LMFDB - Embedded newform 225.4.b.d.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.b (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:	$13.2754297513$
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 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.4.b.d.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{2} -1.00000 q^{4} -20.0000i q^{7} +21.0000i q^{8} +O(q^{10})$	$q+3.00000i q^{2} -1.00000 q^{4} -20.0000i q^{7} +21.0000i q^{8} +24.0000 q^{11} +74.0000i q^{13} +60.0000 q^{14} -71.0000 q^{16} +54.0000i q^{17} +124.000 q^{19} +72.0000i q^{22} +120.000i q^{23} -222.000 q^{26} +20.0000i q^{28} -78.0000 q^{29} +200.000 q^{31} -45.0000i q^{32} -162.000 q^{34} +70.0000i q^{37} +372.000i q^{38} -330.000 q^{41} +92.0000i q^{43} -24.0000 q^{44} -360.000 q^{46} -24.0000i q^{47} -57.0000 q^{49} -74.0000i q^{52} -450.000i q^{53} +420.000 q^{56} -234.000i q^{58} +24.0000 q^{59} -322.000 q^{61} +600.000i q^{62} -433.000 q^{64} +196.000i q^{67} -54.0000i q^{68} +288.000 q^{71} -430.000i q^{73} -210.000 q^{74} -124.000 q^{76} -480.000i q^{77} +520.000 q^{79} -990.000i q^{82} -156.000i q^{83} -276.000 q^{86} +504.000i q^{88} +1026.00 q^{89} +1480.00 q^{91} -120.000i q^{92} +72.0000 q^{94} +286.000i q^{97} -171.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 48 q^{11} + 120 q^{14} - 142 q^{16} + 248 q^{19} - 444 q^{26} - 156 q^{29} + 400 q^{31} - 324 q^{34} - 660 q^{41} - 48 q^{44} - 720 q^{46} - 114 q^{49} + 840 q^{56} + 48 q^{59} - 644 q^{61} - 866 q^{64} + 576 q^{71} - 420 q^{74} - 248 q^{76} + 1040 q^{79} - 552 q^{86} + 2052 q^{89} + 2960 q^{91} + 144 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 48 * q^11 + 120 * q^14 - 142 * q^16 + 248 * q^19 - 444 * q^26 - 156 * q^29 + 400 * q^31 - 324 * q^34 - 660 * q^41 - 48 * q^44 - 720 * q^46 - 114 * q^49 + 840 * q^56 + 48 * q^59 - 644 * q^61 - 866 * q^64 + 576 * q^71 - 420 * q^74 - 248 * q^76 + 1040 * q^79 - 552 * q^86 + 2052 * q^89 + 2960 * q^91 + 144 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

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

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 $

$2$	3.00000i	1.06066i	0.847791	+	0.530330i	$0.177932\pi$
$2$	3.00000i	1.06066i	−0.847791	+	0.530330i	$0.822068\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.125000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 20.0000i	− 1.07990i	−0.841698	−	0.539949i	$-0.818443\pi$
$7$	− 20.0000i	− 1.07990i	0.841698	−	0.539949i	$-0.181557\pi$
$8$	21.0000i	0.928078i
$9$	0	0
$10$	0	0
$11$	24.0000	0.657843	0.328921	−	0.944357i	$-0.393315\pi$
$11$	24.0000	0.657843	0.328921	+	0.944357i	$0.393315\pi$
$12$	0	0
$13$	74.0000i	1.57876i	0.613904	+	0.789381i	$0.289598\pi$
$13$	74.0000i	1.57876i	−0.613904	+	0.789381i	$0.710402\pi$
$14$	60.0000	1.14541
$15$	0	0
$16$	−71.0000	−1.10938
$17$	54.0000i	0.770407i	0.922832	+	0.385204i	$0.125869\pi$
$17$	54.0000i	0.770407i	−0.922832	+	0.385204i	$0.874131\pi$
$18$	0	0
$19$	124.000	1.49724	0.748620	−	0.663000i	$-0.230717\pi$
$19$	124.000	1.49724	0.748620	+	0.663000i	$0.230717\pi$
$20$	0	0
$21$	0	0
$22$	72.0000i	0.697748i
$23$	120.000i	1.08790i	0.839117	+	0.543951i	$0.183072\pi$
$23$	120.000i	1.08790i	−0.839117	+	0.543951i	$0.816928\pi$
$24$	0	0
$25$	0	0
$26$	−222.000	−1.67453
$27$	0	0
$28$	20.0000i	0.134987i
$29$	−78.0000	−0.499456	−0.249728	−	0.968316i	$-0.580341\pi$
$29$	−78.0000	−0.499456	−0.249728	+	0.968316i	$0.580341\pi$
$30$	0	0
$31$	200.000	1.15874	0.579372	−	0.815063i	$-0.303298\pi$
$31$	200.000	1.15874	0.579372	+	0.815063i	$0.303298\pi$
$32$	− 45.0000i	− 0.248592i
$33$	0	0
$34$	−162.000	−0.817140
$35$	0	0
$36$	0	0
$37$	70.0000i	0.311025i	0.987834	+	0.155513i	$0.0497029\pi$
$37$	70.0000i	0.311025i	−0.987834	+	0.155513i	$0.950297\pi$
$38$	372.000i	1.58806i
$39$	0	0
$40$	0	0
$41$	−330.000	−1.25701	−0.628504	−	0.777806i	$-0.716332\pi$
$41$	−330.000	−1.25701	−0.628504	+	0.777806i	$0.716332\pi$
$42$	0	0
$43$	92.0000i	0.326276i	0.986603	+	0.163138i	$0.0521616\pi$
$43$	92.0000i	0.326276i	−0.986603	+	0.163138i	$0.947838\pi$
$44$	−24.0000	−0.0822304
$45$	0	0
$46$	−360.000	−1.15389
$47$	− 24.0000i	− 0.0744843i	−0.999306	−	0.0372421i	$-0.988143\pi$
$47$	− 24.0000i	− 0.0744843i	0.999306	−	0.0372421i	$-0.0118573\pi$
$48$	0	0
$49$	−57.0000	−0.166181
$50$	0	0
$51$	0	0
$52$	− 74.0000i	− 0.197345i
$53$	− 450.000i	− 1.16627i	−0.812376	−	0.583134i	$-0.801826\pi$
$53$	− 450.000i	− 1.16627i	0.812376	−	0.583134i	$-0.198174\pi$
$54$	0	0
$55$	0	0
$56$	420.000	1.00223
$57$	0	0
$58$	− 234.000i	− 0.529754i
$59$	24.0000	0.0529582	0.0264791	−	0.999649i	$-0.491570\pi$
$59$	24.0000	0.0529582	0.0264791	+	0.999649i	$0.491570\pi$
$60$	0	0
$61$	−322.000	−0.675867	−0.337933	−	0.941170i	$-0.609728\pi$
$61$	−322.000	−0.675867	−0.337933	+	0.941170i	$0.609728\pi$
$62$	600.000i	1.22903i
$63$	0	0
$64$	−433.000	−0.845703
$65$	0	0
$66$	0	0
$67$	196.000i	0.357391i	0.983904	+	0.178696i	$0.0571877\pi$
$67$	196.000i	0.357391i	−0.983904	+	0.178696i	$0.942812\pi$
$68$	− 54.0000i	− 0.0963009i
$69$	0	0
$70$	0	0
$71$	288.000	0.481399	0.240699	−	0.970600i	$-0.422623\pi$
$71$	288.000	0.481399	0.240699	+	0.970600i	$0.422623\pi$
$72$	0	0
$73$	− 430.000i	− 0.689420i	−0.938709	−	0.344710i	$-0.887977\pi$
$73$	− 430.000i	− 0.689420i	0.938709	−	0.344710i	$-0.112023\pi$
$74$	−210.000	−0.329892
$75$	0	0
$76$	−124.000	−0.187155
$77$	− 480.000i	− 0.710404i
$78$	0	0
$79$	520.000	0.740564	0.370282	−	0.928919i	$-0.379261\pi$
$79$	520.000	0.740564	0.370282	+	0.928919i	$0.379261\pi$
$80$	0	0
$81$	0	0
$82$	− 990.000i	− 1.33326i
$83$	− 156.000i	− 0.206304i	−0.994666	−	0.103152i	$-0.967107\pi$
$83$	− 156.000i	− 0.206304i	0.994666	−	0.103152i	$-0.0328928\pi$
$84$	0	0
$85$	0	0
$86$	−276.000	−0.346068
$87$	0	0
$88$	504.000i	0.610529i
$89$	1026.00	1.22198	0.610988	−	0.791640i	$-0.290773\pi$
$89$	1026.00	1.22198	0.610988	+	0.791640i	$0.290773\pi$
$90$	0	0
$91$	1480.00	1.70490
$92$	− 120.000i	− 0.135988i
$93$	0	0
$94$	72.0000	0.0790025
$95$	0	0
$96$	0	0
$97$	286.000i	0.299370i	0.988734	+	0.149685i	$0.0478260\pi$
$97$	286.000i	0.299370i	−0.988734	+	0.149685i	$0.952174\pi$
$98$	− 171.000i	− 0.176261i
$99$	0	0
$100$	0	0
$101$	1734.00	1.70831	0.854156	−	0.520017i	$-0.174075\pi$
$101$	1734.00	1.70831	0.854156	+	0.520017i	$0.174075\pi$
$102$	0	0
$103$	452.000i	0.432397i	0.976349	+	0.216198i	$0.0693658\pi$
$103$	452.000i	0.432397i	−0.976349	+	0.216198i	$0.930634\pi$
$104$	−1554.00	−1.46521
$105$	0	0
$106$	1350.00	1.23702
$107$	− 1404.00i	− 1.26850i	−0.773127	−	0.634251i	$-0.781308\pi$
$107$	− 1404.00i	− 1.26850i	0.773127	−	0.634251i	$-0.218692\pi$
$108$	0	0
$109$	1474.00	1.29526	0.647631	−	0.761954i	$-0.275760\pi$
$109$	1474.00	1.29526	0.647631	+	0.761954i	$0.275760\pi$
$110$	0	0
$111$	0	0
$112$	1420.00i	1.19801i
$113$	− 1086.00i	− 0.904091i	−0.891995	−	0.452046i	$-0.850694\pi$
$113$	− 1086.00i	− 0.904091i	0.891995	−	0.452046i	$-0.149306\pi$
$114$	0	0
$115$	0	0
$116$	78.0000	0.0624321
$117$	0	0
$118$	72.0000i	0.0561707i
$119$	1080.00	0.831962
$120$	0	0
$121$	−755.000	−0.567243
$122$	− 966.000i	− 0.716865i
$123$	0	0
$124$	−200.000	−0.144843
$125$	0	0
$126$	0	0
$127$	− 1244.00i	− 0.869190i	−0.900626	−	0.434595i	$-0.856891\pi$
$127$	− 1244.00i	− 0.869190i	0.900626	−	0.434595i	$-0.143109\pi$
$128$	− 1659.00i	− 1.14560i
$129$	0	0
$130$	0	0
$131$	−2328.00	−1.55266	−0.776329	−	0.630327i	$-0.782921\pi$
$131$	−2328.00	−1.55266	−0.776329	+	0.630327i	$0.782921\pi$
$132$	0	0
$133$	− 2480.00i	− 1.61687i
$134$	−588.000	−0.379071
$135$	0	0
$136$	−1134.00	−0.714998
$137$	2118.00i	1.32082i	0.750903	+	0.660412i	$0.229618\pi$
$137$	2118.00i	1.32082i	−0.750903	+	0.660412i	$0.770382\pi$
$138$	0	0
$139$	−2324.00	−1.41812	−0.709062	−	0.705147i	$-0.750881\pi$
$139$	−2324.00	−1.41812	−0.709062	+	0.705147i	$0.750881\pi$
$140$	0	0
$141$	0	0
$142$	864.000i	0.510600i
$143$	1776.00i	1.03858i
$144$	0	0
$145$	0	0
$146$	1290.00	0.731241
$147$	0	0
$148$	− 70.0000i	− 0.0388781i
$149$	258.000	0.141854	0.0709268	−	0.997482i	$-0.477404\pi$
$149$	258.000	0.141854	0.0709268	+	0.997482i	$0.477404\pi$
$150$	0	0
$151$	−808.000	−0.435458	−0.217729	−	0.976009i	$-0.569865\pi$
$151$	−808.000	−0.435458	−0.217729	+	0.976009i	$0.569865\pi$
$152$	2604.00i	1.38955i
$153$	0	0
$154$	1440.00	0.753497
$155$	0	0
$156$	0	0
$157$	− 2378.00i	− 1.20882i	−0.796673	−	0.604411i	$-0.793408\pi$
$157$	− 2378.00i	− 1.20882i	0.796673	−	0.604411i	$-0.206592\pi$
$158$	1560.00i	0.785487i
$159$	0	0
$160$	0	0
$161$	2400.00	1.17482
$162$	0	0
$163$	− 52.0000i	− 0.0249874i	−0.999922	−	0.0124937i	$-0.996023\pi$
$163$	− 52.0000i	− 0.0249874i	0.999922	−	0.0124937i	$-0.00397698\pi$
$164$	330.000	0.157126
$165$	0	0
$166$	468.000	0.218818
$167$	− 3720.00i	− 1.72373i	−0.507141	−	0.861863i	$-0.669298\pi$
$167$	− 3720.00i	− 1.72373i	0.507141	−	0.861863i	$-0.330702\pi$
$168$	0	0
$169$	−3279.00	−1.49249
$170$	0	0
$171$	0	0
$172$	− 92.0000i	− 0.0407845i
$173$	− 426.000i	− 0.187215i	−0.995609	−	0.0936075i	$-0.970160\pi$
$173$	− 426.000i	− 0.187215i	0.995609	−	0.0936075i	$-0.0298399\pi$
$174$	0	0
$175$	0	0
$176$	−1704.00	−0.729795
$177$	0	0
$178$	3078.00i	1.29610i
$179$	−1440.00	−0.601289	−0.300644	−	0.953736i	$-0.597202\pi$
$179$	−1440.00	−0.601289	−0.300644	+	0.953736i	$0.597202\pi$
$180$	0	0
$181$	−3130.00	−1.28537	−0.642683	−	0.766133i	$-0.722179\pi$
$181$	−3130.00	−1.28537	−0.642683	+	0.766133i	$0.722179\pi$
$182$	4440.00i	1.80832i
$183$	0	0
$184$	−2520.00	−1.00966
$185$	0	0
$186$	0	0
$187$	1296.00i	0.506807i
$188$	24.0000i	0.00931053i
$189$	0	0
$190$	0	0
$191$	−3576.00	−1.35471	−0.677357	−	0.735655i	$-0.736875\pi$
$191$	−3576.00	−1.35471	−0.677357	+	0.735655i	$0.736875\pi$
$192$	0	0
$193$	2666.00i	0.994315i	0.867660	+	0.497158i	$0.165623\pi$
$193$	2666.00i	0.994315i	−0.867660	+	0.497158i	$0.834377\pi$
$194$	−858.000	−0.317530
$195$	0	0
$196$	57.0000	0.0207726
$197$	− 2718.00i	− 0.982992i	−0.870880	−	0.491496i	$-0.836450\pi$
$197$	− 2718.00i	− 0.982992i	0.870880	−	0.491496i	$-0.163550\pi$
$198$	0	0
$199$	3832.00	1.36504	0.682521	−	0.730866i	$-0.260884\pi$
$199$	3832.00	1.36504	0.682521	+	0.730866i	$0.260884\pi$
$200$	0	0
$201$	0	0
$202$	5202.00i	1.81194i
$203$	1560.00i	0.539362i
$204$	0	0
$205$	0	0
$206$	−1356.00	−0.458626
$207$	0	0
$208$	− 5254.00i	− 1.75144i
$209$	2976.00	0.984948
$210$	0	0
$211$	1100.00	0.358896	0.179448	−	0.983767i	$-0.442569\pi$
$211$	1100.00	0.358896	0.179448	+	0.983767i	$0.442569\pi$
$212$	450.000i	0.145784i
$213$	0	0
$214$	4212.00	1.34545
$215$	0	0
$216$	0	0
$217$	− 4000.00i	− 1.25133i
$218$	4422.00i	1.37383i
$219$	0	0
$220$	0	0
$221$	−3996.00	−1.21629
$222$	0	0
$223$	1964.00i	0.589772i	0.955532	+	0.294886i	$0.0952817\pi$
$223$	1964.00i	0.589772i	−0.955532	+	0.294886i	$0.904718\pi$
$224$	−900.000	−0.268454
$225$	0	0
$226$	3258.00	0.958933
$227$	660.000i	0.192977i	0.995334	+	0.0964884i	$0.0307611\pi$
$227$	660.000i	0.192977i	−0.995334	+	0.0964884i	$0.969239\pi$
$228$	0	0
$229$	1906.00	0.550009	0.275004	−	0.961443i	$-0.411321\pi$
$229$	1906.00	0.550009	0.275004	+	0.961443i	$0.411321\pi$
$230$	0	0
$231$	0	0
$232$	− 1638.00i	− 0.463534i
$233$	1458.00i	0.409943i	0.978768	+	0.204972i	$0.0657102\pi$
$233$	1458.00i	0.409943i	−0.978768	+	0.204972i	$0.934290\pi$
$234$	0	0
$235$	0	0
$236$	−24.0000	−0.00661978
$237$	0	0
$238$	3240.00i	0.882429i
$239$	1176.00	0.318281	0.159140	−	0.987256i	$-0.449128\pi$
$239$	1176.00	0.318281	0.159140	+	0.987256i	$0.449128\pi$
$240$	0	0
$241$	866.000	0.231469	0.115734	−	0.993280i	$-0.463078\pi$
$241$	866.000	0.231469	0.115734	+	0.993280i	$0.463078\pi$
$242$	− 2265.00i	− 0.601652i
$243$	0	0
$244$	322.000	0.0844834
$245$	0	0
$246$	0	0
$247$	9176.00i	2.36379i
$248$	4200.00i	1.07540i
$249$	0	0
$250$	0	0
$251$	−432.000	−0.108636	−0.0543179	−	0.998524i	$-0.517298\pi$
$251$	−432.000	−0.108636	−0.0543179	+	0.998524i	$0.517298\pi$
$252$	0	0
$253$	2880.00i	0.715668i
$254$	3732.00	0.921915
$255$	0	0
$256$	1513.00	0.369385
$257$	2526.00i	0.613103i	0.951854	+	0.306552i	$0.0991752\pi$
$257$	2526.00i	0.613103i	−0.951854	+	0.306552i	$0.900825\pi$
$258$	0	0
$259$	1400.00	0.335876
$260$	0	0
$261$	0	0
$262$	− 6984.00i	− 1.64684i
$263$	− 5448.00i	− 1.27733i	−0.769484	−	0.638666i	$-0.779487\pi$
$263$	− 5448.00i	− 1.27733i	0.769484	−	0.638666i	$-0.220513\pi$
$264$	0	0
$265$	0	0
$266$	7440.00	1.71495
$267$	0	0
$268$	− 196.000i	− 0.0446739i
$269$	−2574.00	−0.583418	−0.291709	−	0.956507i	$-0.594224\pi$
$269$	−2574.00	−0.583418	−0.291709	+	0.956507i	$0.594224\pi$
$270$	0	0
$271$	−3184.00	−0.713706	−0.356853	−	0.934161i	$-0.616150\pi$
$271$	−3184.00	−0.713706	−0.356853	+	0.934161i	$0.616150\pi$
$272$	− 3834.00i	− 0.854671i
$273$	0	0
$274$	−6354.00	−1.40095
$275$	0	0
$276$	0	0
$277$	− 3962.00i	− 0.859399i	−0.902972	−	0.429699i	$-0.858620\pi$
$277$	− 3962.00i	− 0.859399i	0.902972	−	0.429699i	$-0.141380\pi$
$278$	− 6972.00i	− 1.50415i
$279$	0	0
$280$	0	0
$281$	8286.00	1.75908	0.879540	−	0.475825i	$-0.157851\pi$
$281$	8286.00	1.75908	0.879540	+	0.475825i	$0.157851\pi$
$282$	0	0
$283$	− 2716.00i	− 0.570493i	−0.958454	−	0.285246i	$-0.907925\pi$
$283$	− 2716.00i	− 0.570493i	0.958454	−	0.285246i	$-0.0920754\pi$
$284$	−288.000	−0.0601748
$285$	0	0
$286$	−5328.00	−1.10158
$287$	6600.00i	1.35744i
$288$	0	0
$289$	1997.00	0.406473
$290$	0	0
$291$	0	0
$292$	430.000i	0.0861776i
$293$	− 6018.00i	− 1.19992i	−0.800032	−	0.599958i	$-0.795184\pi$
$293$	− 6018.00i	− 1.19992i	0.800032	−	0.599958i	$-0.204816\pi$
$294$	0	0
$295$	0	0
$296$	−1470.00	−0.288655
$297$	0	0
$298$	774.000i	0.150458i
$299$	−8880.00	−1.71754
$300$	0	0
$301$	1840.00	0.352345
$302$	− 2424.00i	− 0.461873i
$303$	0	0
$304$	−8804.00	−1.66100
$305$	0	0
$306$	0	0
$307$	− 9236.00i	− 1.71702i	−0.512793	−	0.858512i	$-0.671389\pi$
$307$	− 9236.00i	− 1.71702i	0.512793	−	0.858512i	$-0.328611\pi$
$308$	480.000i	0.0888004i
$309$	0	0
$310$	0	0
$311$	−1536.00	−0.280060	−0.140030	−	0.990147i	$-0.544720\pi$
$311$	−1536.00	−0.280060	−0.140030	+	0.990147i	$0.544720\pi$
$312$	0	0
$313$	− 7342.00i	− 1.32586i	−0.748681	−	0.662930i	$-0.769313\pi$
$313$	− 7342.00i	− 1.32586i	0.748681	−	0.662930i	$-0.230687\pi$
$314$	7134.00	1.28215
$315$	0	0
$316$	−520.000	−0.0925705
$317$	− 3894.00i	− 0.689933i	−0.938615	−	0.344967i	$-0.887890\pi$
$317$	− 3894.00i	− 0.689933i	0.938615	−	0.344967i	$-0.112110\pi$
$318$	0	0
$319$	−1872.00	−0.328564
$320$	0	0
$321$	0	0
$322$	7200.00i	1.24609i
$323$	6696.00i	1.15348i
$324$	0	0
$325$	0	0
$326$	156.000	0.0265032
$327$	0	0
$328$	− 6930.00i	− 1.16660i
$329$	−480.000	−0.0804354
$330$	0	0
$331$	3692.00	0.613084	0.306542	−	0.951857i	$-0.400828\pi$
$331$	3692.00	0.613084	0.306542	+	0.951857i	$0.400828\pi$
$332$	156.000i	0.0257880i
$333$	0	0
$334$	11160.0	1.82829
$335$	0	0
$336$	0	0
$337$	8998.00i	1.45446i	0.686395	+	0.727229i	$0.259192\pi$
$337$	8998.00i	1.45446i	−0.686395	+	0.727229i	$0.740808\pi$
$338$	− 9837.00i	− 1.58302i
$339$	0	0
$340$	0	0
$341$	4800.00	0.762271
$342$	0	0
$343$	− 5720.00i	− 0.900440i
$344$	−1932.00	−0.302809
$345$	0	0
$346$	1278.00	0.198571
$347$	5244.00i	0.811276i	0.914034	+	0.405638i	$0.132951\pi$
$347$	5244.00i	0.811276i	−0.914034	+	0.405638i	$0.867049\pi$
$348$	0	0
$349$	−6302.00	−0.966585	−0.483293	−	0.875459i	$-0.660559\pi$
$349$	−6302.00	−0.966585	−0.483293	+	0.875459i	$0.660559\pi$
$350$	0	0
$351$	0	0
$352$	− 1080.00i	− 0.163535i
$353$	− 3414.00i	− 0.514756i	−0.966311	−	0.257378i	$-0.917141\pi$
$353$	− 3414.00i	− 0.514756i	0.966311	−	0.257378i	$-0.0828586\pi$
$354$	0	0
$355$	0	0
$356$	−1026.00	−0.152747
$357$	0	0
$358$	− 4320.00i	− 0.637763i
$359$	4824.00	0.709195	0.354597	−	0.935019i	$-0.384618\pi$
$359$	4824.00	0.709195	0.354597	+	0.935019i	$0.384618\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	− 9390.00i	− 1.36334i
$363$	0	0
$364$	−1480.00	−0.213113
$365$	0	0
$366$	0	0
$367$	3508.00i	0.498954i	0.968381	+	0.249477i	$0.0802587\pi$
$367$	3508.00i	0.498954i	−0.968381	+	0.249477i	$0.919741\pi$
$368$	− 8520.00i	− 1.20689i
$369$	0	0
$370$	0	0
$371$	−9000.00	−1.25945
$372$	0	0
$373$	10802.0i	1.49948i	0.661732	+	0.749740i	$0.269822\pi$
$373$	10802.0i	1.49948i	−0.661732	+	0.749740i	$0.730178\pi$
$374$	−3888.00	−0.537550
$375$	0	0
$376$	504.000	0.0691272
$377$	− 5772.00i	− 0.788523i
$378$	0	0
$379$	−1460.00	−0.197876	−0.0989382	−	0.995094i	$-0.531545\pi$
$379$	−1460.00	−0.197876	−0.0989382	+	0.995094i	$0.531545\pi$
$380$	0	0
$381$	0	0
$382$	− 10728.0i	− 1.43689i
$383$	4872.00i	0.649994i	0.945715	+	0.324997i	$0.105363\pi$
$383$	4872.00i	0.649994i	−0.945715	+	0.324997i	$0.894637\pi$
$384$	0	0
$385$	0	0
$386$	−7998.00	−1.05463
$387$	0	0
$388$	− 286.000i	− 0.0374213i
$389$	−14046.0	−1.83075	−0.915373	−	0.402606i	$-0.868104\pi$
$389$	−14046.0	−1.83075	−0.915373	+	0.402606i	$0.868104\pi$
$390$	0	0
$391$	−6480.00	−0.838127
$392$	− 1197.00i	− 0.154229i
$393$	0	0
$394$	8154.00	1.04262
$395$	0	0
$396$	0	0
$397$	2734.00i	0.345631i	0.984954	+	0.172816i	$0.0552864\pi$
$397$	2734.00i	0.345631i	−0.984954	+	0.172816i	$0.944714\pi$
$398$	11496.0i	1.44785i
$399$	0	0
$400$	0	0
$401$	15942.0	1.98530	0.992650	−	0.121019i	$-0.0386161\pi$
$401$	15942.0	1.98530	0.992650	+	0.121019i	$0.0386161\pi$
$402$	0	0
$403$	14800.0i	1.82938i
$404$	−1734.00	−0.213539
$405$	0	0
$406$	−4680.00	−0.572080
$407$	1680.00i	0.204606i
$408$	0	0
$409$	−8714.00	−1.05350	−0.526748	−	0.850022i	$-0.676589\pi$
$409$	−8714.00	−1.05350	−0.526748	+	0.850022i	$0.676589\pi$
$410$	0	0
$411$	0	0
$412$	− 452.000i	− 0.0540496i
$413$	− 480.000i	− 0.0571895i
$414$	0	0
$415$	0	0
$416$	3330.00	0.392468
$417$	0	0
$418$	8928.00i	1.04470i
$419$	11976.0	1.39634	0.698169	−	0.715933i	$-0.253998\pi$
$419$	11976.0	1.39634	0.698169	+	0.715933i	$0.253998\pi$
$420$	0	0
$421$	11054.0	1.27967	0.639833	−	0.768514i	$-0.279004\pi$
$421$	11054.0	1.27967	0.639833	+	0.768514i	$0.279004\pi$
$422$	3300.00i	0.380667i
$423$	0	0
$424$	9450.00	1.08239
$425$	0	0
$426$	0	0
$427$	6440.00i	0.729868i
$428$	1404.00i	0.158563i
$429$	0	0
$430$	0	0
$431$	−720.000	−0.0804668	−0.0402334	−	0.999190i	$-0.512810\pi$
$431$	−720.000	−0.0804668	−0.0402334	+	0.999190i	$0.512810\pi$
$432$	0	0
$433$	− 15622.0i	− 1.73382i	−0.498462	−	0.866912i	$-0.666102\pi$
$433$	− 15622.0i	− 1.73382i	0.498462	−	0.866912i	$-0.333898\pi$
$434$	12000.0	1.32723
$435$	0	0
$436$	−1474.00	−0.161908
$437$	14880.0i	1.62885i
$438$	0	0
$439$	9880.00	1.07414	0.537069	−	0.843538i	$-0.319531\pi$
$439$	9880.00	1.07414	0.537069	+	0.843538i	$0.319531\pi$
$440$	0	0
$441$	0	0
$442$	− 11988.0i	− 1.29007i
$443$	16116.0i	1.72843i	0.503123	+	0.864215i	$0.332184\pi$
$443$	16116.0i	1.72843i	−0.503123	+	0.864215i	$0.667816\pi$
$444$	0	0
$445$	0	0
$446$	−5892.00	−0.625548
$447$	0	0
$448$	8660.00i	0.913274i
$449$	9018.00	0.947852	0.473926	−	0.880565i	$-0.342836\pi$
$449$	9018.00	0.947852	0.473926	+	0.880565i	$0.342836\pi$
$450$	0	0
$451$	−7920.00	−0.826914
$452$	1086.00i	0.113011i
$453$	0	0
$454$	−1980.00	−0.204683
$455$	0	0
$456$	0	0
$457$	3670.00i	0.375657i	0.982202	+	0.187829i	$0.0601450\pi$
$457$	3670.00i	0.375657i	−0.982202	+	0.187829i	$0.939855\pi$
$458$	5718.00i	0.583372i
$459$	0	0
$460$	0	0
$461$	−17562.0	−1.77428	−0.887141	−	0.461499i	$-0.847312\pi$
$461$	−17562.0	−1.77428	−0.887141	+	0.461499i	$0.847312\pi$
$462$	0	0
$463$	1172.00i	0.117640i	0.998269	+	0.0588202i	$0.0187338\pi$
$463$	1172.00i	0.117640i	−0.998269	+	0.0588202i	$0.981266\pi$
$464$	5538.00	0.554084
$465$	0	0
$466$	−4374.00	−0.434810
$467$	6876.00i	0.681335i	0.940184	+	0.340667i	$0.110653\pi$
$467$	6876.00i	0.681335i	−0.940184	+	0.340667i	$0.889347\pi$
$468$	0	0
$469$	3920.00	0.385946
$470$	0	0
$471$	0	0
$472$	504.000i	0.0491493i
$473$	2208.00i	0.214638i
$474$	0	0
$475$	0	0
$476$	−1080.00	−0.103995
$477$	0	0
$478$	3528.00i	0.337588i
$479$	2280.00	0.217486	0.108743	−	0.994070i	$-0.465317\pi$
$479$	2280.00	0.217486	0.108743	+	0.994070i	$0.465317\pi$
$480$	0	0
$481$	−5180.00	−0.491035
$482$	2598.00i	0.245510i
$483$	0	0
$484$	755.000	0.0709053
$485$	0	0
$486$	0	0
$487$	3076.00i	0.286215i	0.989707	+	0.143108i	$0.0457095\pi$
$487$	3076.00i	0.286215i	−0.989707	+	0.143108i	$0.954290\pi$
$488$	− 6762.00i	− 0.627257i
$489$	0	0
$490$	0	0
$491$	18912.0	1.73826	0.869131	−	0.494582i	$-0.164679\pi$
$491$	18912.0	1.73826	0.869131	+	0.494582i	$0.164679\pi$
$492$	0	0
$493$	− 4212.00i	− 0.384785i
$494$	−27528.0	−2.50717
$495$	0	0
$496$	−14200.0	−1.28548
$497$	− 5760.00i	− 0.519862i
$498$	0	0
$499$	−9956.00	−0.893170	−0.446585	−	0.894741i	$-0.647360\pi$
$499$	−9956.00	−0.893170	−0.446585	+	0.894741i	$0.647360\pi$
$500$	0	0
$501$	0	0
$502$	− 1296.00i	− 0.115226i
$503$	10656.0i	0.944588i	0.881441	+	0.472294i	$0.156574\pi$
$503$	10656.0i	0.944588i	−0.881441	+	0.472294i	$0.843426\pi$
$504$	0	0
$505$	0	0
$506$	−8640.00	−0.759081
$507$	0	0
$508$	1244.00i	0.108649i
$509$	−2766.00	−0.240866	−0.120433	−	0.992721i	$-0.538428\pi$
$509$	−2766.00	−0.240866	−0.120433	+	0.992721i	$0.538428\pi$
$510$	0	0
$511$	−8600.00	−0.744504
$512$	− 8733.00i	− 0.753804i
$513$	0	0
$514$	−7578.00	−0.650294
$515$	0	0
$516$	0	0
$517$	− 576.000i	− 0.0489989i
$518$	4200.00i	0.356250i
$519$	0	0
$520$	0	0
$521$	−10530.0	−0.885466	−0.442733	−	0.896654i	$-0.645991\pi$
$521$	−10530.0	−0.885466	−0.442733	+	0.896654i	$0.645991\pi$
$522$	0	0
$523$	12692.0i	1.06115i	0.847637	+	0.530576i	$0.178024\pi$
$523$	12692.0i	1.06115i	−0.847637	+	0.530576i	$0.821976\pi$
$524$	2328.00	0.194082
$525$	0	0
$526$	16344.0	1.35481
$527$	10800.0i	0.892705i
$528$	0	0
$529$	−2233.00	−0.183529
$530$	0	0
$531$	0	0
$532$	2480.00i	0.202108i
$533$	− 24420.0i	− 1.98452i
$534$	0	0
$535$	0	0
$536$	−4116.00	−0.331687
$537$	0	0
$538$	− 7722.00i	− 0.618809i
$539$	−1368.00	−0.109321
$540$	0	0
$541$	18110.0	1.43920	0.719602	−	0.694386i	$-0.244324\pi$
$541$	18110.0	1.43920	0.719602	+	0.694386i	$0.244324\pi$
$542$	− 9552.00i	− 0.756999i
$543$	0	0
$544$	2430.00	0.191517
$545$	0	0
$546$	0	0
$547$	− 3620.00i	− 0.282962i	−0.989941	−	0.141481i	$-0.954814\pi$
$547$	− 3620.00i	− 0.282962i	0.989941	−	0.141481i	$-0.0451864\pi$
$548$	− 2118.00i	− 0.165103i
$549$	0	0
$550$	0	0
$551$	−9672.00	−0.747806
$552$	0	0
$553$	− 10400.0i	− 0.799734i
$554$	11886.0	0.911530
$555$	0	0
$556$	2324.00	0.177265
$557$	− 14166.0i	− 1.07762i	−0.842428	−	0.538809i	$-0.818875\pi$
$557$	− 14166.0i	− 1.07762i	0.842428	−	0.538809i	$-0.181125\pi$
$558$	0	0
$559$	−6808.00	−0.515112
$560$	0	0
$561$	0	0
$562$	24858.0i	1.86579i
$563$	13404.0i	1.00339i	0.865043	+	0.501697i	$0.167291\pi$
$563$	13404.0i	1.00339i	−0.865043	+	0.501697i	$0.832709\pi$
$564$	0	0
$565$	0	0
$566$	8148.00	0.605099
$567$	0	0
$568$	6048.00i	0.446775i
$569$	−18654.0	−1.37437	−0.687185	−	0.726483i	$-0.741154\pi$
$569$	−18654.0	−1.37437	−0.687185	+	0.726483i	$0.741154\pi$
$570$	0	0
$571$	−7684.00	−0.563162	−0.281581	−	0.959537i	$-0.590859\pi$
$571$	−7684.00	−0.563162	−0.281581	+	0.959537i	$0.590859\pi$
$572$	− 1776.00i	− 0.129822i
$573$	0	0
$574$	−19800.0	−1.43978
$575$	0	0
$576$	0	0
$577$	1726.00i	0.124531i	0.998060	+	0.0622654i	$0.0198325\pi$
$577$	1726.00i	0.124531i	−0.998060	+	0.0622654i	$0.980167\pi$
$578$	5991.00i	0.431129i
$579$	0	0
$580$	0	0
$581$	−3120.00	−0.222787
$582$	0	0
$583$	− 10800.0i	− 0.767222i
$584$	9030.00	0.639836
$585$	0	0
$586$	18054.0	1.27270
$587$	10596.0i	0.745049i	0.928022	+	0.372524i	$0.121508\pi$
$587$	10596.0i	0.745049i	−0.928022	+	0.372524i	$0.878492\pi$
$588$	0	0
$589$	24800.0	1.73492
$590$	0	0
$591$	0	0
$592$	− 4970.00i	− 0.345043i
$593$	− 2862.00i	− 0.198193i	−0.995078	−	0.0990963i	$-0.968405\pi$
$593$	− 2862.00i	− 0.198193i	0.995078	−	0.0990963i	$-0.0315952\pi$
$594$	0	0
$595$	0	0
$596$	−258.000	−0.0177317
$597$	0	0
$598$	− 26640.0i	− 1.82172i
$599$	−23592.0	−1.60925	−0.804627	−	0.593781i	$-0.797635\pi$
$599$	−23592.0	−1.60925	−0.804627	+	0.593781i	$0.797635\pi$
$600$	0	0
$601$	−9574.00	−0.649803	−0.324902	−	0.945748i	$-0.605331\pi$
$601$	−9574.00	−0.649803	−0.324902	+	0.945748i	$0.605331\pi$
$602$	5520.00i	0.373718i
$603$	0	0
$604$	808.000	0.0544322
$605$	0	0
$606$	0	0
$607$	− 17444.0i	− 1.16644i	−0.812314	−	0.583221i	$-0.801792\pi$
$607$	− 17444.0i	− 1.16644i	0.812314	−	0.583221i	$-0.198208\pi$
$608$	− 5580.00i	− 0.372202i
$609$	0	0
$610$	0	0
$611$	1776.00	0.117593
$612$	0	0
$613$	− 2374.00i	− 0.156419i	−0.996937	−	0.0782096i	$-0.975080\pi$
$613$	− 2374.00i	− 0.156419i	0.996937	−	0.0782096i	$-0.0249203\pi$
$614$	27708.0	1.82118
$615$	0	0
$616$	10080.0	0.659310
$617$	− 12162.0i	− 0.793555i	−0.917915	−	0.396778i	$-0.870128\pi$
$617$	− 12162.0i	− 0.793555i	0.917915	−	0.396778i	$-0.129872\pi$
$618$	0	0
$619$	−8804.00	−0.571668	−0.285834	−	0.958279i	$-0.592271\pi$
$619$	−8804.00	−0.571668	−0.285834	+	0.958279i	$0.592271\pi$
$620$	0	0
$621$	0	0
$622$	− 4608.00i	− 0.297048i
$623$	− 20520.0i	− 1.31961i
$624$	0	0
$625$	0	0
$626$	22026.0	1.40629
$627$	0	0
$628$	2378.00i	0.151103i
$629$	−3780.00	−0.239616
$630$	0	0
$631$	−12688.0	−0.800478	−0.400239	−	0.916411i	$-0.631073\pi$
$631$	−12688.0	−0.800478	−0.400239	+	0.916411i	$0.631073\pi$
$632$	10920.0i	0.687301i
$633$	0	0
$634$	11682.0	0.731785
$635$	0	0
$636$	0	0
$637$	− 4218.00i	− 0.262360i
$638$	− 5616.00i	− 0.348495i
$639$	0	0
$640$	0	0
$641$	9150.00	0.563812	0.281906	−	0.959442i	$-0.409033\pi$
$641$	9150.00	0.563812	0.281906	+	0.959442i	$0.409033\pi$
$642$	0	0
$643$	25292.0i	1.55120i	0.631227	+	0.775598i	$0.282552\pi$
$643$	25292.0i	1.55120i	−0.631227	+	0.775598i	$0.717448\pi$
$644$	−2400.00	−0.146853
$645$	0	0
$646$	−20088.0	−1.22345
$647$	− 2736.00i	− 0.166249i	−0.996539	−	0.0831246i	$-0.973510\pi$
$647$	− 2736.00i	− 0.166249i	0.996539	−	0.0831246i	$-0.0264900\pi$
$648$	0	0
$649$	576.000	0.0348382
$650$	0	0
$651$	0	0
$652$	52.0000i	0.00312343i
$653$	− 22218.0i	− 1.33148i	−0.746183	−	0.665741i	$-0.768116\pi$
$653$	− 22218.0i	− 1.33148i	0.746183	−	0.665741i	$-0.231884\pi$
$654$	0	0
$655$	0	0
$656$	23430.0	1.39449
$657$	0	0
$658$	− 1440.00i	− 0.0853147i
$659$	14520.0	0.858299	0.429149	−	0.903234i	$-0.358813\pi$
$659$	14520.0	0.858299	0.429149	+	0.903234i	$0.358813\pi$
$660$	0	0
$661$	−10618.0	−0.624799	−0.312400	−	0.949951i	$-0.601133\pi$
$661$	−10618.0	−0.624799	−0.312400	+	0.949951i	$0.601133\pi$
$662$	11076.0i	0.650273i
$663$	0	0
$664$	3276.00	0.191466
$665$	0	0
$666$	0	0
$667$	− 9360.00i	− 0.543359i
$668$	3720.00i	0.215466i
$669$	0	0
$670$	0	0
$671$	−7728.00	−0.444614
$672$	0	0
$673$	1370.00i	0.0784690i	0.999230	+	0.0392345i	$0.0124919\pi$
$673$	1370.00i	0.0784690i	−0.999230	+	0.0392345i	$0.987508\pi$
$674$	−26994.0	−1.54269
$675$	0	0
$676$	3279.00	0.186561
$677$	− 13758.0i	− 0.781038i	−0.920595	−	0.390519i	$-0.872296\pi$
$677$	− 13758.0i	− 0.781038i	0.920595	−	0.390519i	$-0.127704\pi$
$678$	0	0
$679$	5720.00	0.323289
$680$	0	0
$681$	0	0
$682$	14400.0i	0.808511i
$683$	− 11988.0i	− 0.671608i	−0.941932	−	0.335804i	$-0.890992\pi$
$683$	− 11988.0i	− 0.671608i	0.941932	−	0.335804i	$-0.109008\pi$
$684$	0	0
$685$	0	0
$686$	17160.0	0.955061
$687$	0	0
$688$	− 6532.00i	− 0.361962i
$689$	33300.0	1.84126
$690$	0	0
$691$	32996.0	1.81654	0.908268	−	0.418388i	$-0.137405\pi$
$691$	32996.0	1.81654	0.908268	+	0.418388i	$0.137405\pi$
$692$	426.000i	0.0234019i
$693$	0	0
$694$	−15732.0	−0.860488
$695$	0	0
$696$	0	0
$697$	− 17820.0i	− 0.968408i
$698$	− 18906.0i	− 1.02522i
$699$	0	0
$700$	0	0
$701$	25902.0	1.39558	0.697792	−	0.716300i	$-0.254166\pi$
$701$	25902.0	1.39558	0.697792	+	0.716300i	$0.254166\pi$
$702$	0	0
$703$	8680.00i	0.465679i
$704$	−10392.0	−0.556340
$705$	0	0
$706$	10242.0	0.545981
$707$	− 34680.0i	− 1.84480i
$708$	0	0
$709$	27394.0	1.45106	0.725531	−	0.688189i	$-0.241594\pi$
$709$	27394.0	1.45106	0.725531	+	0.688189i	$0.241594\pi$
$710$	0	0
$711$	0	0
$712$	21546.0i	1.13409i
$713$	24000.0i	1.26060i
$714$	0	0
$715$	0	0
$716$	1440.00	0.0751611
$717$	0	0
$718$	14472.0i	0.752215i
$719$	34848.0	1.80753	0.903763	−	0.428033i	$-0.140793\pi$
$719$	34848.0	1.80753	0.903763	+	0.428033i	$0.140793\pi$
$720$	0	0
$721$	9040.00	0.466945
$722$	25551.0i	1.31705i
$723$	0	0
$724$	3130.00	0.160671
$725$	0	0
$726$	0	0
$727$	− 28028.0i	− 1.42985i	−0.699201	−	0.714925i	$-0.746461\pi$
$727$	− 28028.0i	− 1.42985i	0.699201	−	0.714925i	$-0.253539\pi$
$728$	31080.0i	1.58228i
$729$	0	0
$730$	0	0
$731$	−4968.00	−0.251365
$732$	0	0
$733$	18002.0i	0.907120i	0.891226	+	0.453560i	$0.149846\pi$
$733$	18002.0i	0.907120i	−0.891226	+	0.453560i	$0.850154\pi$
$734$	−10524.0	−0.529221
$735$	0	0
$736$	5400.00	0.270444
$737$	4704.00i	0.235107i
$738$	0	0
$739$	−15284.0	−0.760800	−0.380400	−	0.924822i	$-0.624214\pi$
$739$	−15284.0	−0.760800	−0.380400	+	0.924822i	$0.624214\pi$
$740$	0	0
$741$	0	0
$742$	− 27000.0i	− 1.33585i
$743$	18768.0i	0.926691i	0.886178	+	0.463345i	$0.153351\pi$
$743$	18768.0i	0.926691i	−0.886178	+	0.463345i	$0.846649\pi$
$744$	0	0
$745$	0	0
$746$	−32406.0	−1.59044
$747$	0	0
$748$	− 1296.00i	− 0.0633509i
$749$	−28080.0	−1.36985
$750$	0	0
$751$	8696.00	0.422532	0.211266	−	0.977429i	$-0.432241\pi$
$751$	8696.00	0.422532	0.211266	+	0.977429i	$0.432241\pi$
$752$	1704.00i	0.0826310i
$753$	0	0
$754$	17316.0	0.836355
$755$	0	0
$756$	0	0
$757$	38662.0i	1.85627i	0.372247	+	0.928134i	$0.378587\pi$
$757$	38662.0i	1.85627i	−0.372247	+	0.928134i	$0.621413\pi$
$758$	− 4380.00i	− 0.209880i
$759$	0	0
$760$	0	0
$761$	−23874.0	−1.13723	−0.568615	−	0.822604i	$-0.692521\pi$
$761$	−23874.0	−1.13723	−0.568615	+	0.822604i	$0.692521\pi$
$762$	0	0
$763$	− 29480.0i	− 1.39875i
$764$	3576.00	0.169339
$765$	0	0
$766$	−14616.0	−0.689422
$767$	1776.00i	0.0836084i
$768$	0	0
$769$	−23618.0	−1.10753	−0.553763	−	0.832675i	$-0.686808\pi$
$769$	−23618.0	−1.10753	−0.553763	+	0.832675i	$0.686808\pi$
$770$	0	0
$771$	0	0
$772$	− 2666.00i	− 0.124289i
$773$	− 11538.0i	− 0.536860i	−0.963299	−	0.268430i	$-0.913495\pi$
$773$	− 11538.0i	− 0.536860i	0.963299	−	0.268430i	$-0.0865049\pi$
$774$	0	0
$775$	0	0
$776$	−6006.00	−0.277839
$777$	0	0
$778$	− 42138.0i	− 1.94180i
$779$	−40920.0	−1.88204
$780$	0	0
$781$	6912.00	0.316685
$782$	− 19440.0i	− 0.888968i
$783$	0	0
$784$	4047.00	0.184357
$785$	0	0
$786$	0	0
$787$	14884.0i	0.674152i	0.941478	+	0.337076i	$0.109438\pi$
$787$	14884.0i	0.674152i	−0.941478	+	0.337076i	$0.890562\pi$
$788$	2718.00i	0.122874i
$789$	0	0
$790$	0	0
$791$	−21720.0	−0.976327
$792$	0	0
$793$	− 23828.0i	− 1.06703i
$794$	−8202.00	−0.366597
$795$	0	0
$796$	−3832.00	−0.170630
$797$	− 11334.0i	− 0.503728i	−0.967763	−	0.251864i	$-0.918957\pi$
$797$	− 11334.0i	− 0.503728i	0.967763	−	0.251864i	$-0.0810435\pi$
$798$	0	0
$799$	1296.00	0.0573832
$800$	0	0
$801$	0	0
$802$	47826.0i	2.10573i
$803$	− 10320.0i	− 0.453530i
$804$	0	0
$805$	0	0
$806$	−44400.0	−1.94035
$807$	0	0
$808$	36414.0i	1.58545i
$809$	44730.0	1.94391	0.971955	−	0.235167i	$-0.0755638\pi$
$809$	44730.0	1.94391	0.971955	+	0.235167i	$0.0755638\pi$
$810$	0	0
$811$	−42748.0	−1.85091	−0.925453	−	0.378862i	$-0.876316\pi$
$811$	−42748.0	−1.85091	−0.925453	+	0.378862i	$0.876316\pi$
$812$	− 1560.00i	− 0.0674203i
$813$	0	0
$814$	−5040.00	−0.217017
$815$	0	0
$816$	0	0
$817$	11408.0i	0.488513i
$818$	− 26142.0i	− 1.11740i
$819$	0	0
$820$	0	0
$821$	31686.0	1.34695	0.673477	−	0.739208i	$-0.264800\pi$
$821$	31686.0	1.34695	0.673477	+	0.739208i	$0.264800\pi$
$822$	0	0
$823$	11036.0i	0.467425i	0.972306	+	0.233713i	$0.0750875\pi$
$823$	11036.0i	0.467425i	−0.972306	+	0.233713i	$0.924913\pi$
$824$	−9492.00	−0.401298
$825$	0	0
$826$	1440.00	0.0606586
$827$	25884.0i	1.08836i	0.838968	+	0.544181i	$0.183159\pi$
$827$	25884.0i	1.08836i	−0.838968	+	0.544181i	$0.816841\pi$
$828$	0	0
$829$	−15950.0	−0.668234	−0.334117	−	0.942532i	$-0.608438\pi$
$829$	−15950.0	−0.668234	−0.334117	+	0.942532i	$0.608438\pi$
$830$	0	0
$831$	0	0
$832$	− 32042.0i	− 1.33516i
$833$	− 3078.00i	− 0.128027i
$834$	0	0
$835$	0	0
$836$	−2976.00	−0.123119
$837$	0	0
$838$	35928.0i	1.48104i
$839$	13800.0	0.567853	0.283927	−	0.958846i	$-0.408363\pi$
$839$	13800.0	0.567853	0.283927	+	0.958846i	$0.408363\pi$
$840$	0	0
$841$	−18305.0	−0.750543
$842$	33162.0i	1.35729i
$843$	0	0
$844$	−1100.00	−0.0448620
$845$	0	0
$846$	0	0
$847$	15100.0i	0.612565i
$848$	31950.0i	1.29383i
$849$	0	0
$850$	0	0
$851$	−8400.00	−0.338365
$852$	0	0
$853$	− 27862.0i	− 1.11838i	−0.829040	−	0.559189i	$-0.811113\pi$
$853$	− 27862.0i	− 1.11838i	0.829040	−	0.559189i	$-0.188887\pi$
$854$	−19320.0	−0.774141
$855$	0	0
$856$	29484.0	1.17727
$857$	− 7314.00i	− 0.291530i	−0.989319	−	0.145765i	$-0.953436\pi$
$857$	− 7314.00i	− 0.291530i	0.989319	−	0.145765i	$-0.0465644\pi$
$858$	0	0
$859$	28780.0	1.14314	0.571572	−	0.820552i	$-0.306334\pi$
$859$	28780.0	1.14314	0.571572	+	0.820552i	$0.306334\pi$
$860$	0	0
$861$	0	0
$862$	− 2160.00i	− 0.0853479i
$863$	32688.0i	1.28935i	0.764455	+	0.644677i	$0.223008\pi$
$863$	32688.0i	1.28935i	−0.764455	+	0.644677i	$0.776992\pi$
$864$	0	0
$865$	0	0
$866$	46866.0	1.83900
$867$	0	0
$868$	4000.00i	0.156416i
$869$	12480.0	0.487175
$870$	0	0
$871$	−14504.0	−0.564236
$872$	30954.0i	1.20210i
$873$	0	0
$874$	−44640.0	−1.72766
$875$	0	0
$876$	0	0
$877$	− 36650.0i	− 1.41115i	−0.708633	−	0.705577i	$-0.750688\pi$
$877$	− 36650.0i	− 1.41115i	0.708633	−	0.705577i	$-0.249312\pi$
$878$	29640.0i	1.13930i
$879$	0	0
$880$	0	0
$881$	2646.00	0.101187	0.0505936	−	0.998719i	$-0.483889\pi$
$881$	2646.00	0.101187	0.0505936	+	0.998719i	$0.483889\pi$
$882$	0	0
$883$	10892.0i	0.415113i	0.978223	+	0.207557i	$0.0665511\pi$
$883$	10892.0i	0.415113i	−0.978223	+	0.207557i	$0.933449\pi$
$884$	3996.00	0.152036
$885$	0	0
$886$	−48348.0	−1.83328
$887$	− 43464.0i	− 1.64530i	−0.568550	−	0.822648i	$-0.692496\pi$
$887$	− 43464.0i	− 1.64530i	0.568550	−	0.822648i	$-0.307504\pi$
$888$	0	0
$889$	−24880.0	−0.938637
$890$	0	0
$891$	0	0
$892$	− 1964.00i	− 0.0737215i
$893$	− 2976.00i	− 0.111521i
$894$	0	0
$895$	0	0
$896$	−33180.0	−1.23713
$897$	0	0
$898$	27054.0i	1.00535i
$899$	−15600.0	−0.578742
$900$	0	0
$901$	24300.0	0.898502
$902$	− 23760.0i	− 0.877075i
$903$	0	0
$904$	22806.0	0.839067
$905$	0	0
$906$	0	0
$907$	14884.0i	0.544890i	0.962171	+	0.272445i	$0.0878323\pi$
$907$	14884.0i	0.544890i	−0.962171	+	0.272445i	$0.912168\pi$
$908$	− 660.000i	− 0.0241221i
$909$	0	0
$910$	0	0
$911$	1248.00	0.0453876	0.0226938	−	0.999742i	$-0.492776\pi$
$911$	1248.00	0.0453876	0.0226938	+	0.999742i	$0.492776\pi$
$912$	0	0
$913$	− 3744.00i	− 0.135716i
$914$	−11010.0	−0.398445
$915$	0	0
$916$	−1906.00	−0.0687511
$917$	46560.0i	1.67671i
$918$	0	0
$919$	6640.00	0.238339	0.119169	−	0.992874i	$-0.461977\pi$
$919$	6640.00	0.238339	0.119169	+	0.992874i	$0.461977\pi$
$920$	0	0
$921$	0	0
$922$	− 52686.0i	− 1.88191i
$923$	21312.0i	0.760014i
$924$	0	0
$925$	0	0
$926$	−3516.00	−0.124776
$927$	0	0
$928$	3510.00i	0.124161i
$929$	29946.0	1.05758	0.528792	−	0.848751i	$-0.322645\pi$
$929$	29946.0	1.05758	0.528792	+	0.848751i	$0.322645\pi$
$930$	0	0
$931$	−7068.00	−0.248812
$932$	− 1458.00i	− 0.0512429i
$933$	0	0
$934$	−20628.0	−0.722665
$935$	0	0
$936$	0	0
$937$	− 45002.0i	− 1.56900i	−0.620130	−	0.784499i	$-0.712920\pi$
$937$	− 45002.0i	− 1.56900i	0.620130	−	0.784499i	$-0.287080\pi$
$938$	11760.0i	0.409358i
$939$	0	0
$940$	0	0
$941$	−6090.00	−0.210976	−0.105488	−	0.994421i	$-0.533640\pi$
$941$	−6090.00	−0.210976	−0.105488	+	0.994421i	$0.533640\pi$
$942$	0	0
$943$	− 39600.0i	− 1.36750i
$944$	−1704.00	−0.0587505
$945$	0	0
$946$	−6624.00	−0.227658
$947$	56388.0i	1.93491i	0.253035	+	0.967457i	$0.418571\pi$
$947$	56388.0i	1.93491i	−0.253035	+	0.967457i	$0.581429\pi$
$948$	0	0
$949$	31820.0	1.08843
$950$	0	0
$951$	0	0
$952$	22680.0i	0.772125i
$953$	− 10854.0i	− 0.368936i	−0.982839	−	0.184468i	$-0.940944\pi$
$953$	− 10854.0i	− 0.368936i	0.982839	−	0.184468i	$-0.0590561\pi$
$954$	0	0
$955$	0	0
$956$	−1176.00	−0.0397851
$957$	0	0
$958$	6840.00i	0.230679i
$959$	42360.0	1.42636
$960$	0	0
$961$	10209.0	0.342687
$962$	− 15540.0i	− 0.520821i
$963$	0	0
$964$	−866.000	−0.0289336
$965$	0	0
$966$	0	0
$967$	42316.0i	1.40723i	0.710582	+	0.703615i	$0.248432\pi$
$967$	42316.0i	1.40723i	−0.710582	+	0.703615i	$0.751568\pi$
$968$	− 15855.0i	− 0.526445i
$969$	0	0
$970$	0	0
$971$	−24480.0	−0.809063	−0.404532	−	0.914524i	$-0.632565\pi$
$971$	−24480.0	−0.809063	−0.404532	+	0.914524i	$0.632565\pi$
$972$	0	0
$973$	46480.0i	1.53143i
$974$	−9228.00	−0.303577
$975$	0	0
$976$	22862.0	0.749790
$977$	− 6906.00i	− 0.226144i	−0.993587	−	0.113072i	$-0.963931\pi$
$977$	− 6906.00i	− 0.226144i	0.993587	−	0.113072i	$-0.0360690\pi$
$978$	0	0
$979$	24624.0	0.803868
$980$	0	0
$981$	0	0
$982$	56736.0i	1.84371i
$983$	− 6960.00i	− 0.225829i	−0.993605	−	0.112914i	$-0.963981\pi$
$983$	− 6960.00i	− 0.225829i	0.993605	−	0.112914i	$-0.0360186\pi$
$984$	0	0
$985$	0	0
$986$	12636.0	0.408126
$987$	0	0
$988$	− 9176.00i	− 0.295473i
$989$	−11040.0	−0.354956
$990$	0	0
$991$	47792.0	1.53195	0.765975	−	0.642870i	$-0.222256\pi$
$991$	47792.0	1.53195	0.765975	+	0.642870i	$0.222256\pi$
$992$	− 9000.00i	− 0.288055i
$993$	0	0
$994$	17280.0	0.551397
$995$	0	0
$996$	0	0
$997$	− 9938.00i	− 0.315687i	−0.987464	−	0.157843i	$-0.949546\pi$
$997$	− 9938.00i	− 0.315687i	0.987464	−	0.157843i	$-0.0504541\pi$
$998$	− 29868.0i	− 0.947350i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.b.d.199.2		2
3.2	odd	2		75.4.b.a.49.1		2
5.2	odd	4		45.4.a.b.1.1		1
5.3	odd	4		225.4.a.g.1.1		1
5.4	even	2	inner	225.4.b.d.199.1		2
12.11	even	2		1200.4.f.m.49.2		2
15.2	even	4		15.4.a.b.1.1	✓	1
15.8	even	4		75.4.a.a.1.1		1
15.14	odd	2		75.4.b.a.49.2		2
20.7	even	4		720.4.a.r.1.1		1
35.27	even	4		2205.4.a.c.1.1		1
45.2	even	12		405.4.e.d.271.1		2
45.7	odd	12		405.4.e.k.271.1		2
45.22	odd	12		405.4.e.k.136.1		2
45.32	even	12		405.4.e.d.136.1		2
60.23	odd	4		1200.4.a.o.1.1		1
60.47	odd	4		240.4.a.f.1.1		1
60.59	even	2		1200.4.f.m.49.1		2
105.62	odd	4		735.4.a.i.1.1		1
120.77	even	4		960.4.a.bi.1.1		1
120.107	odd	4		960.4.a.l.1.1		1
165.32	odd	4		1815.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.b.1.1	✓	1	15.2	even	4
45.4.a.b.1.1		1	5.2	odd	4
75.4.a.a.1.1		1	15.8	even	4
75.4.b.a.49.1		2	3.2	odd	2
75.4.b.a.49.2		2	15.14	odd	2
225.4.a.g.1.1		1	5.3	odd	4
225.4.b.d.199.1		2	5.4	even	2	inner
225.4.b.d.199.2		2	1.1	even	1	trivial
240.4.a.f.1.1		1	60.47	odd	4
405.4.e.d.136.1		2	45.32	even	12
405.4.e.d.271.1		2	45.2	even	12
405.4.e.k.136.1		2	45.22	odd	12
405.4.e.k.271.1		2	45.7	odd	12
720.4.a.r.1.1		1	20.7	even	4
735.4.a.i.1.1		1	105.62	odd	4
960.4.a.l.1.1		1	120.107	odd	4
960.4.a.bi.1.1		1	120.77	even	4
1200.4.a.o.1.1		1	60.23	odd	4
1200.4.f.m.49.1		2	60.59	even	2
1200.4.f.m.49.2		2	12.11	even	2
1815.4.a.a.1.1		1	165.32	odd	4
2205.4.a.c.1.1		1	35.27	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{2} -1.00000 q^{4} -20.0000i q^{7} +21.0000i q^{8} +O(q^{10})\)	\(q+3.00000i q^{2} -1.00000 q^{4} -20.0000i q^{7} +21.0000i q^{8} +24.0000 q^{11} +74.0000i q^{13} +60.0000 q^{14} -71.0000 q^{16} +54.0000i q^{17} +124.000 q^{19} +72.0000i q^{22} +120.000i q^{23} -222.000 q^{26} +20.0000i q^{28} -78.0000 q^{29} +200.000 q^{31} -45.0000i q^{32} -162.000 q^{34} +70.0000i q^{37} +372.000i q^{38} -330.000 q^{41} +92.0000i q^{43} -24.0000 q^{44} -360.000 q^{46} -24.0000i q^{47} -57.0000 q^{49} -74.0000i q^{52} -450.000i q^{53} +420.000 q^{56} -234.000i q^{58} +24.0000 q^{59} -322.000 q^{61} +600.000i q^{62} -433.000 q^{64} +196.000i q^{67} -54.0000i q^{68} +288.000 q^{71} -430.000i q^{73} -210.000 q^{74} -124.000 q^{76} -480.000i q^{77} +520.000 q^{79} -990.000i q^{82} -156.000i q^{83} -276.000 q^{86} +504.000i q^{88} +1026.00 q^{89} +1480.00 q^{91} -120.000i q^{92} +72.0000 q^{94} +286.000i q^{97} -171.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 48 q^{11} + 120 q^{14} - 142 q^{16} + 248 q^{19} - 444 q^{26} - 156 q^{29} + 400 q^{31} - 324 q^{34} - 660 q^{41} - 48 q^{44} - 720 q^{46} - 114 q^{49} + 840 q^{56} + 48 q^{59} - 644 q^{61} - 866 q^{64} + 576 q^{71} - 420 q^{74} - 248 q^{76} + 1040 q^{79} - 552 q^{86} + 2052 q^{89} + 2960 q^{91} + 144 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 48 * q^11 + 120 * q^14 - 142 * q^16 + 248 * q^19 - 444 * q^26 - 156 * q^29 + 400 * q^31 - 324 * q^34 - 660 * q^41 - 48 * q^44 - 720 * q^46 - 114 * q^49 + 840 * q^56 + 48 * q^59 - 644 * q^61 - 866 * q^64 + 576 * q^71 - 420 * q^74 - 248 * q^76 + 1040 * q^79 - 552 * q^86 + 2052 * q^89 + 2960 * q^91 + 144 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more