LMFDB - Embedded newform 1200.4.f.e.49.1

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 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -32.0000i q^{7} -9.00000 q^{9} -36.0000 q^{11} +10.0000i q^{13} -78.0000i q^{17} +140.000 q^{19} -96.0000 q^{21} -192.000i q^{23} +27.0000i q^{27} -6.00000 q^{29} +16.0000 q^{31} +108.000i q^{33} -34.0000i q^{37} +30.0000 q^{39} -390.000 q^{41} -52.0000i q^{43} -408.000i q^{47} -681.000 q^{49} -234.000 q^{51} +114.000i q^{53} -420.000i q^{57} +516.000 q^{59} -58.0000 q^{61} +288.000i q^{63} +892.000i q^{67} -576.000 q^{69} +120.000 q^{71} +646.000i q^{73} +1152.00i q^{77} -1168.00 q^{79} +81.0000 q^{81} -732.000i q^{83} +18.0000i q^{87} +1590.00 q^{89} +320.000 q^{91} -48.0000i q^{93} +194.000i q^{97} +324.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} - 72 q^{11} + 280 q^{19} - 192 q^{21} - 12 q^{29} + 32 q^{31} + 60 q^{39} - 780 q^{41} - 1362 q^{49} - 468 q^{51} + 1032 q^{59} - 116 q^{61} - 1152 q^{69} + 240 q^{71} - 2336 q^{79} + 162 q^{81}+ \cdots + 648 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 72 * q^11 + 280 * q^19 - 192 * q^21 - 12 * q^29 + 32 * q^31 + 60 * q^39 - 780 * q^41 - 1362 * q^49 - 468 * q^51 + 1032 * q^59 - 116 * q^61 - 1152 * q^69 + 240 * q^71 - 2336 * q^79 + 162 * q^81 + 3180 * q^89 + 640 * q^91 + 648 * 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
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	0	0
$13$	10.0000i	0.213346i	0.994294	+	0.106673i	$0.0340198\pi$
$13$	10.0000i	0.213346i	−0.994294	+	0.106673i	$0.965980\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 78.0000i	− 1.11281i	−0.830911	−	0.556405i	$-0.812180\pi$
$17$	− 78.0000i	− 1.11281i	0.830911	−	0.556405i	$-0.187820\pi$
$18$	0	0
$19$	140.000	1.69043	0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000	1.69043	0.845216	+	0.534425i	$0.179472\pi$
$20$	0	0
$21$	−96.0000	−0.997567
$22$	0	0
$23$	− 192.000i	− 1.74064i	−0.492485	−	0.870321i	$-0.663911\pi$
$23$	− 192.000i	− 1.74064i	0.492485	−	0.870321i	$-0.336089\pi$
$24$	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
$31$	16.0000	0.0926995	0.0463498	−	0.998925i	$-0.485241\pi$
$31$	16.0000	0.0926995	0.0463498	+	0.998925i	$0.485241\pi$
$32$	0	0
$33$	108.000i	0.569709i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 34.0000i	− 0.151069i	−0.997143	−	0.0755347i	$-0.975934\pi$
$37$	− 34.0000i	− 0.151069i	0.997143	−	0.0755347i	$-0.0240664\pi$
$38$	0	0
$39$	30.0000	0.123176
$40$	0	0
$41$	−390.000	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$42$	0	0
$43$	− 52.0000i	− 0.184417i	−0.995740	−	0.0922084i	$-0.970607\pi$
$43$	− 52.0000i	− 0.184417i	0.995740	−	0.0922084i	$-0.0293926\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 408.000i	− 1.26623i	−0.774057	−	0.633116i	$-0.781776\pi$
$47$	− 408.000i	− 1.26623i	0.774057	−	0.633116i	$-0.218224\pi$
$48$	0	0
$49$	−681.000	−1.98542
$50$	0	0
$51$	−234.000	−0.642481
$52$	0	0
$53$	114.000i	0.295455i	0.989028	+	0.147727i	$0.0471958\pi$
$53$	114.000i	0.295455i	−0.989028	+	0.147727i	$0.952804\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 420.000i	− 0.975971i
$58$	0	0
$59$	516.000	1.13860	0.569301	−	0.822129i	$-0.307214\pi$
$59$	516.000	1.13860	0.569301	+	0.822129i	$0.307214\pi$
$60$	0	0
$61$	−58.0000	−0.121740	−0.0608700	−	0.998146i	$-0.519388\pi$
$61$	−58.0000	−0.121740	−0.0608700	+	0.998146i	$0.519388\pi$
$62$	0	0
$63$	288.000i	0.575946i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	892.000i	1.62649i	0.581918	+	0.813247i	$0.302302\pi$
$67$	892.000i	1.62649i	−0.581918	+	0.813247i	$0.697698\pi$
$68$	0	0
$69$	−576.000	−1.00496
$70$	0	0
$71$	120.000	0.200583	0.100291	−	0.994958i	$-0.468022\pi$
$71$	120.000	0.200583	0.100291	+	0.994958i	$0.468022\pi$
$72$	0	0
$73$	646.000i	1.03573i	0.855461	+	0.517867i	$0.173274\pi$
$73$	646.000i	1.03573i	−0.855461	+	0.517867i	$0.826726\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1152.00i	1.70497i
$78$	0	0
$79$	−1168.00	−1.66342	−0.831711	−	0.555209i	$-0.812638\pi$
$79$	−1168.00	−1.66342	−0.831711	+	0.555209i	$0.812638\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 732.000i	− 0.968041i	−0.875057	−	0.484021i	$-0.839176\pi$
$83$	− 732.000i	− 0.968041i	0.875057	−	0.484021i	$-0.160824\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.0000i	0.0221816i
$88$	0	0
$89$	1590.00	1.89370	0.946852	−	0.321669i	$-0.104244\pi$
$89$	1590.00	1.89370	0.946852	+	0.321669i	$0.104244\pi$
$90$	0	0
$91$	320.000	0.368628
$92$	0	0
$93$	− 48.0000i	− 0.0535201i
$94$	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
$99$	324.000	0.328921
$100$	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
$103$	272.000i	0.260203i	0.991501	+	0.130102i	$0.0415304\pi$
$103$	272.000i	0.260203i	−0.991501	+	0.130102i	$0.958470\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 156.000i	− 0.140945i	−0.997514	−	0.0704724i	$-0.977549\pi$
$107$	− 156.000i	− 0.140945i	0.997514	−	0.0704724i	$-0.0224507\pi$
$108$	0	0
$109$	−1622.00	−1.42532	−0.712658	−	0.701512i	$-0.752509\pi$
$109$	−1622.00	−1.42532	−0.712658	+	0.701512i	$0.752509\pi$
$110$	0	0
$111$	−102.000	−0.0872199
$112$	0	0
$113$	− 1074.00i	− 0.894101i	−0.894509	−	0.447051i	$-0.852474\pi$
$113$	− 1074.00i	− 0.894101i	0.894509	−	0.447051i	$-0.147526\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 90.0000i	− 0.0711154i
$118$	0	0
$119$	−2496.00	−1.92276
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	1170.00i	0.857686i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1528.00i	1.06762i	0.845604	+	0.533811i	$0.179241\pi$
$127$	1528.00i	1.06762i	−0.845604	+	0.533811i	$0.820759\pi$
$128$	0	0
$129$	−156.000	−0.106473
$130$	0	0
$131$	−2412.00	−1.60868	−0.804341	−	0.594168i	$-0.797482\pi$
$131$	−2412.00	−1.60868	−0.804341	+	0.594168i	$0.797482\pi$
$132$	0	0
$133$	− 4480.00i	− 2.92079i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2106.00i	1.31334i	0.754178	+	0.656671i	$0.228036\pi$
$137$	2106.00i	1.31334i	−0.754178	+	0.656671i	$0.771964\pi$
$138$	0	0
$139$	−556.000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−556.000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−1224.00	−0.731060
$142$	0	0
$143$	− 360.000i	− 0.210522i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2043.00i	1.14628i
$148$	0	0
$149$	2418.00	1.32946	0.664732	−	0.747081i	$-0.268546\pi$
$149$	2418.00	1.32946	0.664732	+	0.747081i	$0.268546\pi$
$150$	0	0
$151$	−2840.00	−1.53057	−0.765285	−	0.643692i	$-0.777402\pi$
$151$	−2840.00	−1.53057	−0.765285	+	0.643692i	$0.777402\pi$
$152$	0	0
$153$	702.000i	0.370937i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2054.00i	1.04412i	0.852908	+	0.522061i	$0.174837\pi$
$157$	2054.00i	1.04412i	−0.852908	+	0.522061i	$0.825163\pi$
$158$	0	0
$159$	342.000	0.170581
$160$	0	0
$161$	−6144.00	−3.00755
$162$	0	0
$163$	− 460.000i	− 0.221043i	−0.993874	−	0.110521i	$-0.964748\pi$
$163$	− 460.000i	− 0.221043i	0.993874	−	0.110521i	$-0.0352521\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2016.00i	− 0.934148i	−0.884218	−	0.467074i	$-0.845308\pi$
$167$	− 2016.00i	− 0.934148i	0.884218	−	0.467074i	$-0.154692\pi$
$168$	0	0
$169$	2097.00	0.954483
$170$	0	0
$171$	−1260.00	−0.563477
$172$	0	0
$173$	618.000i	0.271593i	0.990737	+	0.135797i	$0.0433594\pi$
$173$	618.000i	0.271593i	−0.990737	+	0.135797i	$0.956641\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1548.00i	− 0.657372i
$178$	0	0
$179$	−2964.00	−1.23765	−0.618826	−	0.785528i	$-0.712391\pi$
$179$	−2964.00	−1.23765	−0.618826	+	0.785528i	$0.712391\pi$
$180$	0	0
$181$	−370.000	−0.151944	−0.0759721	−	0.997110i	$-0.524206\pi$
$181$	−370.000	−0.151944	−0.0759721	+	0.997110i	$0.524206\pi$
$182$	0	0
$183$	174.000i	0.0702866i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2808.00i	1.09808i
$188$	0	0
$189$	864.000	0.332522
$190$	0	0
$191$	−1104.00	−0.418234	−0.209117	−	0.977891i	$-0.567059\pi$
$191$	−1104.00	−0.418234	−0.209117	+	0.977891i	$0.567059\pi$
$192$	0	0
$193$	2398.00i	0.894362i	0.894444	+	0.447181i	$0.147572\pi$
$193$	2398.00i	0.894362i	−0.894444	+	0.447181i	$0.852428\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1278.00i	0.462202i	0.972930	+	0.231101i	$0.0742327\pi$
$197$	1278.00i	0.462202i	−0.972930	+	0.231101i	$0.925767\pi$
$198$	0	0
$199$	4472.00	1.59302	0.796512	−	0.604623i	$-0.206676\pi$
$199$	4472.00	1.59302	0.796512	+	0.604623i	$0.206676\pi$
$200$	0	0
$201$	2676.00	0.939057
$202$	0	0
$203$	192.000i	0.0663830i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1728.00i	0.580214i
$208$	0	0
$209$	−5040.00	−1.66806
$210$	0	0
$211$	−1340.00	−0.437201	−0.218600	−	0.975814i	$-0.570149\pi$
$211$	−1340.00	−0.437201	−0.218600	+	0.975814i	$0.570149\pi$
$212$	0	0
$213$	− 360.000i	− 0.115807i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 512.000i	− 0.160170i
$218$	0	0
$219$	1938.00	0.597981
$220$	0	0
$221$	780.000	0.237414
$222$	0	0
$223$	2360.00i	0.708687i	0.935115	+	0.354344i	$0.115296\pi$
$223$	2360.00i	0.708687i	−0.935115	+	0.354344i	$0.884704\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1380.00i	− 0.403497i	−0.979437	−	0.201748i	$-0.935338\pi$
$227$	− 1380.00i	− 0.403497i	0.979437	−	0.201748i	$-0.0646624\pi$
$228$	0	0
$229$	−1694.00	−0.488833	−0.244416	−	0.969670i	$-0.578596\pi$
$229$	−1694.00	−0.488833	−0.244416	+	0.969670i	$0.578596\pi$
$230$	0	0
$231$	3456.00	0.984364
$232$	0	0
$233$	5190.00i	1.45926i	0.683841	+	0.729631i	$0.260308\pi$
$233$	5190.00i	1.45926i	−0.683841	+	0.729631i	$0.739692\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3504.00i	0.960377i
$238$	0	0
$239$	2352.00	0.636562	0.318281	−	0.947996i	$-0.396895\pi$
$239$	2352.00	0.636562	0.318281	+	0.947996i	$0.396895\pi$
$240$	0	0
$241$	−3502.00	−0.936032	−0.468016	−	0.883720i	$-0.655031\pi$
$241$	−3502.00	−0.936032	−0.468016	+	0.883720i	$0.655031\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1400.00i	0.360647i
$248$	0	0
$249$	−2196.00	−0.558899
$250$	0	0
$251$	−4788.00	−1.20405	−0.602024	−	0.798478i	$-0.705639\pi$
$251$	−4788.00	−1.20405	−0.602024	+	0.798478i	$0.705639\pi$
$252$	0	0
$253$	6912.00i	1.71760i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1506.00i	0.365532i	0.983156	+	0.182766i	$0.0585051\pi$
$257$	1506.00i	0.365532i	−0.983156	+	0.182766i	$0.941495\pi$
$258$	0	0
$259$	−1088.00	−0.261023
$260$	0	0
$261$	54.0000	0.0128066
$262$	0	0
$263$	− 432.000i	− 0.101286i	−0.998717	−	0.0506431i	$-0.983873\pi$
$263$	− 432.000i	− 0.101286i	0.998717	−	0.0506431i	$-0.0161271\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4770.00i	− 1.09333i
$268$	0	0
$269$	−54.0000	−0.0122395	−0.00611977	−	0.999981i	$-0.501948\pi$
$269$	−54.0000	−0.0122395	−0.00611977	+	0.999981i	$0.501948\pi$
$270$	0	0
$271$	6496.00	1.45610	0.728051	−	0.685522i	$-0.240426\pi$
$271$	6496.00	1.45610	0.728051	+	0.685522i	$0.240426\pi$
$272$	0	0
$273$	− 960.000i	− 0.212827i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 466.000i	− 0.101080i	−0.998722	−	0.0505401i	$-0.983906\pi$
$277$	− 466.000i	− 0.101080i	0.998722	−	0.0505401i	$-0.0160943\pi$
$278$	0	0
$279$	−144.000	−0.0308998
$280$	0	0
$281$	−4854.00	−1.03048	−0.515241	−	0.857045i	$-0.672298\pi$
$281$	−4854.00	−1.03048	−0.515241	+	0.857045i	$0.672298\pi$
$282$	0	0
$283$	− 4516.00i	− 0.948581i	−0.880368	−	0.474290i	$-0.842705\pi$
$283$	− 4516.00i	− 0.948581i	0.880368	−	0.474290i	$-0.157295\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12480.0i	2.56680i
$288$	0	0
$289$	−1171.00	−0.238347
$290$	0	0
$291$	582.000	0.117242
$292$	0	0
$293$	− 8574.00i	− 1.70955i	−0.518998	−	0.854775i	$-0.673695\pi$
$293$	− 8574.00i	− 1.70955i	0.518998	−	0.854775i	$-0.326305\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 972.000i	− 0.189903i
$298$	0	0
$299$	1920.00	0.371359
$300$	0	0
$301$	−1664.00	−0.318642
$302$	0	0
$303$	− 2394.00i	− 0.453900i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3476.00i	− 0.646208i	−0.946363	−	0.323104i	$-0.895274\pi$
$307$	− 3476.00i	− 0.646208i	0.946363	−	0.323104i	$-0.104726\pi$
$308$	0	0
$309$	816.000	0.150229
$310$	0	0
$311$	−2424.00	−0.441969	−0.220985	−	0.975277i	$-0.570927\pi$
$311$	−2424.00	−0.441969	−0.220985	+	0.975277i	$0.570927\pi$
$312$	0	0
$313$	1558.00i	0.281353i	0.990056	+	0.140676i	$0.0449277\pi$
$313$	1558.00i	0.281353i	−0.990056	+	0.140676i	$0.955072\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8538.00i	− 1.51275i	−0.654138	−	0.756375i	$-0.726968\pi$
$317$	− 8538.00i	− 1.51275i	0.654138	−	0.756375i	$-0.273032\pi$
$318$	0	0
$319$	216.000	0.0379112
$320$	0	0
$321$	−468.000	−0.0813745
$322$	0	0
$323$	− 10920.0i	− 1.88113i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4866.00i	0.822906i
$328$	0	0
$329$	−13056.0	−2.18784
$330$	0	0
$331$	988.000	0.164065	0.0820323	−	0.996630i	$-0.473859\pi$
$331$	988.000	0.164065	0.0820323	+	0.996630i	$0.473859\pi$
$332$	0	0
$333$	306.000i	0.0503564i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2546.00i	0.411541i	0.978600	+	0.205771i	$0.0659701\pi$
$337$	2546.00i	0.411541i	−0.978600	+	0.205771i	$0.934030\pi$
$338$	0	0
$339$	−3222.00	−0.516209
$340$	0	0
$341$	−576.000	−0.0914726
$342$	0	0
$343$	10816.0i	1.70265i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8556.00i	− 1.32366i	−0.749654	−	0.661830i	$-0.769780\pi$
$347$	− 8556.00i	− 1.32366i	0.749654	−	0.661830i	$-0.230220\pi$
$348$	0	0
$349$	3706.00	0.568417	0.284209	−	0.958762i	$-0.408269\pi$
$349$	3706.00	0.568417	0.284209	+	0.958762i	$0.408269\pi$
$350$	0	0
$351$	−270.000	−0.0410585
$352$	0	0
$353$	− 11394.0i	− 1.71796i	−0.512005	−	0.858982i	$-0.671097\pi$
$353$	− 11394.0i	− 1.71796i	0.512005	−	0.858982i	$-0.328903\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7488.00i	1.11010i
$358$	0	0
$359$	−264.000	−0.0388117	−0.0194058	−	0.999812i	$-0.506177\pi$
$359$	−264.000	−0.0388117	−0.0194058	+	0.999812i	$0.506177\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	0	0
$363$	105.000i	0.0151820i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10232.0i	− 1.45533i	−0.685933	−	0.727665i	$-0.740606\pi$
$367$	− 10232.0i	− 1.45533i	0.685933	−	0.727665i	$-0.259394\pi$
$368$	0	0
$369$	3510.00	0.495185
$370$	0	0
$371$	3648.00	0.510498
$372$	0	0
$373$	562.000i	0.0780141i	0.999239	+	0.0390070i	$0.0124195\pi$
$373$	562.000i	0.0780141i	−0.999239	+	0.0390070i	$0.987581\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 60.0000i	− 0.00819670i
$378$	0	0
$379$	−7228.00	−0.979624	−0.489812	−	0.871828i	$-0.662935\pi$
$379$	−7228.00	−0.979624	−0.489812	+	0.871828i	$0.662935\pi$
$380$	0	0
$381$	4584.00	0.616392
$382$	0	0
$383$	− 5736.00i	− 0.765263i	−0.923901	−	0.382632i	$-0.875018\pi$
$383$	− 5736.00i	− 0.765263i	0.923901	−	0.382632i	$-0.124982\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	468.000i	0.0614723i
$388$	0	0
$389$	9186.00	1.19730	0.598649	−	0.801012i	$-0.295705\pi$
$389$	9186.00	1.19730	0.598649	+	0.801012i	$0.295705\pi$
$390$	0	0
$391$	−14976.0	−1.93700
$392$	0	0
$393$	7236.00i	0.928773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 394.000i	− 0.0498093i	−0.999690	−	0.0249047i	$-0.992072\pi$
$397$	− 394.000i	− 0.0498093i	0.999690	−	0.0249047i	$-0.00792822\pi$
$398$	0	0
$399$	−13440.0	−1.68632
$400$	0	0
$401$	−1614.00	−0.200996	−0.100498	−	0.994937i	$-0.532044\pi$
$401$	−1614.00	−0.200996	−0.100498	+	0.994937i	$0.532044\pi$
$402$	0	0
$403$	160.000i	0.0197771i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1224.00i	0.149070i
$408$	0	0
$409$	−1034.00	−0.125007	−0.0625037	−	0.998045i	$-0.519909\pi$
$409$	−1034.00	−0.125007	−0.0625037	+	0.998045i	$0.519909\pi$
$410$	0	0
$411$	6318.00	0.758258
$412$	0	0
$413$	− 16512.0i	− 1.96732i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1668.00i	0.195881i
$418$	0	0
$419$	3708.00	0.432333	0.216167	−	0.976356i	$-0.430645\pi$
$419$	3708.00	0.432333	0.216167	+	0.976356i	$0.430645\pi$
$420$	0	0
$421$	−4930.00	−0.570721	−0.285360	−	0.958420i	$-0.592113\pi$
$421$	−4930.00	−0.570721	−0.285360	+	0.958420i	$0.592113\pi$
$422$	0	0
$423$	3672.00i	0.422077i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1856.00i	0.210347i
$428$	0	0
$429$	−1080.00	−0.121545
$430$	0	0
$431$	2592.00	0.289680	0.144840	−	0.989455i	$-0.453733\pi$
$431$	2592.00	0.289680	0.144840	+	0.989455i	$0.453733\pi$
$432$	0	0
$433$	− 2162.00i	− 0.239952i	−0.992777	−	0.119976i	$-0.961718\pi$
$433$	− 2162.00i	− 0.239952i	0.992777	−	0.119976i	$-0.0382817\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 26880.0i	− 2.94244i
$438$	0	0
$439$	1352.00	0.146987	0.0734937	−	0.997296i	$-0.476585\pi$
$439$	1352.00	0.146987	0.0734937	+	0.997296i	$0.476585\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	5532.00i	0.593303i	0.954986	+	0.296652i	$0.0958700\pi$
$443$	5532.00i	0.593303i	−0.954986	+	0.296652i	$0.904130\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 7254.00i	− 0.767567i
$448$	0	0
$449$	3198.00	0.336131	0.168066	−	0.985776i	$-0.446248\pi$
$449$	3198.00	0.336131	0.168066	+	0.985776i	$0.446248\pi$
$450$	0	0
$451$	14040.0	1.46589
$452$	0	0
$453$	8520.00i	0.883674i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1510.00i	− 0.154562i	−0.997009	−	0.0772810i	$-0.975376\pi$
$457$	− 1510.00i	− 0.154562i	0.997009	−	0.0772810i	$-0.0246239\pi$
$458$	0	0
$459$	2106.00	0.214160
$460$	0	0
$461$	16086.0	1.62516	0.812581	−	0.582848i	$-0.198062\pi$
$461$	16086.0	1.62516	0.812581	+	0.582848i	$0.198062\pi$
$462$	0	0
$463$	5384.00i	0.540423i	0.962801	+	0.270211i	$0.0870936\pi$
$463$	5384.00i	0.540423i	−0.962801	+	0.270211i	$0.912906\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2604.00i	0.258027i	0.991643	+	0.129014i	$0.0411811\pi$
$467$	2604.00i	0.258027i	−0.991643	+	0.129014i	$0.958819\pi$
$468$	0	0
$469$	28544.0	2.81032
$470$	0	0
$471$	6162.00	0.602824
$472$	0	0
$473$	1872.00i	0.181976i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1026.00i	− 0.0984849i
$478$	0	0
$479$	11136.0	1.06225	0.531124	−	0.847294i	$-0.321770\pi$
$479$	11136.0	1.06225	0.531124	+	0.847294i	$0.321770\pi$
$480$	0	0
$481$	340.000	0.0322301
$482$	0	0
$483$	18432.0i	1.73641i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 14624.0i	− 1.36073i	−0.732872	−	0.680366i	$-0.761821\pi$
$487$	− 14624.0i	− 1.36073i	0.732872	−	0.680366i	$-0.238179\pi$
$488$	0	0
$489$	−1380.00	−0.127619
$490$	0	0
$491$	−11844.0	−1.08862	−0.544310	−	0.838884i	$-0.683208\pi$
$491$	−11844.0	−1.08862	−0.544310	+	0.838884i	$0.683208\pi$
$492$	0	0
$493$	468.000i	0.0427539i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3840.00i	− 0.346575i
$498$	0	0
$499$	−11284.0	−1.01231	−0.506154	−	0.862443i	$-0.668933\pi$
$499$	−11284.0	−1.01231	−0.506154	+	0.862443i	$0.668933\pi$
$500$	0	0
$501$	−6048.00	−0.539331
$502$	0	0
$503$	4032.00i	0.357412i	0.983903	+	0.178706i	$0.0571910\pi$
$503$	4032.00i	0.357412i	−0.983903	+	0.178706i	$0.942809\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6291.00i	− 0.551071i
$508$	0	0
$509$	17562.0	1.52932	0.764658	−	0.644436i	$-0.222908\pi$
$509$	17562.0	1.52932	0.764658	+	0.644436i	$0.222908\pi$
$510$	0	0
$511$	20672.0	1.78958
$512$	0	0
$513$	3780.00i	0.325324i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14688.0i	1.24947i
$518$	0	0
$519$	1854.00	0.156805
$520$	0	0
$521$	3162.00	0.265892	0.132946	−	0.991123i	$-0.457556\pi$
$521$	3162.00	0.265892	0.132946	+	0.991123i	$0.457556\pi$
$522$	0	0
$523$	6764.00i	0.565524i	0.959190	+	0.282762i	$0.0912507\pi$
$523$	6764.00i	0.565524i	−0.959190	+	0.282762i	$0.908749\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1248.00i	− 0.103157i
$528$	0	0
$529$	−24697.0	−2.02983
$530$	0	0
$531$	−4644.00	−0.379534
$532$	0	0
$533$	− 3900.00i	− 0.316938i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8892.00i	0.714559i
$538$	0	0
$539$	24516.0	1.95914
$540$	0	0
$541$	17798.0	1.41441	0.707205	−	0.707009i	$-0.249956\pi$
$541$	17798.0	1.41441	0.707205	+	0.707009i	$0.249956\pi$
$542$	0	0
$543$	1110.00i	0.0877250i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19996.0i	1.56301i	0.623898	+	0.781506i	$0.285548\pi$
$547$	19996.0i	1.56301i	−0.623898	+	0.781506i	$0.714452\pi$
$548$	0	0
$549$	522.000	0.0405800
$550$	0	0
$551$	−840.000	−0.0649459
$552$	0	0
$553$	37376.0i	2.87412i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11094.0i	0.843928i	0.906613	+	0.421964i	$0.138659\pi$
$557$	11094.0i	0.843928i	−0.906613	+	0.421964i	$0.861341\pi$
$558$	0	0
$559$	520.000	0.0393446
$560$	0	0
$561$	8424.00	0.633978
$562$	0	0
$563$	900.000i	0.0673721i	0.999432	+	0.0336860i	$0.0107246\pi$
$563$	900.000i	0.0673721i	−0.999432	+	0.0336860i	$0.989275\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2592.00i	− 0.191982i
$568$	0	0
$569$	−7914.00	−0.583079	−0.291540	−	0.956559i	$-0.594168\pi$
$569$	−7914.00	−0.583079	−0.291540	+	0.956559i	$0.594168\pi$
$570$	0	0
$571$	2380.00	0.174431	0.0872153	−	0.996189i	$-0.472203\pi$
$571$	2380.00	0.174431	0.0872153	+	0.996189i	$0.472203\pi$
$572$	0	0
$573$	3312.00i	0.241467i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 25726.0i	− 1.85613i	−0.372417	−	0.928065i	$-0.621471\pi$
$577$	− 25726.0i	− 1.85613i	0.372417	−	0.928065i	$-0.378529\pi$
$578$	0	0
$579$	7194.00	0.516360
$580$	0	0
$581$	−23424.0	−1.67262
$582$	0	0
$583$	− 4104.00i	− 0.291544i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3612.00i	− 0.253975i	−0.991904	−	0.126987i	$-0.959469\pi$
$587$	− 3612.00i	− 0.253975i	0.991904	−	0.126987i	$-0.0405308\pi$
$588$	0	0
$589$	2240.00	0.156702
$590$	0	0
$591$	3834.00	0.266852
$592$	0	0
$593$	− 2898.00i	− 0.200686i	−0.994953	−	0.100343i	$-0.968006\pi$
$593$	− 2898.00i	− 0.200686i	0.994953	−	0.100343i	$-0.0319940\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 13416.0i	− 0.919733i
$598$	0	0
$599$	2664.00	0.181716	0.0908582	−	0.995864i	$-0.471039\pi$
$599$	2664.00	0.181716	0.0908582	+	0.995864i	$0.471039\pi$
$600$	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
$603$	− 8028.00i	− 0.542165i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7976.00i	− 0.533337i	−0.963788	−	0.266669i	$-0.914077\pi$
$607$	− 7976.00i	− 0.533337i	0.963788	−	0.266669i	$-0.0859230\pi$
$608$	0	0
$609$	576.000	0.0383263
$610$	0	0
$611$	4080.00	0.270146
$612$	0	0
$613$	− 20414.0i	− 1.34505i	−0.740076	−	0.672523i	$-0.765210\pi$
$613$	− 20414.0i	− 1.34505i	0.740076	−	0.672523i	$-0.234790\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6342.00i	− 0.413808i	−0.978361	−	0.206904i	$-0.933661\pi$
$617$	− 6342.00i	− 0.413808i	0.978361	−	0.206904i	$-0.0663387\pi$
$618$	0	0
$619$	22676.0	1.47242	0.736208	−	0.676755i	$-0.236615\pi$
$619$	22676.0	1.47242	0.736208	+	0.676755i	$0.236615\pi$
$620$	0	0
$621$	5184.00	0.334987
$622$	0	0
$623$	− 50880.0i	− 3.27201i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	15120.0i	0.963054i
$628$	0	0
$629$	−2652.00	−0.168112
$630$	0	0
$631$	7048.00	0.444654	0.222327	−	0.974972i	$-0.428635\pi$
$631$	7048.00	0.444654	0.222327	+	0.974972i	$0.428635\pi$
$632$	0	0
$633$	4020.00i	0.252418i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6810.00i	− 0.423582i
$638$	0	0
$639$	−1080.00	−0.0668609
$640$	0	0
$641$	−20286.0	−1.25000	−0.624999	−	0.780625i	$-0.714901\pi$
$641$	−20286.0	−1.25000	−0.624999	+	0.780625i	$0.714901\pi$
$642$	0	0
$643$	− 16108.0i	− 0.987928i	−0.869482	−	0.493964i	$-0.835548\pi$
$643$	− 16108.0i	− 0.987928i	0.869482	−	0.493964i	$-0.164452\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 27456.0i	− 1.66833i	−0.551518	−	0.834163i	$-0.685951\pi$
$647$	− 27456.0i	− 1.66833i	0.551518	−	0.834163i	$-0.314049\pi$
$648$	0	0
$649$	−18576.0	−1.12353
$650$	0	0
$651$	−1536.00	−0.0924740
$652$	0	0
$653$	12522.0i	0.750419i	0.926940	+	0.375210i	$0.122429\pi$
$653$	12522.0i	0.750419i	−0.926940	+	0.375210i	$0.877571\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5814.00i	− 0.345245i
$658$	0	0
$659$	−16308.0	−0.963990	−0.481995	−	0.876174i	$-0.660088\pi$
$659$	−16308.0	−0.963990	−0.481995	+	0.876174i	$0.660088\pi$
$660$	0	0
$661$	32078.0	1.88758	0.943789	−	0.330547i	$-0.107233\pi$
$661$	32078.0	1.88758	0.943789	+	0.330547i	$0.107233\pi$
$662$	0	0
$663$	− 2340.00i	− 0.137071i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1152.00i	0.0668750i
$668$	0	0
$669$	7080.00	0.409161
$670$	0	0
$671$	2088.00	0.120129
$672$	0	0
$673$	− 4610.00i	− 0.264045i	−0.991247	−	0.132023i	$-0.957853\pi$
$673$	− 4610.00i	− 0.264045i	0.991247	−	0.132023i	$-0.0421472\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10782.0i	0.612091i	0.952017	+	0.306046i	$0.0990060\pi$
$677$	10782.0i	0.612091i	−0.952017	+	0.306046i	$0.900994\pi$
$678$	0	0
$679$	6208.00	0.350871
$680$	0	0
$681$	−4140.00	−0.232959
$682$	0	0
$683$	2892.00i	0.162019i	0.996713	+	0.0810097i	$0.0258145\pi$
$683$	2892.00i	0.162019i	−0.996713	+	0.0810097i	$0.974186\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5082.00i	0.282228i
$688$	0	0
$689$	−1140.00	−0.0630342
$690$	0	0
$691$	29572.0	1.62803	0.814017	−	0.580841i	$-0.197276\pi$
$691$	29572.0	1.62803	0.814017	+	0.580841i	$0.197276\pi$
$692$	0	0
$693$	− 10368.0i	− 0.568323i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	30420.0i	1.65314i
$698$	0	0
$699$	15570.0	0.842506
$700$	0	0
$701$	5766.00	0.310669	0.155334	−	0.987862i	$-0.450354\pi$
$701$	5766.00	0.310669	0.155334	+	0.987862i	$0.450354\pi$
$702$	0	0
$703$	− 4760.00i	− 0.255372i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 25536.0i	− 1.35839i
$708$	0	0
$709$	−3326.00	−0.176178	−0.0880892	−	0.996113i	$-0.528076\pi$
$709$	−3326.00	−0.176178	−0.0880892	+	0.996113i	$0.528076\pi$
$710$	0	0
$711$	10512.0	0.554474
$712$	0	0
$713$	− 3072.00i	− 0.161357i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7056.00i	− 0.367519i
$718$	0	0
$719$	−7728.00	−0.400843	−0.200421	−	0.979710i	$-0.564231\pi$
$719$	−7728.00	−0.400843	−0.200421	+	0.979710i	$0.564231\pi$
$720$	0	0
$721$	8704.00	0.449589
$722$	0	0
$723$	10506.0i	0.540418i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21616.0i	1.10274i	0.834260	+	0.551371i	$0.185895\pi$
$727$	21616.0i	1.10274i	−0.834260	+	0.551371i	$0.814105\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−4056.00	−0.205221
$732$	0	0
$733$	− 10118.0i	− 0.509846i	−0.966961	−	0.254923i	$-0.917950\pi$
$733$	− 10118.0i	− 0.509846i	0.966961	−	0.254923i	$-0.0820501\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 32112.0i	− 1.60497i
$738$	0	0
$739$	10460.0	0.520673	0.260336	−	0.965518i	$-0.416167\pi$
$739$	10460.0	0.520673	0.260336	+	0.965518i	$0.416167\pi$
$740$	0	0
$741$	4200.00	0.208220
$742$	0	0
$743$	17232.0i	0.850849i	0.904994	+	0.425424i	$0.139875\pi$
$743$	17232.0i	0.850849i	−0.904994	+	0.425424i	$0.860125\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6588.00i	0.322680i
$748$	0	0
$749$	−4992.00	−0.243530
$750$	0	0
$751$	−26912.0	−1.30763	−0.653817	−	0.756653i	$-0.726833\pi$
$751$	−26912.0	−1.30763	−0.653817	+	0.756653i	$0.726833\pi$
$752$	0	0
$753$	14364.0i	0.695157i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13838.0i	0.664400i	0.943209	+	0.332200i	$0.107791\pi$
$757$	13838.0i	0.664400i	−0.943209	+	0.332200i	$0.892209\pi$
$758$	0	0
$759$	20736.0	0.991659
$760$	0	0
$761$	−17238.0	−0.821126	−0.410563	−	0.911832i	$-0.634668\pi$
$761$	−17238.0	−0.821126	−0.410563	+	0.911832i	$0.634668\pi$
$762$	0	0
$763$	51904.0i	2.46271i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5160.00i	0.242916i
$768$	0	0
$769$	−21698.0	−1.01749	−0.508745	−	0.860917i	$-0.669890\pi$
$769$	−21698.0	−1.01749	−0.508745	+	0.860917i	$0.669890\pi$
$770$	0	0
$771$	4518.00	0.211040
$772$	0	0
$773$	− 18366.0i	− 0.854565i	−0.904118	−	0.427283i	$-0.859471\pi$
$773$	− 18366.0i	− 0.854565i	0.904118	−	0.427283i	$-0.140529\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3264.00i	0.150702i
$778$	0	0
$779$	−54600.0	−2.51123
$780$	0	0
$781$	−4320.00	−0.197928
$782$	0	0
$783$	− 162.000i	− 0.00739388i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30316.0i	1.37312i	0.727071	+	0.686562i	$0.240881\pi$
$787$	30316.0i	1.37312i	−0.727071	+	0.686562i	$0.759119\pi$
$788$	0	0
$789$	−1296.00	−0.0584776
$790$	0	0
$791$	−34368.0	−1.54486
$792$	0	0
$793$	− 580.000i	− 0.0259728i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7494.00i	0.333063i	0.986036	+	0.166531i	$0.0532567\pi$
$797$	7494.00i	0.333063i	−0.986036	+	0.166531i	$0.946743\pi$
$798$	0	0
$799$	−31824.0	−1.40908
$800$	0	0
$801$	−14310.0	−0.631235
$802$	0	0
$803$	− 23256.0i	− 1.02203i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	162.000i	0.00706651i
$808$	0	0
$809$	11526.0	0.500906	0.250453	−	0.968129i	$-0.419421\pi$
$809$	11526.0	0.500906	0.250453	+	0.968129i	$0.419421\pi$
$810$	0	0
$811$	33820.0	1.46434	0.732171	−	0.681121i	$-0.238507\pi$
$811$	33820.0	1.46434	0.732171	+	0.681121i	$0.238507\pi$
$812$	0	0
$813$	− 19488.0i	− 0.840681i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7280.00i	− 0.311744i
$818$	0	0
$819$	−2880.00	−0.122876
$820$	0	0
$821$	19566.0	0.831739	0.415870	−	0.909424i	$-0.363477\pi$
$821$	19566.0	0.831739	0.415870	+	0.909424i	$0.363477\pi$
$822$	0	0
$823$	− 40096.0i	− 1.69825i	−0.528193	−	0.849124i	$-0.677130\pi$
$823$	− 40096.0i	− 1.69825i	0.528193	−	0.849124i	$-0.322870\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 31884.0i	− 1.34065i	−0.742069	−	0.670324i	$-0.766155\pi$
$827$	− 31884.0i	− 1.34065i	0.742069	−	0.670324i	$-0.233845\pi$
$828$	0	0
$829$	24442.0	1.02401	0.512006	−	0.858982i	$-0.328903\pi$
$829$	24442.0	1.02401	0.512006	+	0.858982i	$0.328903\pi$
$830$	0	0
$831$	−1398.00	−0.0583587
$832$	0	0
$833$	53118.0i	2.20940i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	432.000i	0.0178400i
$838$	0	0
$839$	−31944.0	−1.31446	−0.657228	−	0.753691i	$-0.728271\pi$
$839$	−31944.0	−1.31446	−0.657228	+	0.753691i	$0.728271\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	14562.0i	0.594949i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1120.00i	0.0454352i
$848$	0	0
$849$	−13548.0	−0.547663
$850$	0	0
$851$	−6528.00	−0.262958
$852$	0	0
$853$	− 17486.0i	− 0.701887i	−0.936397	−	0.350943i	$-0.885861\pi$
$853$	− 17486.0i	− 0.701887i	0.936397	−	0.350943i	$-0.114139\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43434.0i	1.73125i	0.500697	+	0.865623i	$0.333077\pi$
$857$	43434.0i	1.73125i	−0.500697	+	0.865623i	$0.666923\pi$
$858$	0	0
$859$	10820.0	0.429771	0.214886	−	0.976639i	$-0.431062\pi$
$859$	10820.0	0.429771	0.214886	+	0.976639i	$0.431062\pi$
$860$	0	0
$861$	37440.0	1.48194
$862$	0	0
$863$	29976.0i	1.18238i	0.806532	+	0.591191i	$0.201342\pi$
$863$	29976.0i	1.18238i	−0.806532	+	0.591191i	$0.798658\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3513.00i	0.137610i
$868$	0	0
$869$	42048.0	1.64140
$870$	0	0
$871$	−8920.00	−0.347007
$872$	0	0
$873$	− 1746.00i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 40522.0i	− 1.56024i	−0.625630	−	0.780120i	$-0.715158\pi$
$877$	− 40522.0i	− 1.56024i	0.625630	−	0.780120i	$-0.284842\pi$
$878$	0	0
$879$	−25722.0	−0.987010
$880$	0	0
$881$	15570.0	0.595422	0.297711	−	0.954656i	$-0.403777\pi$
$881$	15570.0	0.595422	0.297711	+	0.954656i	$0.403777\pi$
$882$	0	0
$883$	− 1084.00i	− 0.0413131i	−0.999787	−	0.0206566i	$-0.993424\pi$
$883$	− 1084.00i	− 0.0413131i	0.999787	−	0.0206566i	$-0.00657566\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8208.00i	0.310708i	0.987859	+	0.155354i	$0.0496518\pi$
$887$	8208.00i	0.310708i	−0.987859	+	0.155354i	$0.950348\pi$
$888$	0	0
$889$	48896.0	1.84468
$890$	0	0
$891$	−2916.00	−0.109640
$892$	0	0
$893$	− 57120.0i	− 2.14048i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5760.00i	− 0.214404i
$898$	0	0
$899$	−96.0000	−0.00356149
$900$	0	0
$901$	8892.00	0.328785
$902$	0	0
$903$	4992.00i	0.183968i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 34076.0i	− 1.24749i	−0.781627	−	0.623746i	$-0.785610\pi$
$907$	− 34076.0i	− 1.24749i	0.781627	−	0.623746i	$-0.214390\pi$
$908$	0	0
$909$	−7182.00	−0.262059
$910$	0	0
$911$	15072.0	0.548142	0.274071	−	0.961709i	$-0.411630\pi$
$911$	15072.0	0.548142	0.274071	+	0.961709i	$0.411630\pi$
$912$	0	0
$913$	26352.0i	0.955229i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	77184.0i	2.77954i
$918$	0	0
$919$	24392.0	0.875536	0.437768	−	0.899088i	$-0.355769\pi$
$919$	24392.0	0.875536	0.437768	+	0.899088i	$0.355769\pi$
$920$	0	0
$921$	−10428.0	−0.373088
$922$	0	0
$923$	1200.00i	0.0427936i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2448.00i	− 0.0867345i
$928$	0	0
$929$	−13602.0	−0.480374	−0.240187	−	0.970727i	$-0.577209\pi$
$929$	−13602.0	−0.480374	−0.240187	+	0.970727i	$0.577209\pi$
$930$	0	0
$931$	−95340.0	−3.35622
$932$	0	0
$933$	7272.00i	0.255171i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 47974.0i	− 1.67262i	−0.548259	−	0.836309i	$-0.684709\pi$
$937$	− 47974.0i	− 1.67262i	0.548259	−	0.836309i	$-0.315291\pi$
$938$	0	0
$939$	4674.00	0.162439
$940$	0	0
$941$	−48330.0	−1.67430	−0.837148	−	0.546976i	$-0.815779\pi$
$941$	−48330.0	−1.67430	−0.837148	+	0.546976i	$0.815779\pi$
$942$	0	0
$943$	74880.0i	2.58582i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9324.00i	0.319946i	0.987121	+	0.159973i	$0.0511408\pi$
$947$	9324.00i	0.319946i	−0.987121	+	0.159973i	$0.948859\pi$
$948$	0	0
$949$	−6460.00	−0.220970
$950$	0	0
$951$	−25614.0	−0.873387
$952$	0	0
$953$	14838.0i	0.504355i	0.967681	+	0.252177i	$0.0811466\pi$
$953$	14838.0i	0.504355i	−0.967681	+	0.252177i	$0.918853\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 648.000i	− 0.0218881i
$958$	0	0
$959$	67392.0	2.26924
$960$	0	0
$961$	−29535.0	−0.991407
$962$	0	0
$963$	1404.00i	0.0469816i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 11360.0i	− 0.377780i	−0.981998	−	0.188890i	$-0.939511\pi$
$967$	− 11360.0i	− 0.377780i	0.981998	−	0.188890i	$-0.0604889\pi$
$968$	0	0
$969$	−32760.0	−1.08607
$970$	0	0
$971$	6972.00	0.230424	0.115212	−	0.993341i	$-0.463245\pi$
$971$	6972.00	0.230424	0.115212	+	0.993341i	$0.463245\pi$
$972$	0	0
$973$	17792.0i	0.586213i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 41166.0i	− 1.34802i	−0.738722	−	0.674011i	$-0.764570\pi$
$977$	− 41166.0i	− 1.34802i	0.738722	−	0.674011i	$-0.235430\pi$
$978$	0	0
$979$	−57240.0	−1.86864
$980$	0	0
$981$	14598.0	0.475105
$982$	0	0
$983$	22464.0i	0.728881i	0.931227	+	0.364441i	$0.118740\pi$
$983$	22464.0i	0.728881i	−0.931227	+	0.364441i	$0.881260\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	39168.0i	1.26315i
$988$	0	0
$989$	−9984.00	−0.321004
$990$	0	0
$991$	10192.0	0.326700	0.163350	−	0.986568i	$-0.447770\pi$
$991$	10192.0	0.326700	0.163350	+	0.986568i	$0.447770\pi$
$992$	0	0
$993$	− 2964.00i	− 0.0947228i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 322.000i	− 0.0102285i	−0.999987	−	0.00511426i	$-0.998372\pi$
$997$	− 322.000i	− 0.0102285i	0.999987	−	0.00511426i	$-0.00162793\pi$
$998$	0	0
$999$	918.000	0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.e.49.1		2
4.3	odd	2		300.4.d.d.49.2		2
5.2	odd	4		1200.4.a.s.1.1		1
5.3	odd	4		240.4.a.j.1.1		1
5.4	even	2	inner	1200.4.f.e.49.2		2
12.11	even	2		900.4.d.b.649.2		2
15.8	even	4		720.4.a.c.1.1		1
20.3	even	4		60.4.a.b.1.1	✓	1
20.7	even	4		300.4.a.e.1.1		1
20.19	odd	2		300.4.d.d.49.1		2
40.3	even	4		960.4.a.bb.1.1		1
40.13	odd	4		960.4.a.a.1.1		1
60.23	odd	4		180.4.a.c.1.1		1
60.47	odd	4		900.4.a.b.1.1		1
60.59	even	2		900.4.d.b.649.1		2
180.23	odd	12		1620.4.i.g.541.1		2
180.43	even	12		1620.4.i.a.1081.1		2
180.83	odd	12		1620.4.i.g.1081.1		2
180.103	even	12		1620.4.i.a.541.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.a.b.1.1	✓	1	20.3	even	4
180.4.a.c.1.1		1	60.23	odd	4
240.4.a.j.1.1		1	5.3	odd	4
300.4.a.e.1.1		1	20.7	even	4
300.4.d.d.49.1		2	20.19	odd	2
300.4.d.d.49.2		2	4.3	odd	2
720.4.a.c.1.1		1	15.8	even	4
900.4.a.b.1.1		1	60.47	odd	4
900.4.d.b.649.1		2	60.59	even	2
900.4.d.b.649.2		2	12.11	even	2
960.4.a.a.1.1		1	40.13	odd	4
960.4.a.bb.1.1		1	40.3	even	4
1200.4.a.s.1.1		1	5.2	odd	4
1200.4.f.e.49.1		2	1.1	even	1	trivial
1200.4.f.e.49.2		2	5.4	even	2	inner
1620.4.i.a.541.1		2	180.103	even	12
1620.4.i.a.1081.1		2	180.43	even	12
1620.4.i.g.541.1		2	180.23	odd	12
1620.4.i.g.1081.1		2	180.83	odd	12

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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -32.0000i q^{7} -9.00000 q^{9} -36.0000 q^{11} +10.0000i q^{13} -78.0000i q^{17} +140.000 q^{19} -96.0000 q^{21} -192.000i q^{23} +27.0000i q^{27} -6.00000 q^{29} +16.0000 q^{31} +108.000i q^{33} -34.0000i q^{37} +30.0000 q^{39} -390.000 q^{41} -52.0000i q^{43} -408.000i q^{47} -681.000 q^{49} -234.000 q^{51} +114.000i q^{53} -420.000i q^{57} +516.000 q^{59} -58.0000 q^{61} +288.000i q^{63} +892.000i q^{67} -576.000 q^{69} +120.000 q^{71} +646.000i q^{73} +1152.00i q^{77} -1168.00 q^{79} +81.0000 q^{81} -732.000i q^{83} +18.0000i q^{87} +1590.00 q^{89} +320.000 q^{91} -48.0000i q^{93} +194.000i q^{97} +324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} - 72 q^{11} + 280 q^{19} - 192 q^{21} - 12 q^{29} + 32 q^{31} + 60 q^{39} - 780 q^{41} - 1362 q^{49} - 468 q^{51} + 1032 q^{59} - 116 q^{61} - 1152 q^{69} + 240 q^{71} - 2336 q^{79} + 162 q^{81}+ \cdots + 648 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 72 * q^11 + 280 * q^19 - 192 * q^21 - 12 * q^29 + 32 * q^31 + 60 * q^39 - 780 * q^41 - 1362 * q^49 - 468 * q^51 + 1032 * q^59 - 116 * q^61 - 1152 * q^69 + 240 * q^71 - 2336 * q^79 + 162 * q^81 + 3180 * q^89 + 640 * q^91 + 648 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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