LMFDB - Embedded newform 1350.4.c.q.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,4,Mod(649,1350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1350.649"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,0,0,0,0,0,0,84,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$79.6525785077$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.q.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +4.00000i q^{7} +8.00000i q^{8} +42.0000 q^{11} +20.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} -93.0000i q^{17} -59.0000 q^{19} -84.0000i q^{22} +9.00000i q^{23} +40.0000 q^{26} -16.0000i q^{28} -120.000 q^{29} +47.0000 q^{31} -32.0000i q^{32} -186.000 q^{34} +262.000i q^{37} +118.000i q^{38} +126.000 q^{41} -178.000i q^{43} -168.000 q^{44} +18.0000 q^{46} -144.000i q^{47} +327.000 q^{49} -80.0000i q^{52} +741.000i q^{53} -32.0000 q^{56} +240.000i q^{58} +444.000 q^{59} +221.000 q^{61} -94.0000i q^{62} -64.0000 q^{64} +538.000i q^{67} +372.000i q^{68} +690.000 q^{71} -1126.00i q^{73} +524.000 q^{74} +236.000 q^{76} +168.000i q^{77} -665.000 q^{79} -252.000i q^{82} +75.0000i q^{83} -356.000 q^{86} +336.000i q^{88} +1086.00 q^{89} -80.0000 q^{91} -36.0000i q^{92} -288.000 q^{94} -1544.00i q^{97} -654.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 84 q^{11} + 16 q^{14} + 32 q^{16} - 118 q^{19} + 80 q^{26} - 240 q^{29} + 94 q^{31} - 372 q^{34} + 252 q^{41} - 336 q^{44} + 36 q^{46} + 654 q^{49} - 64 q^{56} + 888 q^{59} + 442 q^{61}+ \cdots - 576 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 + 84 * q^11 + 16 * q^14 + 32 * q^16 - 118 * q^19 + 80 * q^26 - 240 * q^29 + 94 * q^31 - 372 * q^34 + 252 * q^41 - 336 * q^44 + 36 * q^46 + 654 * q^49 - 64 * q^56 + 888 * q^59 + 442 * q^61 - 128 * q^64 + 1380 * q^71 + 1048 * q^74 + 472 * q^76 - 1330 * q^79 - 712 * q^86 + 2172 * q^89 - 160 * q^91 - 576 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000i	0.215980i	0.994152	+	0.107990i	$0.0344414\pi$
$7$	4.00000i	0.215980i	−0.994152	+	0.107990i	$0.965559\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	42.0000	1.15123	0.575613	−	0.817723i	$-0.304764\pi$
$11$	42.0000	1.15123	0.575613	+	0.817723i	$0.304764\pi$
$12$	0	0
$13$	20.0000i	0.426692i	0.976977	+	0.213346i	$0.0684362\pi$
$13$	20.0000i	0.426692i	−0.976977	+	0.213346i	$0.931564\pi$
$14$	8.00000	0.152721
$15$	0	0
$16$	16.0000	0.250000
$17$	− 93.0000i	− 1.32681i	−0.748259	−	0.663406i	$-0.769110\pi$
$17$	− 93.0000i	− 1.32681i	0.748259	−	0.663406i	$-0.230890\pi$
$18$	0	0
$19$	−59.0000	−0.712396	−0.356198	−	0.934410i	$-0.615927\pi$
$19$	−59.0000	−0.712396	−0.356198	+	0.934410i	$0.615927\pi$
$20$	0	0
$21$	0	0
$22$	− 84.0000i	− 0.814039i
$23$	9.00000i	0.0815926i	0.999167	+	0.0407963i	$0.0129895\pi$
$23$	9.00000i	0.0815926i	−0.999167	+	0.0407963i	$0.987011\pi$
$24$	0	0
$25$	0	0
$26$	40.0000	0.301717
$27$	0	0
$28$	− 16.0000i	− 0.107990i
$29$	−120.000	−0.768395	−0.384197	−	0.923251i	$-0.625522\pi$
$29$	−120.000	−0.768395	−0.384197	+	0.923251i	$0.625522\pi$
$30$	0	0
$31$	47.0000	0.272305	0.136152	−	0.990688i	$-0.456526\pi$
$31$	47.0000	0.272305	0.136152	+	0.990688i	$0.456526\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	−186.000	−0.938198
$35$	0	0
$36$	0	0
$37$	262.000i	1.16412i	0.813145	+	0.582061i	$0.197754\pi$
$37$	262.000i	1.16412i	−0.813145	+	0.582061i	$0.802246\pi$
$38$	118.000i	0.503740i
$39$	0	0
$40$	0	0
$41$	126.000	0.479949	0.239974	−	0.970779i	$-0.422861\pi$
$41$	126.000	0.479949	0.239974	+	0.970779i	$0.422861\pi$
$42$	0	0
$43$	− 178.000i	− 0.631273i	−0.948880	−	0.315637i	$-0.897782\pi$
$43$	− 178.000i	− 0.631273i	0.948880	−	0.315637i	$-0.102218\pi$
$44$	−168.000	−0.575613
$45$	0	0
$46$	18.0000	0.0576947
$47$	− 144.000i	− 0.446906i	−0.974715	−	0.223453i	$-0.928267\pi$
$47$	− 144.000i	− 0.446906i	0.974715	−	0.223453i	$-0.0717328\pi$
$48$	0	0
$49$	327.000	0.953353
$50$	0	0
$51$	0	0
$52$	− 80.0000i	− 0.213346i
$53$	741.000i	1.92046i	0.279217	+	0.960228i	$0.409925\pi$
$53$	741.000i	1.92046i	−0.279217	+	0.960228i	$0.590075\pi$
$54$	0	0
$55$	0	0
$56$	−32.0000	−0.0763604
$57$	0	0
$58$	240.000i	0.543337i
$59$	444.000	0.979727	0.489863	−	0.871799i	$-0.337047\pi$
$59$	444.000	0.979727	0.489863	+	0.871799i	$0.337047\pi$
$60$	0	0
$61$	221.000	0.463871	0.231936	−	0.972731i	$-0.425494\pi$
$61$	221.000	0.463871	0.231936	+	0.972731i	$0.425494\pi$
$62$	− 94.0000i	− 0.192549i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	538.000i	0.981002i	0.871441	+	0.490501i	$0.163186\pi$
$67$	538.000i	0.981002i	−0.871441	+	0.490501i	$0.836814\pi$
$68$	372.000i	0.663406i
$69$	0	0
$70$	0	0
$71$	690.000	1.15335	0.576676	−	0.816973i	$-0.304350\pi$
$71$	690.000	1.15335	0.576676	+	0.816973i	$0.304350\pi$
$72$	0	0
$73$	− 1126.00i	− 1.80532i	−0.430355	−	0.902660i	$-0.641612\pi$
$73$	− 1126.00i	− 1.80532i	0.430355	−	0.902660i	$-0.358388\pi$
$74$	524.000	0.823159
$75$	0	0
$76$	236.000	0.356198
$77$	168.000i	0.248641i
$78$	0	0
$79$	−665.000	−0.947068	−0.473534	−	0.880776i	$-0.657022\pi$
$79$	−665.000	−0.947068	−0.473534	+	0.880776i	$0.657022\pi$
$80$	0	0
$81$	0	0
$82$	− 252.000i	− 0.339375i
$83$	75.0000i	0.0991846i	0.998770	+	0.0495923i	$0.0157922\pi$
$83$	75.0000i	0.0991846i	−0.998770	+	0.0495923i	$0.984208\pi$
$84$	0	0
$85$	0	0
$86$	−356.000	−0.446378
$87$	0	0
$88$	336.000i	0.407020i
$89$	1086.00	1.29344	0.646718	−	0.762729i	$-0.276141\pi$
$89$	1086.00	1.29344	0.646718	+	0.762729i	$0.276141\pi$
$90$	0	0
$91$	−80.0000	−0.0921569
$92$	− 36.0000i	− 0.0407963i
$93$	0	0
$94$	−288.000	−0.316010
$95$	0	0
$96$	0	0
$97$	− 1544.00i	− 1.61618i	−0.589059	−	0.808090i	$-0.700501\pi$
$97$	− 1544.00i	− 1.61618i	0.589059	−	0.808090i	$-0.299499\pi$
$98$	− 654.000i	− 0.674122i
$99$	0	0
$100$	0	0
$101$	−132.000	−0.130044	−0.0650222	−	0.997884i	$-0.520712\pi$
$101$	−132.000	−0.130044	−0.0650222	+	0.997884i	$0.520712\pi$
$102$	0	0
$103$	− 892.000i	− 0.853314i	−0.904413	−	0.426657i	$-0.859691\pi$
$103$	− 892.000i	− 0.853314i	0.904413	−	0.426657i	$-0.140309\pi$
$104$	−160.000	−0.150859
$105$	0	0
$106$	1482.00	1.35797
$107$	1140.00i	1.02998i	0.857196	+	0.514990i	$0.172205\pi$
$107$	1140.00i	1.02998i	−0.857196	+	0.514990i	$0.827795\pi$
$108$	0	0
$109$	1735.00	1.52461	0.762307	−	0.647216i	$-0.224067\pi$
$109$	1735.00	1.52461	0.762307	+	0.647216i	$0.224067\pi$
$110$	0	0
$111$	0	0
$112$	64.0000i	0.0539949i
$113$	− 1434.00i	− 1.19380i	−0.802316	−	0.596900i	$-0.796399\pi$
$113$	− 1434.00i	− 1.19380i	0.802316	−	0.596900i	$-0.203601\pi$
$114$	0	0
$115$	0	0
$116$	480.000	0.384197
$117$	0	0
$118$	− 888.000i	− 0.692771i
$119$	372.000	0.286565
$120$	0	0
$121$	433.000	0.325319
$122$	− 442.000i	− 0.328007i
$123$	0	0
$124$	−188.000	−0.136152
$125$	0	0
$126$	0	0
$127$	− 686.000i	− 0.479312i	−0.970858	−	0.239656i	$-0.922965\pi$
$127$	− 686.000i	− 0.479312i	0.970858	−	0.239656i	$-0.0770347\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−114.000	−0.0760323	−0.0380161	−	0.999277i	$-0.512104\pi$
$131$	−114.000	−0.0760323	−0.0380161	+	0.999277i	$0.512104\pi$
$132$	0	0
$133$	− 236.000i	− 0.153863i
$134$	1076.00	0.693673
$135$	0	0
$136$	744.000	0.469099
$137$	− 159.000i	− 0.0991554i	−0.998770	−	0.0495777i	$-0.984212\pi$
$137$	− 159.000i	− 0.0991554i	0.998770	−	0.0495777i	$-0.0157875\pi$
$138$	0	0
$139$	−2276.00	−1.38883	−0.694417	−	0.719573i	$-0.744337\pi$
$139$	−2276.00	−1.38883	−0.694417	+	0.719573i	$0.744337\pi$
$140$	0	0
$141$	0	0
$142$	− 1380.00i	− 0.815542i
$143$	840.000i	0.491219i
$144$	0	0
$145$	0	0
$146$	−2252.00	−1.27655
$147$	0	0
$148$	− 1048.00i	− 0.582061i
$149$	1398.00	0.768648	0.384324	−	0.923198i	$-0.374434\pi$
$149$	1398.00	0.768648	0.384324	+	0.923198i	$0.374434\pi$
$150$	0	0
$151$	2624.00	1.41416	0.707080	−	0.707134i	$-0.250012\pi$
$151$	2624.00	1.41416	0.707080	+	0.707134i	$0.250012\pi$
$152$	− 472.000i	− 0.251870i
$153$	0	0
$154$	336.000	0.175816
$155$	0	0
$156$	0	0
$157$	394.000i	0.200284i	0.994973	+	0.100142i	$0.0319297\pi$
$157$	394.000i	0.200284i	−0.994973	+	0.100142i	$0.968070\pi$
$158$	1330.00i	0.669678i
$159$	0	0
$160$	0	0
$161$	−36.0000	−0.0176223
$162$	0	0
$163$	− 3346.00i	− 1.60785i	−0.594733	−	0.803923i	$-0.702742\pi$
$163$	− 3346.00i	− 1.60785i	0.594733	−	0.803923i	$-0.297258\pi$
$164$	−504.000	−0.239974
$165$	0	0
$166$	150.000	0.0701341
$167$	1491.00i	0.690881i	0.938441	+	0.345440i	$0.112270\pi$
$167$	1491.00i	0.690881i	−0.938441	+	0.345440i	$0.887730\pi$
$168$	0	0
$169$	1797.00	0.817934
$170$	0	0
$171$	0	0
$172$	712.000i	0.315637i
$173$	2403.00i	1.05605i	0.849229	+	0.528025i	$0.177067\pi$
$173$	2403.00i	1.05605i	−0.849229	+	0.528025i	$0.822933\pi$
$174$	0	0
$175$	0	0
$176$	672.000	0.287806
$177$	0	0
$178$	− 2172.00i	− 0.914597i
$179$	2640.00	1.10236	0.551181	−	0.834386i	$-0.314177\pi$
$179$	2640.00	1.10236	0.551181	+	0.834386i	$0.314177\pi$
$180$	0	0
$181$	1073.00	0.440638	0.220319	−	0.975428i	$-0.429290\pi$
$181$	1073.00	0.440638	0.220319	+	0.975428i	$0.429290\pi$
$182$	160.000i	0.0651648i
$183$	0	0
$184$	−72.0000	−0.0288473
$185$	0	0
$186$	0	0
$187$	− 3906.00i	− 1.52746i
$188$	576.000i	0.223453i
$189$	0	0
$190$	0	0
$191$	1470.00	0.556887	0.278444	−	0.960453i	$-0.410181\pi$
$191$	1470.00	0.556887	0.278444	+	0.960453i	$0.410181\pi$
$192$	0	0
$193$	− 4720.00i	− 1.76038i	−0.474623	−	0.880189i	$-0.657416\pi$
$193$	− 4720.00i	− 1.76038i	0.474623	−	0.880189i	$-0.342584\pi$
$194$	−3088.00	−1.14281
$195$	0	0
$196$	−1308.00	−0.476676
$197$	765.000i	0.276670i	0.990385	+	0.138335i	$0.0441751\pi$
$197$	765.000i	0.276670i	−0.990385	+	0.138335i	$0.955825\pi$
$198$	0	0
$199$	−668.000	−0.237956	−0.118978	−	0.992897i	$-0.537962\pi$
$199$	−668.000	−0.237956	−0.118978	+	0.992897i	$0.537962\pi$
$200$	0	0
$201$	0	0
$202$	264.000i	0.0919553i
$203$	− 480.000i	− 0.165958i
$204$	0	0
$205$	0	0
$206$	−1784.00	−0.603384
$207$	0	0
$208$	320.000i	0.106673i
$209$	−2478.00	−0.820128
$210$	0	0
$211$	4601.00	1.50117	0.750583	−	0.660777i	$-0.229773\pi$
$211$	4601.00	1.50117	0.750583	+	0.660777i	$0.229773\pi$
$212$	− 2964.00i	− 0.960228i
$213$	0	0
$214$	2280.00	0.728307
$215$	0	0
$216$	0	0
$217$	188.000i	0.0588123i
$218$	− 3470.00i	− 1.07806i
$219$	0	0
$220$	0	0
$221$	1860.00	0.566141
$222$	0	0
$223$	− 2158.00i	− 0.648029i	−0.946052	−	0.324014i	$-0.894967\pi$
$223$	− 2158.00i	− 0.648029i	0.946052	−	0.324014i	$-0.105033\pi$
$224$	128.000	0.0381802
$225$	0	0
$226$	−2868.00	−0.844144
$227$	− 3123.00i	− 0.913131i	−0.889690	−	0.456566i	$-0.849079\pi$
$227$	− 3123.00i	− 0.913131i	0.889690	−	0.456566i	$-0.150921\pi$
$228$	0	0
$229$	−2027.00	−0.584925	−0.292463	−	0.956277i	$-0.594475\pi$
$229$	−2027.00	−0.584925	−0.292463	+	0.956277i	$0.594475\pi$
$230$	0	0
$231$	0	0
$232$	− 960.000i	− 0.271668i
$233$	− 438.000i	− 0.123152i	−0.998102	−	0.0615758i	$-0.980387\pi$
$233$	− 438.000i	− 0.123152i	0.998102	−	0.0615758i	$-0.0196126\pi$
$234$	0	0
$235$	0	0
$236$	−1776.00	−0.489863
$237$	0	0
$238$	− 744.000i	− 0.202632i
$239$	−6414.00	−1.73593	−0.867965	−	0.496626i	$-0.834572\pi$
$239$	−6414.00	−1.73593	−0.867965	+	0.496626i	$0.834572\pi$
$240$	0	0
$241$	3431.00	0.917055	0.458527	−	0.888680i	$-0.348377\pi$
$241$	3431.00	0.917055	0.458527	+	0.888680i	$0.348377\pi$
$242$	− 866.000i	− 0.230035i
$243$	0	0
$244$	−884.000	−0.231936
$245$	0	0
$246$	0	0
$247$	− 1180.00i	− 0.303974i
$248$	376.000i	0.0962743i
$249$	0	0
$250$	0	0
$251$	7308.00	1.83776	0.918878	−	0.394541i	$-0.129096\pi$
$251$	7308.00	1.83776	0.918878	+	0.394541i	$0.129096\pi$
$252$	0	0
$253$	378.000i	0.0939314i
$254$	−1372.00	−0.338925
$255$	0	0
$256$	256.000	0.0625000
$257$	3729.00i	0.905092i	0.891741	+	0.452546i	$0.149484\pi$
$257$	3729.00i	0.905092i	−0.891741	+	0.452546i	$0.850516\pi$
$258$	0	0
$259$	−1048.00	−0.251427
$260$	0	0
$261$	0	0
$262$	228.000i	0.0537629i
$263$	− 1956.00i	− 0.458601i	−0.973356	−	0.229301i	$-0.926356\pi$
$263$	− 1956.00i	− 0.458601i	0.973356	−	0.229301i	$-0.0736439\pi$
$264$	0	0
$265$	0	0
$266$	−472.000	−0.108798
$267$	0	0
$268$	− 2152.00i	− 0.490501i
$269$	−990.000	−0.224392	−0.112196	−	0.993686i	$-0.535788\pi$
$269$	−990.000	−0.224392	−0.112196	+	0.993686i	$0.535788\pi$
$270$	0	0
$271$	8495.00	1.90419	0.952093	−	0.305808i	$-0.0989266\pi$
$271$	8495.00	1.90419	0.952093	+	0.305808i	$0.0989266\pi$
$272$	− 1488.00i	− 0.331703i
$273$	0	0
$274$	−318.000	−0.0701134
$275$	0	0
$276$	0	0
$277$	1366.00i	0.296300i	0.988965	+	0.148150i	$0.0473318\pi$
$277$	1366.00i	0.296300i	−0.988965	+	0.148150i	$0.952668\pi$
$278$	4552.00i	0.982053i
$279$	0	0
$280$	0	0
$281$	5520.00	1.17187	0.585935	−	0.810358i	$-0.300727\pi$
$281$	5520.00	1.17187	0.585935	+	0.810358i	$0.300727\pi$
$282$	0	0
$283$	5438.00i	1.14225i	0.820865	+	0.571123i	$0.193492\pi$
$283$	5438.00i	1.14225i	−0.820865	+	0.571123i	$0.806508\pi$
$284$	−2760.00	−0.576676
$285$	0	0
$286$	1680.00	0.347344
$287$	504.000i	0.103659i
$288$	0	0
$289$	−3736.00	−0.760432
$290$	0	0
$291$	0	0
$292$	4504.00i	0.902660i
$293$	− 8253.00i	− 1.64555i	−0.568369	−	0.822774i	$-0.692425\pi$
$293$	− 8253.00i	− 1.64555i	0.568369	−	0.822774i	$-0.307575\pi$
$294$	0	0
$295$	0	0
$296$	−2096.00	−0.411579
$297$	0	0
$298$	− 2796.00i	− 0.543517i
$299$	−180.000	−0.0348149
$300$	0	0
$301$	712.000	0.136342
$302$	− 5248.00i	− 0.999962i
$303$	0	0
$304$	−944.000	−0.178099
$305$	0	0
$306$	0	0
$307$	− 9290.00i	− 1.72706i	−0.504295	−	0.863531i	$-0.668248\pi$
$307$	− 9290.00i	− 1.72706i	0.504295	−	0.863531i	$-0.331752\pi$
$308$	− 672.000i	− 0.124321i
$309$	0	0
$310$	0	0
$311$	−8112.00	−1.47907	−0.739533	−	0.673121i	$-0.764953\pi$
$311$	−8112.00	−1.47907	−0.739533	+	0.673121i	$0.764953\pi$
$312$	0	0
$313$	− 7900.00i	− 1.42663i	−0.700845	−	0.713314i	$-0.747193\pi$
$313$	− 7900.00i	− 1.42663i	0.700845	−	0.713314i	$-0.252807\pi$
$314$	788.000	0.141622
$315$	0	0
$316$	2660.00	0.473534
$317$	4419.00i	0.782952i	0.920188	+	0.391476i	$0.128035\pi$
$317$	4419.00i	0.782952i	−0.920188	+	0.391476i	$0.871965\pi$
$318$	0	0
$319$	−5040.00	−0.884595
$320$	0	0
$321$	0	0
$322$	72.0000i	0.0124609i
$323$	5487.00i	0.945216i
$324$	0	0
$325$	0	0
$326$	−6692.00	−1.13692
$327$	0	0
$328$	1008.00i	0.169687i
$329$	576.000	0.0965225
$330$	0	0
$331$	−8200.00	−1.36167	−0.680835	−	0.732437i	$-0.738383\pi$
$331$	−8200.00	−1.36167	−0.680835	+	0.732437i	$0.738383\pi$
$332$	− 300.000i	− 0.0495923i
$333$	0	0
$334$	2982.00	0.488526
$335$	0	0
$336$	0	0
$337$	9556.00i	1.54465i	0.635225	+	0.772327i	$0.280907\pi$
$337$	9556.00i	1.54465i	−0.635225	+	0.772327i	$0.719093\pi$
$338$	− 3594.00i	− 0.578366i
$339$	0	0
$340$	0	0
$341$	1974.00	0.313484
$342$	0	0
$343$	2680.00i	0.421885i
$344$	1424.00	0.223189
$345$	0	0
$346$	4806.00	0.746740
$347$	10116.0i	1.56500i	0.622650	+	0.782500i	$0.286056\pi$
$347$	10116.0i	1.56500i	−0.622650	+	0.782500i	$0.713944\pi$
$348$	0	0
$349$	6751.00	1.03545	0.517726	−	0.855546i	$-0.326779\pi$
$349$	6751.00	1.03545	0.517726	+	0.855546i	$0.326779\pi$
$350$	0	0
$351$	0	0
$352$	− 1344.00i	− 0.203510i
$353$	− 4062.00i	− 0.612460i	−0.951958	−	0.306230i	$-0.900932\pi$
$353$	− 4062.00i	− 0.612460i	0.951958	−	0.306230i	$-0.0990677\pi$
$354$	0	0
$355$	0	0
$356$	−4344.00	−0.646718
$357$	0	0
$358$	− 5280.00i	− 0.779488i
$359$	8778.00	1.29049	0.645244	−	0.763977i	$-0.276756\pi$
$359$	8778.00	1.29049	0.645244	+	0.763977i	$0.276756\pi$
$360$	0	0
$361$	−3378.00	−0.492492
$362$	− 2146.00i	− 0.311578i
$363$	0	0
$364$	320.000	0.0460785
$365$	0	0
$366$	0	0
$367$	− 956.000i	− 0.135975i	−0.997686	−	0.0679875i	$-0.978342\pi$
$367$	− 956.000i	− 0.135975i	0.997686	−	0.0679875i	$-0.0216578\pi$
$368$	144.000i	0.0203981i
$369$	0	0
$370$	0	0
$371$	−2964.00	−0.414780
$372$	0	0
$373$	2300.00i	0.319275i	0.987176	+	0.159637i	$0.0510325\pi$
$373$	2300.00i	0.319275i	−0.987176	+	0.159637i	$0.948968\pi$
$374$	−7812.00	−1.08008
$375$	0	0
$376$	1152.00	0.158005
$377$	− 2400.00i	− 0.327868i
$378$	0	0
$379$	−29.0000	−0.00393042	−0.00196521	−	0.999998i	$-0.500626\pi$
$379$	−29.0000	−0.00393042	−0.00196521	+	0.999998i	$0.500626\pi$
$380$	0	0
$381$	0	0
$382$	− 2940.00i	− 0.393779i
$383$	− 8127.00i	− 1.08426i	−0.840296	−	0.542128i	$-0.817619\pi$
$383$	− 8127.00i	− 1.08426i	0.840296	−	0.542128i	$-0.182381\pi$
$384$	0	0
$385$	0	0
$386$	−9440.00	−1.24478
$387$	0	0
$388$	6176.00i	0.808090i
$389$	−7938.00	−1.03463	−0.517317	−	0.855794i	$-0.673069\pi$
$389$	−7938.00	−1.03463	−0.517317	+	0.855794i	$0.673069\pi$
$390$	0	0
$391$	837.000	0.108258
$392$	2616.00i	0.337061i
$393$	0	0
$394$	1530.00	0.195635
$395$	0	0
$396$	0	0
$397$	− 272.000i	− 0.0343861i	−0.999852	−	0.0171931i	$-0.994527\pi$
$397$	− 272.000i	− 0.0343861i	0.999852	−	0.0171931i	$-0.00547299\pi$
$398$	1336.00i	0.168260i
$399$	0	0
$400$	0	0
$401$	−4554.00	−0.567122	−0.283561	−	0.958954i	$-0.591516\pi$
$401$	−4554.00	−0.567122	−0.283561	+	0.958954i	$0.591516\pi$
$402$	0	0
$403$	940.000i	0.116190i
$404$	528.000	0.0650222
$405$	0	0
$406$	−960.000	−0.117350
$407$	11004.0i	1.34017i
$408$	0	0
$409$	−1001.00	−0.121018	−0.0605089	−	0.998168i	$-0.519272\pi$
$409$	−1001.00	−0.121018	−0.0605089	+	0.998168i	$0.519272\pi$
$410$	0	0
$411$	0	0
$412$	3568.00i	0.426657i
$413$	1776.00i	0.211601i
$414$	0	0
$415$	0	0
$416$	640.000	0.0754293
$417$	0	0
$418$	4956.00i	0.579918i
$419$	−1794.00	−0.209171	−0.104585	−	0.994516i	$-0.533352\pi$
$419$	−1794.00	−0.209171	−0.104585	+	0.994516i	$0.533352\pi$
$420$	0	0
$421$	−16129.0	−1.86717	−0.933586	−	0.358354i	$-0.883338\pi$
$421$	−16129.0	−1.86717	−0.933586	+	0.358354i	$0.883338\pi$
$422$	− 9202.00i	− 1.06148i
$423$	0	0
$424$	−5928.00	−0.678984
$425$	0	0
$426$	0	0
$427$	884.000i	0.100187i
$428$	− 4560.00i	− 0.514990i
$429$	0	0
$430$	0	0
$431$	13356.0	1.49266	0.746329	−	0.665577i	$-0.231814\pi$
$431$	13356.0	1.49266	0.746329	+	0.665577i	$0.231814\pi$
$432$	0	0
$433$	− 11500.0i	− 1.27634i	−0.769896	−	0.638169i	$-0.779692\pi$
$433$	− 11500.0i	− 1.27634i	0.769896	−	0.638169i	$-0.220308\pi$
$434$	376.000	0.0415866
$435$	0	0
$436$	−6940.00	−0.762307
$437$	− 531.000i	− 0.0581263i
$438$	0	0
$439$	11149.0	1.21210	0.606051	−	0.795426i	$-0.292753\pi$
$439$	11149.0	1.21210	0.606051	+	0.795426i	$0.292753\pi$
$440$	0	0
$441$	0	0
$442$	− 3720.00i	− 0.400322i
$443$	− 3849.00i	− 0.412803i	−0.978467	−	0.206401i	$-0.933825\pi$
$443$	− 3849.00i	− 0.412803i	0.978467	−	0.206401i	$-0.0661752\pi$
$444$	0	0
$445$	0	0
$446$	−4316.00	−0.458225
$447$	0	0
$448$	− 256.000i	− 0.0269975i
$449$	18048.0	1.89697	0.948483	−	0.316828i	$-0.102618\pi$
$449$	18048.0	1.89697	0.948483	+	0.316828i	$0.102618\pi$
$450$	0	0
$451$	5292.00	0.552529
$452$	5736.00i	0.596900i
$453$	0	0
$454$	−6246.00	−0.645681
$455$	0	0
$456$	0	0
$457$	4264.00i	0.436458i	0.975898	+	0.218229i	$0.0700280\pi$
$457$	4264.00i	0.436458i	−0.975898	+	0.218229i	$0.929972\pi$
$458$	4054.00i	0.413605i
$459$	0	0
$460$	0	0
$461$	10242.0	1.03475	0.517373	−	0.855760i	$-0.326910\pi$
$461$	10242.0	1.03475	0.517373	+	0.855760i	$0.326910\pi$
$462$	0	0
$463$	3302.00i	0.331441i	0.986173	+	0.165720i	$0.0529949\pi$
$463$	3302.00i	0.331441i	−0.986173	+	0.165720i	$0.947005\pi$
$464$	−1920.00	−0.192099
$465$	0	0
$466$	−876.000	−0.0870814
$467$	− 1923.00i	− 0.190548i	−0.995451	−	0.0952739i	$-0.969627\pi$
$467$	− 1923.00i	− 0.190548i	0.995451	−	0.0952739i	$-0.0303727\pi$
$468$	0	0
$469$	−2152.00	−0.211877
$470$	0	0
$471$	0	0
$472$	3552.00i	0.346386i
$473$	− 7476.00i	− 0.726738i
$474$	0	0
$475$	0	0
$476$	−1488.00	−0.143282
$477$	0	0
$478$	12828.0i	1.22749i
$479$	15246.0	1.45430	0.727148	−	0.686481i	$-0.240846\pi$
$479$	15246.0	1.45430	0.727148	+	0.686481i	$0.240846\pi$
$480$	0	0
$481$	−5240.00	−0.496722
$482$	− 6862.00i	− 0.648455i
$483$	0	0
$484$	−1732.00	−0.162660
$485$	0	0
$486$	0	0
$487$	8206.00i	0.763551i	0.924255	+	0.381776i	$0.124687\pi$
$487$	8206.00i	0.763551i	−0.924255	+	0.381776i	$0.875313\pi$
$488$	1768.00i	0.164003i
$489$	0	0
$490$	0	0
$491$	16806.0	1.54469	0.772346	−	0.635202i	$-0.219083\pi$
$491$	16806.0	1.54469	0.772346	+	0.635202i	$0.219083\pi$
$492$	0	0
$493$	11160.0i	1.01952i
$494$	−2360.00	−0.214942
$495$	0	0
$496$	752.000	0.0680762
$497$	2760.00i	0.249100i
$498$	0	0
$499$	5425.00	0.486686	0.243343	−	0.969940i	$-0.421756\pi$
$499$	5425.00	0.486686	0.243343	+	0.969940i	$0.421756\pi$
$500$	0	0
$501$	0	0
$502$	− 14616.0i	− 1.29949i
$503$	19665.0i	1.74318i	0.490236	+	0.871589i	$0.336910\pi$
$503$	19665.0i	1.74318i	−0.490236	+	0.871589i	$0.663090\pi$
$504$	0	0
$505$	0	0
$506$	756.000	0.0664196
$507$	0	0
$508$	2744.00i	0.239656i
$509$	−14724.0	−1.28218	−0.641090	−	0.767466i	$-0.721518\pi$
$509$	−14724.0	−1.28218	−0.641090	+	0.767466i	$0.721518\pi$
$510$	0	0
$511$	4504.00	0.389912
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	7458.00	0.639997
$515$	0	0
$516$	0	0
$517$	− 6048.00i	− 0.514489i
$518$	2096.00i	0.177786i
$519$	0	0
$520$	0	0
$521$	−2058.00	−0.173057	−0.0865284	−	0.996249i	$-0.527577\pi$
$521$	−2058.00	−0.173057	−0.0865284	+	0.996249i	$0.527577\pi$
$522$	0	0
$523$	11912.0i	0.995938i	0.867195	+	0.497969i	$0.165921\pi$
$523$	11912.0i	0.995938i	−0.867195	+	0.497969i	$0.834079\pi$
$524$	456.000	0.0380161
$525$	0	0
$526$	−3912.00	−0.324280
$527$	− 4371.00i	− 0.361297i
$528$	0	0
$529$	12086.0	0.993343
$530$	0	0
$531$	0	0
$532$	944.000i	0.0769316i
$533$	2520.00i	0.204790i
$534$	0	0
$535$	0	0
$536$	−4304.00	−0.346837
$537$	0	0
$538$	1980.00i	0.158669i
$539$	13734.0	1.09752
$540$	0	0
$541$	−5170.00	−0.410861	−0.205430	−	0.978672i	$-0.565859\pi$
$541$	−5170.00	−0.410861	−0.205430	+	0.978672i	$0.565859\pi$
$542$	− 16990.0i	− 1.34646i
$543$	0	0
$544$	−2976.00	−0.234550
$545$	0	0
$546$	0	0
$547$	4186.00i	0.327204i	0.986526	+	0.163602i	$0.0523112\pi$
$547$	4186.00i	0.327204i	−0.986526	+	0.163602i	$0.947689\pi$
$548$	636.000i	0.0495777i
$549$	0	0
$550$	0	0
$551$	7080.00	0.547401
$552$	0	0
$553$	− 2660.00i	− 0.204547i
$554$	2732.00	0.209515
$555$	0	0
$556$	9104.00	0.694417
$557$	13026.0i	0.990896i	0.868637	+	0.495448i	$0.164996\pi$
$557$	13026.0i	0.990896i	−0.868637	+	0.495448i	$0.835004\pi$
$558$	0	0
$559$	3560.00	0.269359
$560$	0	0
$561$	0	0
$562$	− 11040.0i	− 0.828638i
$563$	10668.0i	0.798584i	0.916824	+	0.399292i	$0.130744\pi$
$563$	10668.0i	0.798584i	−0.916824	+	0.399292i	$0.869256\pi$
$564$	0	0
$565$	0	0
$566$	10876.0	0.807690
$567$	0	0
$568$	5520.00i	0.407771i
$569$	−15372.0	−1.13256	−0.566281	−	0.824212i	$-0.691618\pi$
$569$	−15372.0	−1.13256	−0.566281	+	0.824212i	$0.691618\pi$
$570$	0	0
$571$	−14989.0	−1.09855	−0.549273	−	0.835643i	$-0.685095\pi$
$571$	−14989.0	−1.09855	−0.549273	+	0.835643i	$0.685095\pi$
$572$	− 3360.00i	− 0.245610i
$573$	0	0
$574$	1008.00	0.0732981
$575$	0	0
$576$	0	0
$577$	1066.00i	0.0769119i	0.999260	+	0.0384559i	$0.0122439\pi$
$577$	1066.00i	0.0769119i	−0.999260	+	0.0384559i	$0.987756\pi$
$578$	7472.00i	0.537706i
$579$	0	0
$580$	0	0
$581$	−300.000	−0.0214219
$582$	0	0
$583$	31122.0i	2.21088i
$584$	9008.00	0.638277
$585$	0	0
$586$	−16506.0	−1.16358
$587$	− 621.000i	− 0.0436651i	−0.999762	−	0.0218325i	$-0.993050\pi$
$587$	− 621.000i	− 0.0436651i	0.999762	−	0.0218325i	$-0.00695007\pi$
$588$	0	0
$589$	−2773.00	−0.193989
$590$	0	0
$591$	0	0
$592$	4192.00i	0.291031i
$593$	20187.0i	1.39794i	0.715149	+	0.698972i	$0.246359\pi$
$593$	20187.0i	1.39794i	−0.715149	+	0.698972i	$0.753641\pi$
$594$	0	0
$595$	0	0
$596$	−5592.00	−0.384324
$597$	0	0
$598$	360.000i	0.0246179i
$599$	−18228.0	−1.24337	−0.621683	−	0.783269i	$-0.713551\pi$
$599$	−18228.0	−1.24337	−0.621683	+	0.783269i	$0.713551\pi$
$600$	0	0
$601$	−11743.0	−0.797017	−0.398508	−	0.917165i	$-0.630472\pi$
$601$	−11743.0	−0.797017	−0.398508	+	0.917165i	$0.630472\pi$
$602$	− 1424.00i	− 0.0964085i
$603$	0	0
$604$	−10496.0	−0.707080
$605$	0	0
$606$	0	0
$607$	24418.0i	1.63278i	0.577503	+	0.816389i	$0.304027\pi$
$607$	24418.0i	1.63278i	−0.577503	+	0.816389i	$0.695973\pi$
$608$	1888.00i	0.125935i
$609$	0	0
$610$	0	0
$611$	2880.00	0.190691
$612$	0	0
$613$	2672.00i	0.176054i	0.996118	+	0.0880270i	$0.0280562\pi$
$613$	2672.00i	0.176054i	−0.996118	+	0.0880270i	$0.971944\pi$
$614$	−18580.0	−1.22122
$615$	0	0
$616$	−1344.00	−0.0879080
$617$	− 8601.00i	− 0.561205i	−0.959824	−	0.280602i	$-0.909466\pi$
$617$	− 8601.00i	− 0.561205i	0.959824	−	0.280602i	$-0.0905342\pi$
$618$	0	0
$619$	−21308.0	−1.38359	−0.691794	−	0.722095i	$-0.743179\pi$
$619$	−21308.0	−1.38359	−0.691794	+	0.722095i	$0.743179\pi$
$620$	0	0
$621$	0	0
$622$	16224.0i	1.04586i
$623$	4344.00i	0.279356i
$624$	0	0
$625$	0	0
$626$	−15800.0	−1.00878
$627$	0	0
$628$	− 1576.00i	− 0.100142i
$629$	24366.0	1.54457
$630$	0	0
$631$	−19015.0	−1.19964	−0.599822	−	0.800134i	$-0.704762\pi$
$631$	−19015.0	−1.19964	−0.599822	+	0.800134i	$0.704762\pi$
$632$	− 5320.00i	− 0.334839i
$633$	0	0
$634$	8838.00	0.553631
$635$	0	0
$636$	0	0
$637$	6540.00i	0.406788i
$638$	10080.0i	0.625503i
$639$	0	0
$640$	0	0
$641$	4416.00	0.272108	0.136054	−	0.990701i	$-0.456558\pi$
$641$	4416.00	0.272108	0.136054	+	0.990701i	$0.456558\pi$
$642$	0	0
$643$	7580.00i	0.464893i	0.972609	+	0.232446i	$0.0746730\pi$
$643$	7580.00i	0.464893i	−0.972609	+	0.232446i	$0.925327\pi$
$644$	144.000	0.00881117
$645$	0	0
$646$	10974.0	0.668369
$647$	− 14901.0i	− 0.905439i	−0.891653	−	0.452719i	$-0.850454\pi$
$647$	− 14901.0i	− 0.905439i	0.891653	−	0.452719i	$-0.149546\pi$
$648$	0	0
$649$	18648.0	1.12789
$650$	0	0
$651$	0	0
$652$	13384.0i	0.803923i
$653$	− 12915.0i	− 0.773971i	−0.922086	−	0.386985i	$-0.873516\pi$
$653$	− 12915.0i	− 0.773971i	0.922086	−	0.386985i	$-0.126484\pi$
$654$	0	0
$655$	0	0
$656$	2016.00	0.119987
$657$	0	0
$658$	− 1152.00i	− 0.0682517i
$659$	28128.0	1.66269	0.831344	−	0.555758i	$-0.187572\pi$
$659$	28128.0	1.66269	0.831344	+	0.555758i	$0.187572\pi$
$660$	0	0
$661$	−8362.00	−0.492049	−0.246024	−	0.969264i	$-0.579124\pi$
$661$	−8362.00	−0.492049	−0.246024	+	0.969264i	$0.579124\pi$
$662$	16400.0i	0.962846i
$663$	0	0
$664$	−600.000	−0.0350670
$665$	0	0
$666$	0	0
$667$	− 1080.00i	− 0.0626953i
$668$	− 5964.00i	− 0.345440i
$669$	0	0
$670$	0	0
$671$	9282.00	0.534020
$672$	0	0
$673$	29708.0i	1.70157i	0.525511	+	0.850787i	$0.323874\pi$
$673$	29708.0i	1.70157i	−0.525511	+	0.850787i	$0.676126\pi$
$674$	19112.0	1.09224
$675$	0	0
$676$	−7188.00	−0.408967
$677$	6762.00i	0.383877i	0.981407	+	0.191939i	$0.0614774\pi$
$677$	6762.00i	0.383877i	−0.981407	+	0.191939i	$0.938523\pi$
$678$	0	0
$679$	6176.00	0.349062
$680$	0	0
$681$	0	0
$682$	− 3948.00i	− 0.221667i
$683$	19155.0i	1.07313i	0.843860	+	0.536563i	$0.180278\pi$
$683$	19155.0i	1.07313i	−0.843860	+	0.536563i	$0.819722\pi$
$684$	0	0
$685$	0	0
$686$	5360.00	0.298317
$687$	0	0
$688$	− 2848.00i	− 0.157818i
$689$	−14820.0	−0.819444
$690$	0	0
$691$	−22975.0	−1.26485	−0.632424	−	0.774622i	$-0.717940\pi$
$691$	−22975.0	−1.26485	−0.632424	+	0.774622i	$0.717940\pi$
$692$	− 9612.00i	− 0.528025i
$693$	0	0
$694$	20232.0	1.10662
$695$	0	0
$696$	0	0
$697$	− 11718.0i	− 0.636802i
$698$	− 13502.0i	− 0.732175i
$699$	0	0
$700$	0	0
$701$	6450.00	0.347522	0.173761	−	0.984788i	$-0.444408\pi$
$701$	6450.00	0.347522	0.173761	+	0.984788i	$0.444408\pi$
$702$	0	0
$703$	− 15458.0i	− 0.829317i
$704$	−2688.00	−0.143903
$705$	0	0
$706$	−8124.00	−0.433075
$707$	− 528.000i	− 0.0280870i
$708$	0	0
$709$	−34538.0	−1.82948	−0.914740	−	0.404042i	$-0.867605\pi$
$709$	−34538.0	−1.82948	−0.914740	+	0.404042i	$0.867605\pi$
$710$	0	0
$711$	0	0
$712$	8688.00i	0.457299i
$713$	423.000i	0.0222181i
$714$	0	0
$715$	0	0
$716$	−10560.0	−0.551181
$717$	0	0
$718$	− 17556.0i	− 0.912513i
$719$	27114.0	1.40637	0.703186	−	0.711006i	$-0.251760\pi$
$719$	27114.0	1.40637	0.703186	+	0.711006i	$0.251760\pi$
$720$	0	0
$721$	3568.00	0.184299
$722$	6756.00i	0.348244i
$723$	0	0
$724$	−4292.00	−0.220319
$725$	0	0
$726$	0	0
$727$	− 236.000i	− 0.0120396i	−0.999982	−	0.00601978i	$-0.998084\pi$
$727$	− 236.000i	− 0.0120396i	0.999982	−	0.00601978i	$-0.00191617\pi$
$728$	− 640.000i	− 0.0325824i
$729$	0	0
$730$	0	0
$731$	−16554.0	−0.837581
$732$	0	0
$733$	27128.0i	1.36698i	0.729960	+	0.683489i	$0.239538\pi$
$733$	27128.0i	1.36698i	−0.729960	+	0.683489i	$0.760462\pi$
$734$	−1912.00	−0.0961488
$735$	0	0
$736$	288.000	0.0144237
$737$	22596.0i	1.12935i
$738$	0	0
$739$	−5249.00	−0.261282	−0.130641	−	0.991430i	$-0.541704\pi$
$739$	−5249.00	−0.261282	−0.130641	+	0.991430i	$0.541704\pi$
$740$	0	0
$741$	0	0
$742$	5928.00i	0.293293i
$743$	− 13896.0i	− 0.686130i	−0.939312	−	0.343065i	$-0.888535\pi$
$743$	− 13896.0i	− 0.686130i	0.939312	−	0.343065i	$-0.111465\pi$
$744$	0	0
$745$	0	0
$746$	4600.00	0.225761
$747$	0	0
$748$	15624.0i	0.763730i
$749$	−4560.00	−0.222455
$750$	0	0
$751$	27665.0	1.34422	0.672111	−	0.740451i	$-0.265388\pi$
$751$	27665.0	1.34422	0.672111	+	0.740451i	$0.265388\pi$
$752$	− 2304.00i	− 0.111726i
$753$	0	0
$754$	−4800.00	−0.231838
$755$	0	0
$756$	0	0
$757$	8122.00i	0.389959i	0.980807	+	0.194980i	$0.0624641\pi$
$757$	8122.00i	0.389959i	−0.980807	+	0.194980i	$0.937536\pi$
$758$	58.0000i	0.00277923i
$759$	0	0
$760$	0	0
$761$	−10584.0	−0.504165	−0.252083	−	0.967706i	$-0.581115\pi$
$761$	−10584.0	−0.504165	−0.252083	+	0.967706i	$0.581115\pi$
$762$	0	0
$763$	6940.00i	0.329286i
$764$	−5880.00	−0.278444
$765$	0	0
$766$	−16254.0	−0.766685
$767$	8880.00i	0.418042i
$768$	0	0
$769$	18619.0	0.873106	0.436553	−	0.899679i	$-0.356199\pi$
$769$	18619.0	0.873106	0.436553	+	0.899679i	$0.356199\pi$
$770$	0	0
$771$	0	0
$772$	18880.0i	0.880189i
$773$	− 22251.0i	− 1.03533i	−0.855582	−	0.517667i	$-0.826801\pi$
$773$	− 22251.0i	− 1.03533i	0.855582	−	0.517667i	$-0.173199\pi$
$774$	0	0
$775$	0	0
$776$	12352.0	0.571406
$777$	0	0
$778$	15876.0i	0.731597i
$779$	−7434.00	−0.341914
$780$	0	0
$781$	28980.0	1.32777
$782$	− 1674.00i	− 0.0765500i
$783$	0	0
$784$	5232.00	0.238338
$785$	0	0
$786$	0	0
$787$	− 24854.0i	− 1.12573i	−0.826549	−	0.562865i	$-0.809699\pi$
$787$	− 24854.0i	− 1.12573i	0.826549	−	0.562865i	$-0.190301\pi$
$788$	− 3060.00i	− 0.138335i
$789$	0	0
$790$	0	0
$791$	5736.00	0.257837
$792$	0	0
$793$	4420.00i	0.197930i
$794$	−544.000	−0.0243147
$795$	0	0
$796$	2672.00	0.118978
$797$	3681.00i	0.163598i	0.996649	+	0.0817991i	$0.0260666\pi$
$797$	3681.00i	0.163598i	−0.996649	+	0.0817991i	$0.973933\pi$
$798$	0	0
$799$	−13392.0	−0.592960
$800$	0	0
$801$	0	0
$802$	9108.00i	0.401016i
$803$	− 47292.0i	− 2.07833i
$804$	0	0
$805$	0	0
$806$	1880.00	0.0821590
$807$	0	0
$808$	− 1056.00i	− 0.0459777i
$809$	−5142.00	−0.223465	−0.111732	−	0.993738i	$-0.535640\pi$
$809$	−5142.00	−0.223465	−0.111732	+	0.993738i	$0.535640\pi$
$810$	0	0
$811$	−18484.0	−0.800322	−0.400161	−	0.916445i	$-0.631046\pi$
$811$	−18484.0	−0.800322	−0.400161	+	0.916445i	$0.631046\pi$
$812$	1920.00i	0.0829788i
$813$	0	0
$814$	22008.0	0.947641
$815$	0	0
$816$	0	0
$817$	10502.0i	0.449717i
$818$	2002.00i	0.0855725i
$819$	0	0
$820$	0	0
$821$	−25014.0	−1.06333	−0.531665	−	0.846954i	$-0.678434\pi$
$821$	−25014.0	−1.06333	−0.531665	+	0.846954i	$0.678434\pi$
$822$	0	0
$823$	− 32146.0i	− 1.36153i	−0.732502	−	0.680765i	$-0.761648\pi$
$823$	− 32146.0i	− 1.36153i	0.732502	−	0.680765i	$-0.238352\pi$
$824$	7136.00	0.301692
$825$	0	0
$826$	3552.00	0.149625
$827$	− 10977.0i	− 0.461557i	−0.973006	−	0.230779i	$-0.925873\pi$
$827$	− 10977.0i	− 0.461557i	0.973006	−	0.230779i	$-0.0741273\pi$
$828$	0	0
$829$	−36602.0	−1.53346	−0.766731	−	0.641969i	$-0.778118\pi$
$829$	−36602.0	−1.53346	−0.766731	+	0.641969i	$0.778118\pi$
$830$	0	0
$831$	0	0
$832$	− 1280.00i	− 0.0533366i
$833$	− 30411.0i	− 1.26492i
$834$	0	0
$835$	0	0
$836$	9912.00	0.410064
$837$	0	0
$838$	3588.00i	0.147906i
$839$	11076.0	0.455764	0.227882	−	0.973689i	$-0.426820\pi$
$839$	11076.0	0.455764	0.227882	+	0.973689i	$0.426820\pi$
$840$	0	0
$841$	−9989.00	−0.409570
$842$	32258.0i	1.32029i
$843$	0	0
$844$	−18404.0	−0.750583
$845$	0	0
$846$	0	0
$847$	1732.00i	0.0702624i
$848$	11856.0i	0.480114i
$849$	0	0
$850$	0	0
$851$	−2358.00	−0.0949838
$852$	0	0
$853$	36848.0i	1.47908i	0.673115	+	0.739538i	$0.264956\pi$
$853$	36848.0i	1.47908i	−0.673115	+	0.739538i	$0.735044\pi$
$854$	1768.00	0.0708428
$855$	0	0
$856$	−9120.00	−0.364153
$857$	− 26961.0i	− 1.07464i	−0.843377	−	0.537322i	$-0.819436\pi$
$857$	− 26961.0i	− 1.07464i	0.843377	−	0.537322i	$-0.180564\pi$
$858$	0	0
$859$	415.000	0.0164838	0.00824192	−	0.999966i	$-0.497376\pi$
$859$	415.000	0.0164838	0.00824192	+	0.999966i	$0.497376\pi$
$860$	0	0
$861$	0	0
$862$	− 26712.0i	− 1.05547i
$863$	45501.0i	1.79475i	0.441265	+	0.897377i	$0.354530\pi$
$863$	45501.0i	1.79475i	−0.441265	+	0.897377i	$0.645470\pi$
$864$	0	0
$865$	0	0
$866$	−23000.0	−0.902508
$867$	0	0
$868$	− 752.000i	− 0.0294062i
$869$	−27930.0	−1.09029
$870$	0	0
$871$	−10760.0	−0.418586
$872$	13880.0i	0.539032i
$873$	0	0
$874$	−1062.00	−0.0411015
$875$	0	0
$876$	0	0
$877$	− 35042.0i	− 1.34924i	−0.738165	−	0.674620i	$-0.764307\pi$
$877$	− 35042.0i	− 1.34924i	0.738165	−	0.674620i	$-0.235693\pi$
$878$	− 22298.0i	− 0.857085i
$879$	0	0
$880$	0	0
$881$	−1080.00	−0.0413009	−0.0206505	−	0.999787i	$-0.506574\pi$
$881$	−1080.00	−0.0413009	−0.0206505	+	0.999787i	$0.506574\pi$
$882$	0	0
$883$	− 20164.0i	− 0.768485i	−0.923232	−	0.384243i	$-0.874463\pi$
$883$	− 20164.0i	− 0.768485i	0.923232	−	0.384243i	$-0.125537\pi$
$884$	−7440.00	−0.283070
$885$	0	0
$886$	−7698.00	−0.291895
$887$	20067.0i	0.759621i	0.925064	+	0.379811i	$0.124011\pi$
$887$	20067.0i	0.759621i	−0.925064	+	0.379811i	$0.875989\pi$
$888$	0	0
$889$	2744.00	0.103522
$890$	0	0
$891$	0	0
$892$	8632.00i	0.324014i
$893$	8496.00i	0.318374i
$894$	0	0
$895$	0	0
$896$	−512.000	−0.0190901
$897$	0	0
$898$	− 36096.0i	− 1.34136i
$899$	−5640.00	−0.209238
$900$	0	0
$901$	68913.0	2.54809
$902$	− 10584.0i	− 0.390697i
$903$	0	0
$904$	11472.0	0.422072
$905$	0	0
$906$	0	0
$907$	26524.0i	0.971020i	0.874231	+	0.485510i	$0.161366\pi$
$907$	26524.0i	0.971020i	−0.874231	+	0.485510i	$0.838634\pi$
$908$	12492.0i	0.456566i
$909$	0	0
$910$	0	0
$911$	−35568.0	−1.29355	−0.646773	−	0.762683i	$-0.723882\pi$
$911$	−35568.0	−1.29355	−0.646773	+	0.762683i	$0.723882\pi$
$912$	0	0
$913$	3150.00i	0.114184i
$914$	8528.00	0.308623
$915$	0	0
$916$	8108.00	0.292463
$917$	− 456.000i	− 0.0164214i
$918$	0	0
$919$	23704.0	0.850841	0.425420	−	0.904996i	$-0.360126\pi$
$919$	23704.0	0.850841	0.425420	+	0.904996i	$0.360126\pi$
$920$	0	0
$921$	0	0
$922$	− 20484.0i	− 0.731675i
$923$	13800.0i	0.492126i
$924$	0	0
$925$	0	0
$926$	6604.00	0.234364
$927$	0	0
$928$	3840.00i	0.135834i
$929$	−40590.0	−1.43349	−0.716746	−	0.697334i	$-0.754369\pi$
$929$	−40590.0	−1.43349	−0.716746	+	0.697334i	$0.754369\pi$
$930$	0	0
$931$	−19293.0	−0.679165
$932$	1752.00i	0.0615758i
$933$	0	0
$934$	−3846.00	−0.134738
$935$	0	0
$936$	0	0
$937$	12964.0i	0.451991i	0.974128	+	0.225995i	$0.0725634\pi$
$937$	12964.0i	0.451991i	−0.974128	+	0.225995i	$0.927437\pi$
$938$	4304.00i	0.149819i
$939$	0	0
$940$	0	0
$941$	−29922.0	−1.03659	−0.518294	−	0.855203i	$-0.673433\pi$
$941$	−29922.0	−1.03659	−0.518294	+	0.855203i	$0.673433\pi$
$942$	0	0
$943$	1134.00i	0.0391603i
$944$	7104.00	0.244932
$945$	0	0
$946$	−14952.0	−0.513881
$947$	5241.00i	0.179841i	0.995949	+	0.0899206i	$0.0286613\pi$
$947$	5241.00i	0.179841i	−0.995949	+	0.0899206i	$0.971339\pi$
$948$	0	0
$949$	22520.0	0.770316
$950$	0	0
$951$	0	0
$952$	2976.00i	0.101316i
$953$	− 26214.0i	− 0.891033i	−0.895274	−	0.445517i	$-0.853020\pi$
$953$	− 26214.0i	− 0.891033i	0.895274	−	0.445517i	$-0.146980\pi$
$954$	0	0
$955$	0	0
$956$	25656.0	0.867965
$957$	0	0
$958$	− 30492.0i	− 1.02834i
$959$	636.000	0.0214155
$960$	0	0
$961$	−27582.0	−0.925850
$962$	10480.0i	0.351236i
$963$	0	0
$964$	−13724.0	−0.458527
$965$	0	0
$966$	0	0
$967$	− 18278.0i	− 0.607840i	−0.952698	−	0.303920i	$-0.901705\pi$
$967$	− 18278.0i	− 0.607840i	0.952698	−	0.303920i	$-0.0982955\pi$
$968$	3464.00i	0.115018i
$969$	0	0
$970$	0	0
$971$	24942.0	0.824333	0.412166	−	0.911109i	$-0.364772\pi$
$971$	24942.0	0.824333	0.412166	+	0.911109i	$0.364772\pi$
$972$	0	0
$973$	− 9104.00i	− 0.299960i
$974$	16412.0	0.539912
$975$	0	0
$976$	3536.00	0.115968
$977$	− 11226.0i	− 0.367607i	−0.982963	−	0.183803i	$-0.941159\pi$
$977$	− 11226.0i	− 0.367607i	0.982963	−	0.183803i	$-0.0588409\pi$
$978$	0	0
$979$	45612.0	1.48904
$980$	0	0
$981$	0	0
$982$	− 33612.0i	− 1.09226i
$983$	23073.0i	0.748641i	0.927299	+	0.374321i	$0.122124\pi$
$983$	23073.0i	0.748641i	−0.927299	+	0.374321i	$0.877876\pi$
$984$	0	0
$985$	0	0
$986$	22320.0	0.720906
$987$	0	0
$988$	4720.00i	0.151987i
$989$	1602.00	0.0515072
$990$	0	0
$991$	22037.0	0.706386	0.353193	−	0.935551i	$-0.385096\pi$
$991$	22037.0	0.706386	0.353193	+	0.935551i	$0.385096\pi$
$992$	− 1504.00i	− 0.0481371i
$993$	0	0
$994$	5520.00	0.176141
$995$	0	0
$996$	0	0
$997$	− 19082.0i	− 0.606151i	−0.952966	−	0.303076i	$-0.901986\pi$
$997$	− 19082.0i	− 0.606151i	0.952966	−	0.303076i	$-0.0980135\pi$
$998$	− 10850.0i	− 0.344139i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.q.649.1		2
3.2	odd	2		1350.4.c.d.649.2		2
5.2	odd	4		270.4.a.i.1.1	yes	1
5.3	odd	4		1350.4.a.g.1.1		1
5.4	even	2	inner	1350.4.c.q.649.2		2
15.2	even	4		270.4.a.e.1.1	✓	1
15.8	even	4		1350.4.a.u.1.1		1
15.14	odd	2		1350.4.c.d.649.1		2
20.7	even	4		2160.4.a.e.1.1		1
45.2	even	12		810.4.e.q.271.1		2
45.7	odd	12		810.4.e.h.271.1		2
45.22	odd	12		810.4.e.h.541.1		2
45.32	even	12		810.4.e.q.541.1		2
60.47	odd	4		2160.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.e.1.1	✓	1	15.2	even	4
270.4.a.i.1.1	yes	1	5.2	odd	4
810.4.e.h.271.1		2	45.7	odd	12
810.4.e.h.541.1		2	45.22	odd	12
810.4.e.q.271.1		2	45.2	even	12
810.4.e.q.541.1		2	45.32	even	12
1350.4.a.g.1.1		1	5.3	odd	4
1350.4.a.u.1.1		1	15.8	even	4
1350.4.c.d.649.1		2	15.14	odd	2
1350.4.c.d.649.2		2	3.2	odd	2
1350.4.c.q.649.1		2	1.1	even	1	trivial
1350.4.c.q.649.2		2	5.4	even	2	inner
2160.4.a.e.1.1		1	20.7	even	4
2160.4.a.o.1.1		1	60.47	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +4.00000i q^{7} +8.00000i q^{8} +42.0000 q^{11} +20.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} -93.0000i q^{17} -59.0000 q^{19} -84.0000i q^{22} +9.00000i q^{23} +40.0000 q^{26} -16.0000i q^{28} -120.000 q^{29} +47.0000 q^{31} -32.0000i q^{32} -186.000 q^{34} +262.000i q^{37} +118.000i q^{38} +126.000 q^{41} -178.000i q^{43} -168.000 q^{44} +18.0000 q^{46} -144.000i q^{47} +327.000 q^{49} -80.0000i q^{52} +741.000i q^{53} -32.0000 q^{56} +240.000i q^{58} +444.000 q^{59} +221.000 q^{61} -94.0000i q^{62} -64.0000 q^{64} +538.000i q^{67} +372.000i q^{68} +690.000 q^{71} -1126.00i q^{73} +524.000 q^{74} +236.000 q^{76} +168.000i q^{77} -665.000 q^{79} -252.000i q^{82} +75.0000i q^{83} -356.000 q^{86} +336.000i q^{88} +1086.00 q^{89} -80.0000 q^{91} -36.0000i q^{92} -288.000 q^{94} -1544.00i q^{97} -654.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 84 q^{11} + 16 q^{14} + 32 q^{16} - 118 q^{19} + 80 q^{26} - 240 q^{29} + 94 q^{31} - 372 q^{34} + 252 q^{41} - 336 q^{44} + 36 q^{46} + 654 q^{49} - 64 q^{56} + 888 q^{59} + 442 q^{61}+ \cdots - 576 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 + 84 * q^11 + 16 * q^14 + 32 * q^16 - 118 * q^19 + 80 * q^26 - 240 * q^29 + 94 * q^31 - 372 * q^34 + 252 * q^41 - 336 * q^44 + 36 * q^46 + 654 * q^49 - 64 * q^56 + 888 * q^59 + 442 * q^61 - 128 * q^64 + 1380 * q^71 + 1048 * q^74 + 472 * q^76 - 1330 * q^79 - 712 * q^86 + 2172 * q^89 - 160 * q^91 - 576 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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