$
$18$	0	0
$19$	−76.0000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−76.0000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	96.0000	0.997567
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	−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	$-0.265405\pi$
$31$	232.000	1.34414	0.672071	+	0.740486i	$0.265405\pi$
$32$	0	0
$33$	180.000i	0.949514i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	134.000i	0.595391i	0.954661	+	0.297695i	$0.0962180\pi$
$37$	134.000i	0.595391i	−0.954661	+	0.297695i	$0.903782\pi$
$38$	0	0
$39$	−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	$0.352967\pi$
$42$	0	0
$43$	− 412.000i	− 1.46115i	−0.682833	−	0.730575i	$-0.739252\pi$
$43$	− 412.000i	− 1.46115i	0.682833	−	0.730575i	$-0.260748\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	360.000i	1.11726i	0.829416	+	0.558632i	$0.188674\pi$
$47$	360.000i	1.11726i	−0.829416	+	0.558632i	$0.811326\pi$
$48$	0	0
$49$	−681.000	−1.98542
$50$	0	0
$51$	−126.000	−0.345952
$52$	0	0
$53$	− 222.000i	− 0.575359i	−0.957727	−	0.287680i	$-0.907116\pi$
$53$	− 222.000i	− 0.575359i	0.957727	−	0.287680i	$-0.0928838\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 228.000i	− 0.529813i
$58$	0	0
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	0	0
$61$	−490.000	−1.02849	−0.514246	−	0.857642i	$-0.671928\pi$
$61$	−490.000	−1.02849	−0.514246	+	0.857642i	$0.671928\pi$
$62$	0	0
$63$	288.000i	0.575946i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 812.000i	− 1.48062i	−0.672265	−	0.740310i	$-0.734679\pi$
$67$	− 812.000i	− 1.48062i	0.672265	−	0.740310i	$-0.265321\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−120.000	−0.200583	−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000	−0.200583	−0.100291	+	0.994958i	$0.531978\pi$
$72$	0	0
$73$	− 746.000i	− 1.19606i	−0.801472	−	0.598032i	$-0.795949\pi$
$73$	− 746.000i	− 1.19606i	0.801472	−	0.598032i	$-0.204051\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1920.00i	− 2.84161i
$78$	0	0
$79$	152.000	0.216473	0.108236	−	0.994125i	$-0.465480\pi$
$79$	152.000	0.216473	0.108236	+	0.994125i	$0.465480\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 804.000i	− 1.06326i	−0.846977	−	0.531629i	$-0.821580\pi$
$83$	− 804.000i	− 1.06326i	0.846977	−	0.531629i	$-0.178420\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 18.0000i	− 0.0221816i
$88$	0	0
$89$	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
$94$	0	0
$95$	0	0
$96$	0	0
$97$	194.000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	194.000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	−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	$0.371406\pi$
$102$	0	0
$103$	1088.00i	1.04081i	0.853918	+	0.520407i	$0.174220\pi$
$103$	1088.00i	1.04081i	−0.853918	+	0.520407i	$0.825780\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1716.00i	− 1.55039i	−0.631721	−	0.775196i	$-0.717651\pi$
$107$	− 1716.00i	− 1.55039i	0.631721	−	0.775196i	$-0.282349\pi$
$108$	0	0
$109$	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
$112$	0	0
$113$	− 426.000i	− 0.354643i	−0.984153	−	0.177322i	$-0.943257\pi$
$113$	− 426.000i	− 0.354643i	0.984153	−	0.177322i	$-0.0567433\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 306.000i	− 0.241792i
$118$	0	0
$119$	1344.00	1.03533
$120$	0	0
$121$	2269.00	1.70473
$122$	0	0
$123$	702.000i	0.514611i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 200.000i	− 0.139741i	−0.997556	−	0.0698706i	$-0.977741\pi$
$127$	− 200.000i	− 0.139741i	0.997556	−	0.0698706i	$-0.0222586\pi$
$128$	0	0
$129$	1236.00	0.843595
$130$	0	0
$131$	−60.0000	−0.0400170	−0.0200085	−	0.999800i	$-0.506369\pi$
$131$	−60.0000	−0.0400170	−0.0200085	+	0.999800i	$0.506369\pi$
$132$	0	0
$133$	2432.00i	1.58557i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	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	$-0.832856\pi$
$139$	−2836.00	−1.73055	−0.865275	+	0.501298i	$0.832856\pi$
$140$	0	0
$141$	−1080.00	−0.645053
$142$	0	0
$143$	2040.00i	1.19296i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2043.00i	− 1.14628i
$148$	0	0
$149$	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	$-0.290274\pi$
$151$	2272.00	1.22446	0.612228	+	0.790682i	$0.290274\pi$
$152$	0	0
$153$	− 378.000i	− 0.199735i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1694.00i	0.861120i	0.902562	+	0.430560i	$0.141684\pi$
$157$	1694.00i	0.861120i	−0.902562	+	0.430560i	$0.858316\pi$
$158$	0	0
$159$	666.000	0.332184
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 52.0000i	− 0.0249874i	−0.999922	−	0.0124937i	$-0.996023\pi$
$163$	− 52.0000i	− 0.0249874i	0.999922	−	0.0124937i	$-0.00397698\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1200.00i	0.556041i	0.960575	+	0.278020i	$0.0896783\pi$
$167$	1200.00i	0.556041i	−0.960575	+	0.278020i	$0.910322\pi$
$168$	0	0
$169$	1041.00	0.473828
$170$	0	0
$171$	684.000	0.305888
$172$	0	0
$173$	− 54.0000i	− 0.0237315i	−0.999930	−	0.0118657i	$-0.996223\pi$
$173$	− 54.0000i	− 0.0237315i	0.999930	−	0.0118657i	$-0.00377707\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1980.00i	0.840824i
$178$	0	0
$179$	876.000	0.365784	0.182892	−	0.983133i	$-0.441454\pi$
$179$	876.000	0.365784	0.182892	+	0.983133i	$0.441454\pi$
$180$	0	0
$181$	3854.00	1.58268	0.791341	−	0.611375i	$-0.209383\pi$
$181$	3854.00	1.58268	0.791341	+	0.611375i	$0.209383\pi$
$182$	0	0
$183$	− 1470.00i	− 0.593801i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2520.00i	0.985458i
$188$	0	0
$189$	−864.000	−0.332522
$190$	0	0
$191$	2784.00	1.05468	0.527338	−	0.849656i	$-0.323190\pi$
$191$	2784.00	1.05468	0.527338	+	0.849656i	$0.323190\pi$
$192$	0	0
$193$	− 914.000i	− 0.340887i	−0.985367	−	0.170443i	$-0.945480\pi$
$193$	− 914.000i	− 0.340887i	0.985367	−	0.170443i	$-0.0545200\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5202.00i	− 1.88136i	−0.339300	−	0.940678i	$-0.610190\pi$
$197$	− 5202.00i	− 1.88136i	0.339300	−	0.940678i	$-0.389810\pi$
$198$	0	0
$199$	3152.00	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$199$	3152.00	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$200$	0	0
$201$	2436.00	0.854837
$202$	0	0
$203$	192.000i	0.0663830i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4560.00	−1.50920
$210$	0	0
$211$	−740.000	−0.241439	−0.120720	−	0.992687i	$-0.538520\pi$
$211$	−740.000	−0.241439	−0.120720	+	0.992687i	$0.538520\pi$
$212$	0	0
$213$	− 360.000i	− 0.115807i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7424.00i	− 2.32246i
$218$	0	0
$219$	2238.00	0.690548
$220$	0	0
$221$	−1428.00	−0.434650
$222$	0	0
$223$	− 520.000i	− 0.156151i	−0.996947	−	0.0780757i	$-0.975122\pi$
$223$	− 520.000i	− 0.156151i	0.996947	−	0.0780757i	$-0.0248776\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 396.000i	− 0.115786i	−0.998323	−	0.0578930i	$-0.981562\pi$
$227$	− 396.000i	− 0.115786i	0.998323	−	0.0578930i	$-0.0184382\pi$
$228$	0	0
$229$	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
$232$	0	0
$233$	− 4866.00i	− 1.36816i	−0.729405	−	0.684082i	$-0.760203\pi$
$233$	− 4866.00i	− 1.36816i	0.729405	−	0.684082i	$-0.239797\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	456.000i	0.124981i
$238$	0	0
$239$	−1824.00	−0.493660	−0.246830	−	0.969059i	$-0.579389\pi$
$239$	−1824.00	−0.493660	−0.246830	+	0.969059i	$0.579389\pi$
$240$	0	0
$241$	6482.00	1.73254	0.866270	−	0.499575i	$-0.166511\pi$
$241$	6482.00	1.73254	0.866270	+	0.499575i	$0.166511\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2584.00i	− 0.665652i
$248$	0	0
$249$	2412.00	0.613873
$250$	0	0
$251$	−1476.00	−0.371172	−0.185586	−	0.982628i	$-0.559418\pi$
$251$	−1476.00	−0.371172	−0.185586	+	0.982628i	$0.559418\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4314.00i	1.04708i	0.852001	+	0.523541i	$0.175389\pi$
$257$	4314.00i	1.04708i	−0.852001	+	0.523541i	$0.824611\pi$
$258$	0	0
$259$	4288.00	1.02874
$260$	0	0
$261$	54.0000	0.0128066
$262$	0	0
$263$	− 5280.00i	− 1.23794i	−0.785414	−	0.618971i	$-0.787550\pi$
$263$	− 5280.00i	− 1.23794i	0.785414	−	0.618971i	$-0.212450\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2034.00i	0.466213i
$268$	0	0
$269$	−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	$-0.572841\pi$
$271$	−2024.00	−0.453687	−0.226844	+	0.973931i	$0.572841\pi$
$272$	0	0
$273$	3264.00i	0.723613i
$274$	0	0
$275$	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	$0.928491\pi$
$278$	0	0
$279$	−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	$0.782296\pi$
$282$	0	0
$283$	− 3724.00i	− 0.782222i	−0.920344	−	0.391111i	$-0.872091\pi$
$283$	− 3724.00i	− 0.782222i	0.920344	−	0.391111i	$-0.127909\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 7488.00i	− 1.54008i
$288$	0	0
$289$	3149.00	0.640953
$290$	0	0
$291$	−582.000	−0.117242
$292$	0	0
$293$	7218.00i	1.43918i	0.694399	+	0.719591i	$0.255670\pi$
$293$	7218.00i	1.43918i	−0.694399	+	0.719591i	$0.744330\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1620.00i	− 0.316505i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−13184.0	−2.52463
$302$	0	0
$303$	2394.00i	0.453900i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2540.00i	− 0.472200i	−0.971729	−	0.236100i	$-0.924131\pi$
$307$	− 2540.00i	− 0.472200i	0.971729	−	0.236100i	$-0.0758693\pi$
$308$	0	0
$309$	−3264.00	−0.600914
$310$	0	0
$311$	−1560.00	−0.284436	−0.142218	−	0.989835i	$-0.545423\pi$
$311$	−1560.00	−0.284436	−0.142218	+	0.989835i	$0.545423\pi$
$312$	0	0
$313$	934.000i	0.168667i	0.996438	+	0.0843335i	$0.0268761\pi$
$313$	934.000i	0.168667i	−0.996438	+	0.0843335i	$0.973124\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1674.00i	− 0.296597i	−0.988943	−	0.148298i	$-0.952620\pi$
$317$	− 1674.00i	− 0.296597i	0.988943	−	0.148298i	$-0.0473796\pi$
$318$	0	0
$319$	−360.000	−0.0631854
$320$	0	0
$321$	5148.00	0.895119
$322$	0	0
$323$	− 3192.00i	− 0.549869i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2910.00i	0.492120i
$328$	0	0
$329$	11520.0	1.93045
$330$	0	0
$331$	3988.00	0.662237	0.331118	−	0.943589i	$-0.392574\pi$
$331$	3988.00	0.662237	0.331118	+	0.943589i	$0.392574\pi$
$332$	0	0
$333$	− 1206.00i	− 0.198464i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000i	0 0.000323285i	1.00000		0.000161642i	$5.14524e-5\pi$
$337$	2.00000i	0 0.000323285i	−1.00000		0.000161642i	$0.999949\pi$
$338$	0	0
$339$	1278.00	0.204753
$340$	0	0
$341$	13920.0	2.21059
$342$	0	0
$343$	10816.0i	1.70265i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1764.00i	− 0.272901i	−0.990647	−	0.136450i	$-0.956431\pi$
$347$	− 1764.00i	− 0.272901i	0.990647	−	0.136450i	$-0.0435694\pi$
$348$	0	0
$349$	−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
$352$	0	0
$353$	− 138.000i	− 0.0208074i	−0.999946	−	0.0104037i	$-0.996688\pi$
$353$	− 138.000i	− 0.0208074i	0.999946	−	0.0104037i	$-0.00331165\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4032.00i	0.597748i
$358$	0	0
$359$	−11976.0	−1.76064	−0.880319	−	0.474382i	$-0.842672\pi$
$359$	−11976.0	−1.76064	−0.880319	+	0.474382i	$0.842672\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	0	0
$363$	6807.00i	0.984228i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9704.00i	− 1.38023i	−0.723699	−	0.690115i	$-0.757560\pi$
$367$	− 9704.00i	− 1.38023i	0.723699	−	0.690115i	$-0.242440\pi$
$368$	0	0
$369$	−2106.00	−0.297111
$370$	0	0
$371$	−7104.00	−0.994128
$372$	0	0
$373$	8122.00i	1.12746i	0.825960	+	0.563728i	$0.190633\pi$
$373$	8122.00i	1.12746i	−0.825960	+	0.563728i	$0.809367\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 204.000i	− 0.0278688i
$378$	0	0
$379$	3404.00	0.461350	0.230675	−	0.973031i	$-0.425907\pi$
$379$	3404.00	0.461350	0.230675	+	0.973031i	$0.425907\pi$
$380$	0	0
$381$	600.000	0.0806796
$382$	0	0
$383$	− 2520.00i	− 0.336204i	−0.985770	−	0.168102i	$-0.946236\pi$
$383$	− 2520.00i	− 0.336204i	0.985770	−	0.168102i	$-0.0537637\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3708.00i	0.487050i
$388$	0	0
$389$	−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
$392$	0	0
$393$	− 180.000i	− 0.0231038i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4354.00i	− 0.550431i	−0.961383	−	0.275215i	$-0.911251\pi$
$397$	− 4354.00i	− 0.550431i	0.961383	−	0.275215i	$-0.0887492\pi$
$398$	0	0
$399$	−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	$0.667033\pi$
$402$	0	0
$403$	7888.00i	0.975011i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8040.00i	0.979184i
$408$	0	0
$409$	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
$412$	0	0
$413$	− 21120.0i	− 2.51634i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8508.00i	− 0.999133i
$418$	0	0
$419$	11580.0	1.35017	0.675084	−	0.737741i	$-0.264108\pi$
$419$	11580.0	1.35017	0.675084	+	0.737741i	$0.264108\pi$
$420$	0	0
$421$	−370.000	−0.0428330	−0.0214165	−	0.999771i	$-0.506818\pi$
$421$	−370.000	−0.0428330	−0.0214165	+	0.999771i	$0.506818\pi$
$422$	0	0
$423$	− 3240.00i	− 0.372421i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15680.0i	1.77707i
$428$	0	0
$429$	−6120.00	−0.688756
$430$	0	0
$431$	−5040.00	−0.563267	−0.281634	−	0.959522i	$-0.590876\pi$
$431$	−5040.00	−0.563267	−0.281634	+	0.959522i	$0.590876\pi$
$432$	0	0
$433$	3742.00i	0.415310i	0.978202	+	0.207655i	$0.0665831\pi$
$433$	3742.00i	0.415310i	−0.978202	+	0.207655i	$0.933417\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6208.00	−0.674924	−0.337462	−	0.941339i	$-0.609568\pi$
$439$	−6208.00	−0.674924	−0.337462	+	0.941339i	$0.609568\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	− 15564.0i	− 1.66923i	−0.550835	−	0.834614i	$-0.685691\pi$
$443$	− 15564.0i	− 1.66923i	0.550835	−	0.834614i	$-0.314309\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4662.00i	0.493300i
$448$	0	0
$449$	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
$452$	0	0
$453$	6816.00i	0.706940i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9722.00i	0.995133i	0.867426	+	0.497567i	$0.165773\pi$
$457$	9722.00i	0.995133i	−0.867426	+	0.497567i	$0.834227\pi$
$458$	0	0
$459$	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	$0.685413\pi$
$462$	0	0
$463$	15128.0i	1.51848i	0.650809	+	0.759242i	$0.274430\pi$
$463$	15128.0i	1.51848i	−0.650809	+	0.759242i	$0.725570\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 10668.0i	− 1.05708i	−0.848909	−	0.528540i	$-0.822740\pi$
$467$	− 10668.0i	− 1.05708i	0.848909	−	0.528540i	$-0.177260\pi$
$468$	0	0
$469$	−25984.0	−2.55827
$470$	0	0
$471$	−5082.00	−0.497168
$472$	0	0
$473$	− 24720.0i	− 2.40302i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1998.00i	0.191786i
$478$	0	0
$479$	15264.0	1.45601	0.728006	−	0.685571i	$-0.240447\pi$
$479$	15264.0	1.45601	0.728006	+	0.685571i	$0.240447\pi$
$480$	0	0
$481$	−4556.00	−0.431883
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5776.00i	0.537445i	0.963218	+	0.268722i	$0.0866014\pi$
$487$	5776.00i	0.537445i	−0.963218	+	0.268722i	$0.913399\pi$
$488$	0	0
$489$	156.000	0.0144265
$490$	0	0
$491$	−14244.0	−1.30921	−0.654606	−	0.755971i	$-0.727165\pi$
$491$	−14244.0	−1.30921	−0.654606	+	0.755971i	$0.727165\pi$
$492$	0	0
$493$	− 252.000i	− 0.0230213i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3840.00i	0.346575i
$498$	0	0
$499$	−17116.0	−1.53551	−0.767753	−	0.640746i	$-0.778625\pi$
$499$	−17116.0	−1.53551	−0.767753	+	0.640746i	$0.778625\pi$
$500$	0	0
$501$	−3600.00	−0.321030
$502$	0	0
$503$	− 16848.0i	− 1.49347i	−0.665122	−	0.746735i	$-0.731620\pi$
$503$	− 16848.0i	− 1.49347i	0.665122	−	0.746735i	$-0.268380\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3123.00i	0.273565i
$508$	0	0
$509$	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
$512$	0	0
$513$	2052.00i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	21600.0i	1.83746i
$518$	0	0
$519$	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	$0.790631\pi$
$522$	0	0
$523$	− 15340.0i	− 1.28255i	−0.767313	−	0.641273i	$-0.778407\pi$
$523$	− 15340.0i	− 1.28255i	0.767313	−	0.641273i	$-0.221593\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9744.00i	0.805418i
$528$	0	0
$529$	12167.0	1.00000
$530$	0	0
$531$	−5940.00	−0.485450
$532$	0	0
$533$	7956.00i	0.646553i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2628.00i	0.211185i
$538$	0	0
$539$	−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	$0.228616\pi$
$542$	0	0
$543$	11562.0i	0.913762i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10036.0i	0.784476i	0.919864	+	0.392238i	$0.128299\pi$
$547$	10036.0i	0.784476i	−0.919864	+	0.392238i	$0.871701\pi$
$548$	0	0
$549$	4410.00	0.342831
$550$	0	0
$551$	456.000	0.0352564
$552$	0	0
$553$	− 4864.00i	− 0.374030i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10326.0i	0.785506i	0.919644	+	0.392753i	$0.128477\pi$
$557$	10326.0i	0.785506i	−0.919644	+	0.392753i	$0.871523\pi$
$558$	0	0
$559$	14008.0	1.05988
$560$	0	0
$561$	−7560.00	−0.568954
$562$	0	0
$563$	4524.00i	0.338657i	0.985560	+	0.169328i	$0.0541599\pi$
$563$	4524.00i	0.338657i	−0.985560	+	0.169328i	$0.945840\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2592.00i	− 0.191982i
$568$	0	0
$569$	−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	$-0.577997\pi$
$571$	−6620.00	−0.485181	−0.242591	+	0.970129i	$0.577997\pi$
$572$	0	0
$573$	8352.00i	0.608918i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8834.00i	0.637373i	0.947860	+	0.318687i	$0.103242\pi$
$577$	8834.00i	0.637373i	−0.947860	+	0.318687i	$0.896758\pi$
$578$	0	0
$579$	2742.00	0.196811
$580$	0	0
$581$	−25728.0	−1.83714
$582$	0	0
$583$	− 13320.0i	− 0.946240i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3636.00i	− 0.255662i	−0.991796	−	0.127831i	$-0.959198\pi$
$587$	− 3636.00i	− 0.255662i	0.991796	−	0.127831i	$-0.0408016\pi$
$588$	0	0
$589$	−17632.0	−1.23347
$590$	0	0
$591$	15606.0	1.08620
$592$	0	0
$593$	− 6570.00i	− 0.454971i	−0.973782	−	0.227485i	$-0.926950\pi$
$593$	− 6570.00i	− 0.454971i	0.973782	−	0.227485i	$-0.0730504\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9456.00i	0.648255i
$598$	0	0
$599$	16584.0	1.13123	0.565613	−	0.824671i	$-0.308640\pi$
$599$	16584.0	1.13123	0.565613	+	0.824671i	$0.308640\pi$
$600$	0	0
$601$	−502.000	−0.0340716	−0.0170358	−	0.999855i	$-0.505423\pi$
$601$	−502.000	−0.0340716	−0.0170358	+	0.999855i	$0.505423\pi$
$602$	0	0
$603$	7308.00i	0.493540i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18568.0i	1.24160i	0.783969	+	0.620801i	$0.213192\pi$
$607$	18568.0i	1.24160i	−0.783969	+	0.620801i	$0.786808\pi$
$608$	0	0
$609$	−576.000	−0.0383263
$610$	0	0
$611$	−12240.0	−0.810438
$612$	0	0
$613$	13114.0i	0.864061i	0.901859	+	0.432031i	$0.142203\pi$
$613$	13114.0i	0.864061i	−0.901859	+	0.432031i	$0.857797\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5250.00i	0.342556i	0.985223	+	0.171278i	$0.0547896\pi$
$617$	5250.00i	0.342556i	−0.985223	+	0.171278i	$0.945210\pi$
$618$	0	0
$619$	−10804.0	−0.701534	−0.350767	−	0.936463i	$-0.614079\pi$
$619$	−10804.0	−0.701534	−0.350767	+	0.936463i	$0.614079\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 21696.0i	− 1.39524i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 13680.0i	− 0.871334i
$628$	0	0
$629$	−5628.00	−0.356762
$630$	0	0
$631$	27088.0	1.70896	0.854482	−	0.519481i	$-0.173875\pi$
$631$	27088.0	1.70896	0.854482	+	0.519481i	$0.173875\pi$
$632$	0	0
$633$	− 2220.00i	− 0.139395i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 23154.0i	− 1.44018i
$638$	0	0
$639$	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	$0.301792\pi$
$642$	0	0
$643$	20108.0i	1.23325i	0.787256	+	0.616627i	$0.211501\pi$
$643$	20108.0i	1.23325i	−0.787256	+	0.616627i	$0.788499\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7152.00i	0.434581i	0.976107	+	0.217291i	$0.0697219\pi$
$647$	7152.00i	0.434581i	−0.976107	+	0.217291i	$0.930278\pi$
$648$	0	0
$649$	39600.0	2.39512
$650$	0	0
$651$	22272.0	1.34087
$652$	0	0
$653$	31626.0i	1.89528i	0.319333	+	0.947642i	$0.396541\pi$
$653$	31626.0i	1.89528i	−0.319333	+	0.947642i	$0.603459\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6714.00i	0.398688i
$658$	0	0
$659$	28092.0	1.66056	0.830280	−	0.557347i	$-0.188181\pi$
$659$	28092.0	1.66056	0.830280	+	0.557347i	$0.188181\pi$
$660$	0	0
$661$	−13186.0	−0.775909	−0.387955	−	0.921678i	$-0.626818\pi$
$661$	−13186.0	−0.775909	−0.387955	+	0.921678i	$0.626818\pi$
$662$	0	0
$663$	− 4284.00i	− 0.250945i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1560.00	0.0901541
$670$	0	0
$671$	−29400.0	−1.69147
$672$	0	0
$673$	− 5138.00i	− 0.294287i	−0.989115	−	0.147144i	$-0.952992\pi$
$673$	− 5138.00i	− 0.294287i	0.989115	−	0.147144i	$-0.0470080\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6078.00i	0.345047i	0.985005	+	0.172523i	$0.0551920\pi$
$677$	6078.00i	0.345047i	−0.985005	+	0.172523i	$0.944808\pi$
$678$	0	0
$679$	6208.00	0.350871
$680$	0	0
$681$	1188.00	0.0668491
$682$	0	0
$683$	32244.0i	1.80642i	0.429203	+	0.903208i	$0.358795\pi$
$683$	32244.0i	1.80642i	−0.429203	+	0.903208i	$0.641205\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3990.00i	0.221584i
$688$	0	0
$689$	7548.00	0.417353
$690$	0	0
$691$	−4484.00	−0.246859	−0.123429	−	0.992353i	$-0.539389\pi$
$691$	−4484.00	−0.246859	−0.123429	+	0.992353i	$0.539389\pi$
$692$	0	0
$693$	17280.0i	0.947205i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9828.00i	0.534092i
$698$	0	0
$699$	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	$0.805842\pi$
$702$	0	0
$703$	− 10184.0i	− 0.546368i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 25536.0i	− 1.35839i
$708$	0	0
$709$	−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
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 5472.00i	− 0.285015i
$718$	0	0
$719$	13920.0	0.722014	0.361007	−	0.932563i	$-0.382433\pi$
$719$	13920.0	0.722014	0.361007	+	0.932563i	$0.382433\pi$
$720$	0	0
$721$	34816.0	1.79836
$722$	0	0
$723$	19446.0i	1.00028i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9376.00i	0.478317i	0.970981	+	0.239159i	$0.0768716\pi$
$727$	9376.00i	0.478317i	−0.970981	+	0.239159i	$0.923128\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	17304.0	0.875529
$732$	0	0
$733$	− 6014.00i	− 0.303045i	−0.988454	−	0.151523i	$-0.951582\pi$
$733$	− 6014.00i	− 0.303045i	0.988454	−	0.151523i	$-0.0484176\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 48720.0i	− 2.43504i
$738$	0	0
$739$	−7468.00	−0.371739	−0.185869	−	0.982574i	$-0.559510\pi$
$739$	−7468.00	−0.371739	−0.185869	+	0.982574i	$0.559510\pi$
$740$	0	0
$741$	7752.00	0.384314
$742$	0	0
$743$	31248.0i	1.54290i	0.636287	+	0.771452i	$0.280469\pi$
$743$	31248.0i	1.54290i	−0.636287	+	0.771452i	$0.719531\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7236.00i	0.354420i
$748$	0	0
$749$	−54912.0	−2.67883
$750$	0	0
$751$	−32840.0	−1.59567	−0.797835	−	0.602875i	$-0.794022\pi$
$751$	−32840.0	−1.59567	−0.797835	+	0.602875i	$0.794022\pi$
$752$	0	0
$753$	− 4428.00i	− 0.214297i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 19066.0i	− 0.915410i	−0.889104	−	0.457705i	$-0.848672\pi$
$757$	− 19066.0i	− 0.915410i	0.889104	−	0.457705i	$-0.151328\pi$
$758$	0	0
$759$	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	$0.447773\pi$
$762$	0	0
$763$	− 31040.0i	− 1.47277i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22440.0i	1.05640i
$768$	0	0
$769$	−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
$772$	0	0
$773$	− 14286.0i	− 0.664724i	−0.943152	−	0.332362i	$-0.892154\pi$
$773$	− 14286.0i	− 0.664724i	0.943152	−	0.332362i	$-0.107846\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12864.0i	0.593943i
$778$	0	0
$779$	−17784.0	−0.817943
$780$	0	0
$781$	−7200.00	−0.329880
$782$	0	0
$783$	162.000i	0.00739388i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18868.0i	0.854602i	0.904109	+	0.427301i	$0.140535\pi$
$787$	18868.0i	0.854602i	−0.904109	+	0.427301i	$0.859465\pi$
$788$	0	0
$789$	15840.0	0.714726
$790$	0	0
$791$	−13632.0	−0.612766
$792$	0	0
$793$	− 16660.0i	− 0.746045i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21690.0i	− 0.963989i	−0.876174	−	0.481994i	$-0.839913\pi$
$797$	− 21690.0i	− 0.963989i	0.876174	−	0.481994i	$-0.160087\pi$
$798$	0	0
$799$	−15120.0	−0.669471
$800$	0	0
$801$	−6102.00	−0.269168
$802$	0	0
$803$	− 44760.0i	− 1.96706i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 16578.0i	− 0.723139i
$808$	0	0
$809$	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	$-0.481770\pi$
$811$	2644.00	0.114480	0.0572401	+	0.998360i	$0.481770\pi$
$812$	0	0
$813$	− 6072.00i	− 0.261936i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	31312.0i	1.34084i
$818$	0	0
$819$	−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	$0.797471\pi$
$822$	0	0
$823$	− 880.000i	− 0.0372720i	−0.999826	−	0.0186360i	$-0.994068\pi$
$823$	− 880.000i	− 0.0372720i	0.999826	−	0.0186360i	$-0.00593237\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12876.0i	0.541406i	0.962663	+	0.270703i	$0.0872561\pi$
$827$	12876.0i	0.541406i	−0.962663	+	0.270703i	$0.912744\pi$
$828$	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
$832$	0	0
$833$	− 28602.0i	− 1.18968i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 6264.00i	− 0.258680i
$838$	0	0
$839$	−40584.0	−1.66998	−0.834991	−	0.550263i	$-0.814527\pi$
$839$	−40584.0	−1.66998	−0.834991	+	0.550263i	$0.814527\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	− 21906.0i	− 0.894997i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 72608.0i	− 2.94550i
$848$	0	0
$849$	11172.0	0.451616
$850$	0	0
$851$	0	0
$852$	0	0
$853$	25738.0i	1.03312i	0.856251	+	0.516561i	$0.172788\pi$
$853$	25738.0i	1.03312i	−0.856251	+	0.516561i	$0.827212\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13314.0i	0.530686i	0.964154	+	0.265343i	$0.0854851\pi$
$857$	13314.0i	0.530686i	−0.964154	+	0.265343i	$0.914515\pi$
$858$	0	0
$859$	24524.0	0.974096	0.487048	−	0.873375i	$-0.338074\pi$
$859$	24524.0	0.974096	0.487048	+	0.873375i	$0.338074\pi$
$860$	0	0
$861$	22464.0	0.889165
$862$	0	0
$863$	5592.00i	0.220572i	0.993900	+	0.110286i	$0.0351767\pi$
$863$	5592.00i	0.220572i	−0.993900	+	0.110286i	$0.964823\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9447.00i	0.370054i
$868$	0	0
$869$	9120.00	0.356012
$870$	0	0
$871$	27608.0	1.07401
$872$	0	0
$873$	− 1746.00i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14386.0i	− 0.553912i	−0.960883	−	0.276956i	$-0.910674\pi$
$877$	− 14386.0i	− 0.553912i	0.960883	−	0.276956i	$-0.0893256\pi$
$878$	0	0
$879$	−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	$0.143051\pi$
$882$	0	0
$883$	51548.0i	1.96458i	0.187354	+	0.982292i	$0.440009\pi$
$883$	51548.0i	1.96458i	−0.187354	+	0.982292i	$0.559991\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 34080.0i	− 1.29007i	−0.764152	−	0.645036i	$-0.776842\pi$
$887$	− 34080.0i	− 1.29007i	0.764152	−	0.645036i	$-0.223158\pi$
$888$	0	0
$889$	−6400.00	−0.241450
$890$	0	0
$891$	4860.00	0.182734
$892$	0	0
$893$	− 27360.0i	− 1.02527i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1392.00	−0.0516416
$900$	0	0
$901$	9324.00	0.344759
$902$	0	0
$903$	− 39552.0i	− 1.45759i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 25748.0i	− 0.942611i	−0.881970	−	0.471306i	$-0.843783\pi$
$907$	− 25748.0i	− 0.942611i	0.881970	−	0.471306i	$-0.156217\pi$
$908$	0	0
$909$	−7182.00	−0.262059
$910$	0	0
$911$	24768.0	0.900769	0.450384	−	0.892835i	$-0.351287\pi$
$911$	24768.0	0.900769	0.450384	+	0.892835i	$0.351287\pi$
$912$	0	0
$913$	− 48240.0i	− 1.74864i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1920.00i	0.0691428i
$918$	0	0
$919$	−31264.0	−1.12220	−0.561101	−	0.827747i	$-0.689622\pi$
$919$	−31264.0	−1.12220	−0.561101	+	0.827747i	$0.689622\pi$
$920$	0	0
$921$	7620.00	0.272625
$922$	0	0
$923$	− 4080.00i	− 0.145498i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 9792.00i	− 0.346938i
$928$	0	0
$929$	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
$932$	0	0
$933$	− 4680.00i	− 0.164219i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	28922.0i	1.00837i	0.863596	+	0.504184i	$0.168207\pi$
$937$	28922.0i	1.00837i	−0.863596	+	0.504184i	$0.831793\pi$
$938$	0	0
$939$	−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	$0.330959\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2868.00i	0.0984134i	0.998789	+	0.0492067i	$0.0156693\pi$
$947$	2868.00i	0.0984134i	−0.998789	+	0.0492067i	$0.984331\pi$
$948$	0	0
$949$	25364.0	0.867598
$950$	0	0
$951$	5022.00	0.171240
$952$	0	0
$953$	− 24018.0i	− 0.816390i	−0.912895	−	0.408195i	$-0.866158\pi$
$953$	− 24018.0i	− 0.816390i	0.912895	−	0.408195i	$-0.133842\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1080.00i	− 0.0364801i
$958$	0	0
$959$	20544.0	0.691763
$960$	0	0
$961$	24033.0	0.806720
$962$	0	0
$963$	15444.0i	0.516797i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 25712.0i	− 0.855059i	−0.904001	−	0.427530i	$-0.859384\pi$
$967$	− 25712.0i	− 0.855059i	0.904001	−	0.427530i	$-0.140616\pi$
$968$	0	0
$969$	9576.00	0.317467
$970$	0	0
$971$	12396.0	0.409688	0.204844	−	0.978795i	$-0.434331\pi$
$971$	12396.0	0.409688	0.204844	+	0.978795i	$0.434331\pi$
$972$	0	0
$973$	90752.0i	2.99011i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 46614.0i	− 1.52642i	−0.646150	−	0.763211i	$-0.723622\pi$
$977$	− 46614.0i	− 1.52642i	0.646150	−	0.763211i	$-0.276378\pi$
$978$	0	0
$979$	40680.0	1.32803
$980$	0	0
$981$	−8730.00	−0.284126
$982$	0	0
$983$	− 672.000i	− 0.0218041i	−0.999941	−	0.0109021i	$-0.996530\pi$
$983$	− 672.000i	− 0.0218041i	0.999941	−	0.0109021i	$-0.00347031\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	34560.0i	1.11455i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	38776.0	1.24295	0.621473	−	0.783435i	$-0.286534\pi$
$991$	38776.0	1.24295	0.621473	+	0.783435i	$0.286534\pi$
$992$	0	0
$993$	11964.0i	0.382342i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	30422.0i	0.966374i	0.875517	+	0.483187i	$0.160521\pi$
$997$	30422.0i	0.966374i	−0.875517	+	0.483187i	$0.839479\pi$
$998$	0	0
$999$	3618.00	0.114583

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.u.49.2		2
4.3	odd	2		150.4.c.a.49.1		2
5.2	odd	4		1200.4.a.bk.1.1		1
5.3	odd	4		240.4.a.c.1.1		1
5.4	even	2	inner	1200.4.f.u.49.1		2
12.11	even	2		450.4.c.k.199.2		2
15.8	even	4		720.4.a.b.1.1		1
20.3	even	4		30.4.a.a.1.1	✓	1
20.7	even	4		150.4.a.e.1.1		1
20.19	odd	2		150.4.c.a.49.2		2
40.3	even	4		960.4.a.j.1.1		1
40.13	odd	4		960.4.a.s.1.1		1
60.23	odd	4		90.4.a.d.1.1		1
60.47	odd	4		450.4.a.b.1.1		1
60.59	even	2		450.4.c.k.199.1		2
140.83	odd	4		1470.4.a.a.1.1		1
180.23	odd	12		810.4.e.e.541.1		2
180.43	even	12		810.4.e.m.271.1		2
180.83	odd	12		810.4.e.e.271.1		2
180.103	even	12		810.4.e.m.541.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.a.1.1	✓	1	20.3	even	4
90.4.a.d.1.1		1	60.23	odd	4
150.4.a.e.1.1		1	20.7	even	4
150.4.c.a.49.1		2	4.3	odd	2
150.4.c.a.49.2		2	20.19	odd	2
240.4.a.c.1.1		1	5.3	odd	4
450.4.a.b.1.1		1	60.47	odd	4
450.4.c.k.199.1		2	60.59	even	2
450.4.c.k.199.2		2	12.11	even	2
720.4.a.b.1.1		1	15.8	even	4
810.4.e.e.271.1		2	180.83	odd	12
810.4.e.e.541.1		2	180.23	odd	12
810.4.e.m.271.1		2	180.43	even	12
810.4.e.m.541.1		2	180.103	even	12
960.4.a.j.1.1		1	40.3	even	4
960.4.a.s.1.1		1	40.13	odd	4
1200.4.a.bk.1.1		1	5.2	odd	4
1200.4.f.u.49.1		2	5.4	even	2	inner
1200.4.f.u.49.2		2	1.1	even	1	trivial
1470.4.a.a.1.1		1	140.83	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -32.0000i q^{7} -9.00000 q^{9} +60.0000 q^{11} +34.0000i q^{13} +42.0000i q^{17} -76.0000 q^{19} +96.0000 q^{21} -27.0000i q^{27} -6.00000 q^{29} +232.000 q^{31} +180.000i q^{33} +134.000i q^{37} -102.000 q^{39} +234.000 q^{41} -412.000i q^{43} +360.000i q^{47} -681.000 q^{49} -126.000 q^{51} -222.000i q^{53} -228.000i q^{57} +660.000 q^{59} -490.000 q^{61} +288.000i q^{63} -812.000i q^{67} -120.000 q^{71} -746.000i q^{73} -1920.00i q^{77} +152.000 q^{79} +81.0000 q^{81} -804.000i q^{83} -18.0000i q^{87} +678.000 q^{89} +1088.00 q^{91} +696.000i q^{93} +194.000i q^{97} -540.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 120 q^{11} - 152 q^{19} + 192 q^{21} - 12 q^{29} + 464 q^{31} - 204 q^{39} + 468 q^{41} - 1362 q^{49} - 252 q^{51} + 1320 q^{59} - 980 q^{61} - 240 q^{71} + 304 q^{79} + 162 q^{81} + 1356 q^{89}+ \cdots - 1080 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 120 * q^11 - 152 * q^19 + 192 * q^21 - 12 * q^29 + 464 * q^31 - 204 * q^39 + 468 * q^41 - 1362 * q^49 - 252 * q^51 + 1320 * q^59 - 980 * q^61 - 240 * q^71 + 304 * q^79 + 162 * q^81 + 1356 * q^89 + 2176 * q^91 - 1080 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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