LMFDB - Embedded newform 1350.4.c.n.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$79.6525785077$
Analytic rank:	$1$
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.n.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} -8.00000i q^{7} +8.00000i q^{8} +18.0000 q^{11} +8.00000i q^{13} -16.0000 q^{14} +16.0000 q^{16} -15.0000i q^{17} -23.0000 q^{19} -36.0000i q^{22} +63.0000i q^{23} +16.0000 q^{26} +32.0000i q^{28} -156.000 q^{29} -85.0000 q^{31} -32.0000i q^{32} -30.0000 q^{34} -74.0000i q^{37} +46.0000i q^{38} +246.000 q^{41} -190.000i q^{43} -72.0000 q^{44} +126.000 q^{46} -288.000i q^{47} +279.000 q^{49} -32.0000i q^{52} -177.000i q^{53} +64.0000 q^{56} +312.000i q^{58} -792.000 q^{59} -907.000 q^{61} +170.000i q^{62} -64.0000 q^{64} +322.000i q^{67} +60.0000i q^{68} -270.000 q^{71} +254.000i q^{73} -148.000 q^{74} +92.0000 q^{76} -144.000i q^{77} +1123.00 q^{79} -492.000i q^{82} -771.000i q^{83} -380.000 q^{86} +144.000i q^{88} +198.000 q^{89} +64.0000 q^{91} -252.000i q^{92} -576.000 q^{94} +1192.00i q^{97} -558.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 36 q^{11} - 32 q^{14} + 32 q^{16} - 46 q^{19} + 32 q^{26} - 312 q^{29} - 170 q^{31} - 60 q^{34} + 492 q^{41} - 144 q^{44} + 252 q^{46} + 558 q^{49} + 128 q^{56} - 1584 q^{59} - 1814 q^{61}+ \cdots - 1152 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 + 36 * q^11 - 32 * q^14 + 32 * q^16 - 46 * q^19 + 32 * q^26 - 312 * q^29 - 170 * q^31 - 60 * q^34 + 492 * q^41 - 144 * q^44 + 252 * q^46 + 558 * q^49 + 128 * q^56 - 1584 * q^59 - 1814 * q^61 - 128 * q^64 - 540 * q^71 - 296 * q^74 + 184 * q^76 + 2246 * q^79 - 760 * q^86 + 396 * q^89 + 128 * q^91 - 1152 * 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
$7$	− 8.00000i	− 0.431959i	−0.976398	−	0.215980i	$-0.930705\pi$
$7$	− 8.00000i	− 0.431959i	0.976398	−	0.215980i	$-0.0692945\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	18.0000	0.493382	0.246691	−	0.969094i	$-0.420657\pi$
$11$	18.0000	0.493382	0.246691	+	0.969094i	$0.420657\pi$
$12$	0	0
$13$	8.00000i	0.170677i	0.996352	+	0.0853385i	$0.0271972\pi$
$13$	8.00000i	0.170677i	−0.996352	+	0.0853385i	$0.972803\pi$
$14$	−16.0000	−0.305441
$15$	0	0
$16$	16.0000	0.250000
$17$	− 15.0000i	− 0.214002i	−0.994259	−	0.107001i	$-0.965875\pi$
$17$	− 15.0000i	− 0.214002i	0.994259	−	0.107001i	$-0.0341248\pi$
$18$	0	0
$19$	−23.0000	−0.277714	−0.138857	−	0.990312i	$-0.544343\pi$
$19$	−23.0000	−0.277714	−0.138857	+	0.990312i	$0.544343\pi$
$20$	0	0
$21$	0	0
$22$	− 36.0000i	− 0.348874i
$23$	63.0000i	0.571148i	0.958357	+	0.285574i	$0.0921843\pi$
$23$	63.0000i	0.571148i	−0.958357	+	0.285574i	$0.907816\pi$
$24$	0	0
$25$	0	0
$26$	16.0000	0.120687
$27$	0	0
$28$	32.0000i	0.215980i
$29$	−156.000	−0.998913	−0.499456	−	0.866339i	$-0.666467\pi$
$29$	−156.000	−0.998913	−0.499456	+	0.866339i	$0.666467\pi$
$30$	0	0
$31$	−85.0000	−0.492466	−0.246233	−	0.969211i	$-0.579193\pi$
$31$	−85.0000	−0.492466	−0.246233	+	0.969211i	$0.579193\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	−30.0000	−0.151322
$35$	0	0
$36$	0	0
$37$	− 74.0000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 74.0000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	46.0000i	0.196373i
$39$	0	0
$40$	0	0
$41$	246.000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	246.000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 190.000i	− 0.673831i	−0.941535	−	0.336915i	$-0.890616\pi$
$43$	− 190.000i	− 0.673831i	0.941535	−	0.336915i	$-0.109384\pi$
$44$	−72.0000	−0.246691
$45$	0	0
$46$	126.000	0.403863
$47$	− 288.000i	− 0.893811i	−0.894581	−	0.446906i	$-0.852526\pi$
$47$	− 288.000i	− 0.893811i	0.894581	−	0.446906i	$-0.147474\pi$
$48$	0	0
$49$	279.000	0.813411
$50$	0	0
$51$	0	0
$52$	− 32.0000i	− 0.0853385i
$53$	− 177.000i	− 0.458732i	−0.973340	−	0.229366i	$-0.926335\pi$
$53$	− 177.000i	− 0.458732i	0.973340	−	0.229366i	$-0.0736653\pi$
$54$	0	0
$55$	0	0
$56$	64.0000	0.152721
$57$	0	0
$58$	312.000i	0.706338i
$59$	−792.000	−1.74762	−0.873810	−	0.486267i	$-0.838358\pi$
$59$	−792.000	−1.74762	−0.873810	+	0.486267i	$0.838358\pi$
$60$	0	0
$61$	−907.000	−1.90376	−0.951881	−	0.306469i	$-0.900853\pi$
$61$	−907.000	−1.90376	−0.951881	+	0.306469i	$0.900853\pi$
$62$	170.000i	0.348226i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	322.000i	0.587143i	0.955937	+	0.293571i	$0.0948438\pi$
$67$	322.000i	0.587143i	−0.955937	+	0.293571i	$0.905156\pi$
$68$	60.0000i	0.107001i
$69$	0	0
$70$	0	0
$71$	−270.000	−0.451311	−0.225656	−	0.974207i	$-0.572452\pi$
$71$	−270.000	−0.451311	−0.225656	+	0.974207i	$0.572452\pi$
$72$	0	0
$73$	254.000i	0.407239i	0.979050	+	0.203620i	$0.0652706\pi$
$73$	254.000i	0.407239i	−0.979050	+	0.203620i	$0.934729\pi$
$74$	−148.000	−0.232495
$75$	0	0
$76$	92.0000	0.138857
$77$	− 144.000i	− 0.213121i
$78$	0	0
$79$	1123.00	1.59933	0.799667	−	0.600444i	$-0.205009\pi$
$79$	1123.00	1.59933	0.799667	+	0.600444i	$0.205009\pi$
$80$	0	0
$81$	0	0
$82$	− 492.000i	− 0.662589i
$83$	− 771.000i	− 1.01962i	−0.860288	−	0.509809i	$-0.829716\pi$
$83$	− 771.000i	− 1.01962i	0.860288	−	0.509809i	$-0.170284\pi$
$84$	0	0
$85$	0	0
$86$	−380.000	−0.476470
$87$	0	0
$88$	144.000i	0.174437i
$89$	198.000	0.235820	0.117910	−	0.993024i	$-0.462381\pi$
$89$	198.000	0.235820	0.117910	+	0.993024i	$0.462381\pi$
$90$	0	0
$91$	64.0000	0.0737255
$92$	− 252.000i	− 0.285574i
$93$	0	0
$94$	−576.000	−0.632020
$95$	0	0
$96$	0	0
$97$	1192.00i	1.24772i	0.781534	+	0.623862i	$0.214437\pi$
$97$	1192.00i	1.24772i	−0.781534	+	0.623862i	$0.785563\pi$
$98$	− 558.000i	− 0.575168i
$99$	0	0
$100$	0	0
$101$	−1692.00	−1.66693	−0.833467	−	0.552570i	$-0.813647\pi$
$101$	−1692.00	−1.66693	−0.833467	+	0.552570i	$0.813647\pi$
$102$	0	0
$103$	1748.00i	1.67219i	0.548585	+	0.836095i	$0.315167\pi$
$103$	1748.00i	1.67219i	−0.548585	+	0.836095i	$0.684833\pi$
$104$	−64.0000	−0.0603434
$105$	0	0
$106$	−354.000	−0.324373
$107$	948.000i	0.856510i	0.903658	+	0.428255i	$0.140872\pi$
$107$	948.000i	0.856510i	−0.903658	+	0.428255i	$0.859128\pi$
$108$	0	0
$109$	−593.000	−0.521093	−0.260546	−	0.965461i	$-0.583903\pi$
$109$	−593.000	−0.521093	−0.260546	+	0.965461i	$0.583903\pi$
$110$	0	0
$111$	0	0
$112$	− 128.000i	− 0.107990i
$113$	− 1062.00i	− 0.884111i	−0.896988	−	0.442056i	$-0.854249\pi$
$113$	− 1062.00i	− 0.884111i	0.896988	−	0.442056i	$-0.145751\pi$
$114$	0	0
$115$	0	0
$116$	624.000	0.499456
$117$	0	0
$118$	1584.00i	1.23575i
$119$	−120.000	−0.0924402
$120$	0	0
$121$	−1007.00	−0.756574
$122$	1814.00i	1.34616i
$123$	0	0
$124$	340.000	0.246233
$125$	0	0
$126$	0	0
$127$	− 326.000i	− 0.227778i	−0.993493	−	0.113889i	$-0.963669\pi$
$127$	− 326.000i	− 0.227778i	0.993493	−	0.113889i	$-0.0363308\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−990.000	−0.660280	−0.330140	−	0.943932i	$-0.607096\pi$
$131$	−990.000	−0.660280	−0.330140	+	0.943932i	$0.607096\pi$
$132$	0	0
$133$	184.000i	0.119961i
$134$	644.000	0.415173
$135$	0	0
$136$	120.000	0.0756611
$137$	147.000i	0.0916720i	0.998949	+	0.0458360i	$0.0145952\pi$
$137$	147.000i	0.0916720i	−0.998949	+	0.0458360i	$0.985405\pi$
$138$	0	0
$139$	−1604.00	−0.978773	−0.489387	−	0.872067i	$-0.662779\pi$
$139$	−1604.00	−0.978773	−0.489387	+	0.872067i	$0.662779\pi$
$140$	0	0
$141$	0	0
$142$	540.000i	0.319125i
$143$	144.000i	0.0842090i
$144$	0	0
$145$	0	0
$146$	508.000	0.287962
$147$	0	0
$148$	296.000i	0.164399i
$149$	−1218.00	−0.669681	−0.334840	−	0.942275i	$-0.608682\pi$
$149$	−1218.00	−0.669681	−0.334840	+	0.942275i	$0.608682\pi$
$150$	0	0
$151$	−2248.00	−1.21152	−0.605760	−	0.795647i	$-0.707131\pi$
$151$	−2248.00	−1.21152	−0.605760	+	0.795647i	$0.707131\pi$
$152$	− 184.000i	− 0.0981866i
$153$	0	0
$154$	−288.000	−0.150699
$155$	0	0
$156$	0	0
$157$	2998.00i	1.52399i	0.647583	+	0.761995i	$0.275780\pi$
$157$	2998.00i	1.52399i	−0.647583	+	0.761995i	$0.724220\pi$
$158$	− 2246.00i	− 1.13090i
$159$	0	0
$160$	0	0
$161$	504.000	0.246713
$162$	0	0
$163$	3470.00i	1.66743i	0.552194	+	0.833716i	$0.313791\pi$
$163$	3470.00i	1.66743i	−0.552194	+	0.833716i	$0.686209\pi$
$164$	−984.000	−0.468521
$165$	0	0
$166$	−1542.00	−0.720978
$167$	− 387.000i	− 0.179323i	−0.995972	−	0.0896616i	$-0.971421\pi$
$167$	− 387.000i	− 0.179323i	0.995972	−	0.0896616i	$-0.0285785\pi$
$168$	0	0
$169$	2133.00	0.970869
$170$	0	0
$171$	0	0
$172$	760.000i	0.336915i
$173$	− 855.000i	− 0.375748i	−0.982193	−	0.187874i	$-0.939840\pi$
$173$	− 855.000i	− 0.375748i	0.982193	−	0.187874i	$-0.0601597\pi$
$174$	0	0
$175$	0	0
$176$	288.000	0.123346
$177$	0	0
$178$	− 396.000i	− 0.166750i
$179$	264.000	0.110236	0.0551181	−	0.998480i	$-0.482446\pi$
$179$	264.000	0.110236	0.0551181	+	0.998480i	$0.482446\pi$
$180$	0	0
$181$	−2551.00	−1.04759	−0.523797	−	0.851843i	$-0.675485\pi$
$181$	−2551.00	−1.04759	−0.523797	+	0.851843i	$0.675485\pi$
$182$	− 128.000i	− 0.0521318i
$183$	0	0
$184$	−504.000	−0.201931
$185$	0	0
$186$	0	0
$187$	− 270.000i	− 0.105585i
$188$	1152.00i	0.446906i
$189$	0	0
$190$	0	0
$191$	−2238.00	−0.847832	−0.423916	−	0.905701i	$-0.639345\pi$
$191$	−2238.00	−0.847832	−0.423916	+	0.905701i	$0.639345\pi$
$192$	0	0
$193$	2180.00i	0.813056i	0.913638	+	0.406528i	$0.133261\pi$
$193$	2180.00i	0.813056i	−0.913638	+	0.406528i	$0.866739\pi$
$194$	2384.00	0.882274
$195$	0	0
$196$	−1116.00	−0.406706
$197$	− 2577.00i	− 0.931998i	−0.884785	−	0.465999i	$-0.845695\pi$
$197$	− 2577.00i	− 0.931998i	0.884785	−	0.465999i	$-0.154305\pi$
$198$	0	0
$199$	−1412.00	−0.502985	−0.251493	−	0.967859i	$-0.580921\pi$
$199$	−1412.00	−0.502985	−0.251493	+	0.967859i	$0.580921\pi$
$200$	0	0
$201$	0	0
$202$	3384.00i	1.17870i
$203$	1248.00i	0.431490i
$204$	0	0
$205$	0	0
$206$	3496.00	1.18242
$207$	0	0
$208$	128.000i	0.0426692i
$209$	−414.000	−0.137019
$210$	0	0
$211$	−307.000	−0.100165	−0.0500823	−	0.998745i	$-0.515948\pi$
$211$	−307.000	−0.100165	−0.0500823	+	0.998745i	$0.515948\pi$
$212$	708.000i	0.229366i
$213$	0	0
$214$	1896.00	0.605644
$215$	0	0
$216$	0	0
$217$	680.000i	0.212725i
$218$	1186.00i	0.368468i
$219$	0	0
$220$	0	0
$221$	120.000	0.0365252
$222$	0	0
$223$	5234.00i	1.57172i	0.618402	+	0.785862i	$0.287781\pi$
$223$	5234.00i	1.57172i	−0.618402	+	0.785862i	$0.712219\pi$
$224$	−256.000	−0.0763604
$225$	0	0
$226$	−2124.00	−0.625161
$227$	− 1509.00i	− 0.441215i	−0.975363	−	0.220608i	$-0.929196\pi$
$227$	− 1509.00i	− 0.441215i	0.975363	−	0.220608i	$-0.0708040\pi$
$228$	0	0
$229$	−1211.00	−0.349455	−0.174727	−	0.984617i	$-0.555904\pi$
$229$	−1211.00	−0.349455	−0.174727	+	0.984617i	$0.555904\pi$
$230$	0	0
$231$	0	0
$232$	− 1248.00i	− 0.353169i
$233$	6246.00i	1.75618i	0.478499	+	0.878088i	$0.341181\pi$
$233$	6246.00i	1.75618i	−0.478499	+	0.878088i	$0.658819\pi$
$234$	0	0
$235$	0	0
$236$	3168.00	0.873810
$237$	0	0
$238$	240.000i	0.0653651i
$239$	−4650.00	−1.25851	−0.629254	−	0.777200i	$-0.716640\pi$
$239$	−4650.00	−1.25851	−0.629254	+	0.777200i	$0.716640\pi$
$240$	0	0
$241$	−3145.00	−0.840611	−0.420306	−	0.907383i	$-0.638077\pi$
$241$	−3145.00	−0.840611	−0.420306	+	0.907383i	$0.638077\pi$
$242$	2014.00i	0.534979i
$243$	0	0
$244$	3628.00	0.951881
$245$	0	0
$246$	0	0
$247$	− 184.000i	− 0.0473994i
$248$	− 680.000i	− 0.174113i
$249$	0	0
$250$	0	0
$251$	1020.00	0.256501	0.128251	−	0.991742i	$-0.459064\pi$
$251$	1020.00	0.256501	0.128251	+	0.991742i	$0.459064\pi$
$252$	0	0
$253$	1134.00i	0.281794i
$254$	−652.000	−0.161063
$255$	0	0
$256$	256.000	0.0625000
$257$	− 6741.00i	− 1.63616i	−0.575107	−	0.818078i	$-0.695040\pi$
$257$	− 6741.00i	− 1.63616i	0.575107	−	0.818078i	$-0.304960\pi$
$258$	0	0
$259$	−592.000	−0.142027
$260$	0	0
$261$	0	0
$262$	1980.00i	0.466889i
$263$	2340.00i	0.548633i	0.961639	+	0.274317i	$0.0884517\pi$
$263$	2340.00i	0.548633i	−0.961639	+	0.274317i	$0.911548\pi$
$264$	0	0
$265$	0	0
$266$	368.000	0.0848253
$267$	0	0
$268$	− 1288.00i	− 0.293571i
$269$	6198.00	1.40483	0.702414	−	0.711769i	$-0.252106\pi$
$269$	6198.00	1.40483	0.702414	+	0.711769i	$0.252106\pi$
$270$	0	0
$271$	875.000	0.196135	0.0980673	−	0.995180i	$-0.468734\pi$
$271$	875.000	0.196135	0.0980673	+	0.995180i	$0.468734\pi$
$272$	− 240.000i	− 0.0535005i
$273$	0	0
$274$	294.000	0.0648219
$275$	0	0
$276$	0	0
$277$	− 5486.00i	− 1.18997i	−0.803737	−	0.594985i	$-0.797158\pi$
$277$	− 5486.00i	− 1.18997i	0.803737	−	0.594985i	$-0.202842\pi$
$278$	3208.00i	0.692097i
$279$	0	0
$280$	0	0
$281$	−3204.00	−0.680194	−0.340097	−	0.940390i	$-0.610460\pi$
$281$	−3204.00	−0.680194	−0.340097	+	0.940390i	$0.610460\pi$
$282$	0	0
$283$	7322.00i	1.53798i	0.639262	+	0.768989i	$0.279240\pi$
$283$	7322.00i	1.53798i	−0.639262	+	0.768989i	$0.720760\pi$
$284$	1080.00	0.225656
$285$	0	0
$286$	288.000	0.0595447
$287$	− 1968.00i	− 0.404764i
$288$	0	0
$289$	4688.00	0.954203
$290$	0	0
$291$	0	0
$292$	− 1016.00i	− 0.203620i
$293$	1353.00i	0.269772i	0.990861	+	0.134886i	$0.0430668\pi$
$293$	1353.00i	0.269772i	−0.990861	+	0.134886i	$0.956933\pi$
$294$	0	0
$295$	0	0
$296$	592.000	0.116248
$297$	0	0
$298$	2436.00i	0.473536i
$299$	−504.000	−0.0974818
$300$	0	0
$301$	−1520.00	−0.291068
$302$	4496.00i	0.856675i
$303$	0	0
$304$	−368.000	−0.0694284
$305$	0	0
$306$	0	0
$307$	− 1658.00i	− 0.308231i	−0.988053	−	0.154116i	$-0.950747\pi$
$307$	− 1658.00i	− 0.308231i	0.988053	−	0.154116i	$-0.0492529\pi$
$308$	576.000i	0.106561i
$309$	0	0
$310$	0	0
$311$	−1044.00	−0.190353	−0.0951765	−	0.995460i	$-0.530342\pi$
$311$	−1044.00	−0.190353	−0.0951765	+	0.995460i	$0.530342\pi$
$312$	0	0
$313$	2588.00i	0.467356i	0.972314	+	0.233678i	$0.0750761\pi$
$313$	2588.00i	0.467356i	−0.972314	+	0.233678i	$0.924924\pi$
$314$	5996.00	1.07762
$315$	0	0
$316$	−4492.00	−0.799667
$317$	1449.00i	0.256732i	0.991727	+	0.128366i	$0.0409732\pi$
$317$	1449.00i	0.256732i	−0.991727	+	0.128366i	$0.959027\pi$
$318$	0	0
$319$	−2808.00	−0.492846
$320$	0	0
$321$	0	0
$322$	− 1008.00i	− 0.174452i
$323$	345.000i	0.0594313i
$324$	0	0
$325$	0	0
$326$	6940.00	1.17905
$327$	0	0
$328$	1968.00i	0.331295i
$329$	−2304.00	−0.386090
$330$	0	0
$331$	4880.00	0.810360	0.405180	−	0.914237i	$-0.367209\pi$
$331$	4880.00	0.810360	0.405180	+	0.914237i	$0.367209\pi$
$332$	3084.00i	0.509809i
$333$	0	0
$334$	−774.000	−0.126801
$335$	0	0
$336$	0	0
$337$	7744.00i	1.25176i	0.779920	+	0.625879i	$0.215260\pi$
$337$	7744.00i	1.25176i	−0.779920	+	0.625879i	$0.784740\pi$
$338$	− 4266.00i	− 0.686508i
$339$	0	0
$340$	0	0
$341$	−1530.00	−0.242974
$342$	0	0
$343$	− 4976.00i	− 0.783320i
$344$	1520.00	0.238235
$345$	0	0
$346$	−1710.00	−0.265694
$347$	− 804.000i	− 0.124383i	−0.998064	−	0.0621916i	$-0.980191\pi$
$347$	− 804.000i	− 0.124383i	0.998064	−	0.0621916i	$-0.0198090\pi$
$348$	0	0
$349$	2815.00	0.431758	0.215879	−	0.976420i	$-0.430738\pi$
$349$	2815.00	0.431758	0.215879	+	0.976420i	$0.430738\pi$
$350$	0	0
$351$	0	0
$352$	− 576.000i	− 0.0872185i
$353$	− 3738.00i	− 0.563608i	−0.959472	−	0.281804i	$-0.909067\pi$
$353$	− 3738.00i	− 0.563608i	0.959472	−	0.281804i	$-0.0909328\pi$
$354$	0	0
$355$	0	0
$356$	−792.000	−0.117910
$357$	0	0
$358$	− 528.000i	− 0.0779488i
$359$	11022.0	1.62039	0.810193	−	0.586163i	$-0.199362\pi$
$359$	11022.0	1.62039	0.810193	+	0.586163i	$0.199362\pi$
$360$	0	0
$361$	−6330.00	−0.922875
$362$	5102.00i	0.740760i
$363$	0	0
$364$	−256.000	−0.0368628
$365$	0	0
$366$	0	0
$367$	− 7544.00i	− 1.07301i	−0.843898	−	0.536504i	$-0.819745\pi$
$367$	− 7544.00i	− 1.07301i	0.843898	−	0.536504i	$-0.180255\pi$
$368$	1008.00i	0.142787i
$369$	0	0
$370$	0	0
$371$	−1416.00	−0.198154
$372$	0	0
$373$	− 5404.00i	− 0.750157i	−0.926993	−	0.375078i	$-0.877616\pi$
$373$	− 5404.00i	− 0.750157i	0.926993	−	0.375078i	$-0.122384\pi$
$374$	−540.000	−0.0746597
$375$	0	0
$376$	2304.00	0.316010
$377$	− 1248.00i	− 0.170491i
$378$	0	0
$379$	2335.00	0.316467	0.158233	−	0.987402i	$-0.449420\pi$
$379$	2335.00	0.316467	0.158233	+	0.987402i	$0.449420\pi$
$380$	0	0
$381$	0	0
$382$	4476.00i	0.599508i
$383$	− 6633.00i	− 0.884936i	−0.896784	−	0.442468i	$-0.854103\pi$
$383$	− 6633.00i	− 0.884936i	0.896784	−	0.442468i	$-0.145897\pi$
$384$	0	0
$385$	0	0
$386$	4360.00	0.574918
$387$	0	0
$388$	− 4768.00i	− 0.623862i
$389$	7566.00	0.986148	0.493074	−	0.869987i	$-0.335873\pi$
$389$	7566.00	0.986148	0.493074	+	0.869987i	$0.335873\pi$
$390$	0	0
$391$	945.000	0.122227
$392$	2232.00i	0.287584i
$393$	0	0
$394$	−5154.00	−0.659022
$395$	0	0
$396$	0	0
$397$	7420.00i	0.938033i	0.883189	+	0.469017i	$0.155392\pi$
$397$	7420.00i	0.938033i	−0.883189	+	0.469017i	$0.844608\pi$
$398$	2824.00i	0.355664i
$399$	0	0
$400$	0	0
$401$	−8502.00	−1.05878	−0.529389	−	0.848379i	$-0.677579\pi$
$401$	−8502.00	−1.05878	−0.529389	+	0.848379i	$0.677579\pi$
$402$	0	0
$403$	− 680.000i	− 0.0840526i
$404$	6768.00	0.833467
$405$	0	0
$406$	2496.00	0.305109
$407$	− 1332.00i	− 0.162223i
$408$	0	0
$409$	1903.00	0.230067	0.115033	−	0.993362i	$-0.463303\pi$
$409$	1903.00	0.230067	0.115033	+	0.993362i	$0.463303\pi$
$410$	0	0
$411$	0	0
$412$	− 6992.00i	− 0.836095i
$413$	6336.00i	0.754901i
$414$	0	0
$415$	0	0
$416$	256.000	0.0301717
$417$	0	0
$418$	828.000i	0.0968871i
$419$	−13482.0	−1.57193	−0.785965	−	0.618271i	$-0.787834\pi$
$419$	−13482.0	−1.57193	−0.785965	+	0.618271i	$0.787834\pi$
$420$	0	0
$421$	−1537.00	−0.177931	−0.0889653	−	0.996035i	$-0.528356\pi$
$421$	−1537.00	−0.177931	−0.0889653	+	0.996035i	$0.528356\pi$
$422$	614.000i	0.0708271i
$423$	0	0
$424$	1416.00	0.162186
$425$	0	0
$426$	0	0
$427$	7256.00i	0.822348i
$428$	− 3792.00i	− 0.428255i
$429$	0	0
$430$	0	0
$431$	−10368.0	−1.15872	−0.579361	−	0.815071i	$-0.696698\pi$
$431$	−10368.0	−1.15872	−0.579361	+	0.815071i	$0.696698\pi$
$432$	0	0
$433$	− 13168.0i	− 1.46146i	−0.682665	−	0.730732i	$-0.739179\pi$
$433$	− 13168.0i	− 1.46146i	0.682665	−	0.730732i	$-0.260821\pi$
$434$	1360.00	0.150420
$435$	0	0
$436$	2372.00	0.260546
$437$	− 1449.00i	− 0.158616i
$438$	0	0
$439$	−7319.00	−0.795710	−0.397855	−	0.917448i	$-0.630245\pi$
$439$	−7319.00	−0.795710	−0.397855	+	0.917448i	$0.630245\pi$
$440$	0	0
$441$	0	0
$442$	− 240.000i	− 0.0258272i
$443$	− 4119.00i	− 0.441760i	−0.975301	−	0.220880i	$-0.929107\pi$
$443$	− 4119.00i	− 0.441760i	0.975301	−	0.220880i	$-0.0708929\pi$
$444$	0	0
$445$	0	0
$446$	10468.0	1.11138
$447$	0	0
$448$	512.000i	0.0539949i
$449$	5388.00	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	5388.00	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	4428.00	0.462320
$452$	4248.00i	0.442056i
$453$	0	0
$454$	−3018.00	−0.311986
$455$	0	0
$456$	0	0
$457$	2752.00i	0.281692i	0.990032	+	0.140846i	$0.0449822\pi$
$457$	2752.00i	0.281692i	−0.990032	+	0.140846i	$0.955018\pi$
$458$	2422.00i	0.247102i
$459$	0	0
$460$	0	0
$461$	−4314.00	−0.435842	−0.217921	−	0.975966i	$-0.569927\pi$
$461$	−4314.00	−0.435842	−0.217921	+	0.975966i	$0.569927\pi$
$462$	0	0
$463$	− 5794.00i	− 0.581577i	−0.956787	−	0.290788i	$-0.906082\pi$
$463$	− 5794.00i	− 0.581577i	0.956787	−	0.290788i	$-0.0939176\pi$
$464$	−2496.00	−0.249728
$465$	0	0
$466$	12492.0	1.24180
$467$	− 6309.00i	− 0.625151i	−0.949893	−	0.312576i	$-0.898808\pi$
$467$	− 6309.00i	− 0.625151i	0.949893	−	0.312576i	$-0.101192\pi$
$468$	0	0
$469$	2576.00	0.253622
$470$	0	0
$471$	0	0
$472$	− 6336.00i	− 0.617877i
$473$	− 3420.00i	− 0.332456i
$474$	0	0
$475$	0	0
$476$	480.000	0.0462201
$477$	0	0
$478$	9300.00i	0.889900i
$479$	−14826.0	−1.41423	−0.707116	−	0.707097i	$-0.750004\pi$
$479$	−14826.0	−1.41423	−0.707116	+	0.707097i	$0.750004\pi$
$480$	0	0
$481$	592.000	0.0561182
$482$	6290.00i	0.594402i
$483$	0	0
$484$	4028.00	0.378287
$485$	0	0
$486$	0	0
$487$	− 6758.00i	− 0.628818i	−0.949288	−	0.314409i	$-0.898194\pi$
$487$	− 6758.00i	− 0.628818i	0.949288	−	0.314409i	$-0.101806\pi$
$488$	− 7256.00i	− 0.673081i
$489$	0	0
$490$	0	0
$491$	−14574.0	−1.33954	−0.669771	−	0.742567i	$-0.733608\pi$
$491$	−14574.0	−1.33954	−0.669771	+	0.742567i	$0.733608\pi$
$492$	0	0
$493$	2340.00i	0.213769i
$494$	−368.000	−0.0335164
$495$	0	0
$496$	−1360.00	−0.123117
$497$	2160.00i	0.194948i
$498$	0	0
$499$	−12611.0	−1.13135	−0.565677	−	0.824627i	$-0.691385\pi$
$499$	−12611.0	−1.13135	−0.565677	+	0.824627i	$0.691385\pi$
$500$	0	0
$501$	0	0
$502$	− 2040.00i	− 0.181374i
$503$	15639.0i	1.38630i	0.720794	+	0.693150i	$0.243778\pi$
$503$	15639.0i	1.38630i	−0.720794	+	0.693150i	$0.756222\pi$
$504$	0	0
$505$	0	0
$506$	2268.00	0.199259
$507$	0	0
$508$	1304.00i	0.113889i
$509$	−15420.0	−1.34279	−0.671394	−	0.741100i	$-0.734304\pi$
$509$	−15420.0	−1.34279	−0.671394	+	0.741100i	$0.734304\pi$
$510$	0	0
$511$	2032.00	0.175911
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	−13482.0	−1.15694
$515$	0	0
$516$	0	0
$517$	− 5184.00i	− 0.440990i
$518$	1184.00i	0.100429i
$519$	0	0
$520$	0	0
$521$	10494.0	0.882439	0.441219	−	0.897399i	$-0.354546\pi$
$521$	10494.0	0.882439	0.441219	+	0.897399i	$0.354546\pi$
$522$	0	0
$523$	− 10708.0i	− 0.895274i	−0.894215	−	0.447637i	$-0.852266\pi$
$523$	− 10708.0i	− 0.895274i	0.894215	−	0.447637i	$-0.147734\pi$
$524$	3960.00	0.330140
$525$	0	0
$526$	4680.00	0.387942
$527$	1275.00i	0.105389i
$528$	0	0
$529$	8198.00	0.673790
$530$	0	0
$531$	0	0
$532$	− 736.000i	− 0.0599805i
$533$	1968.00i	0.159932i
$534$	0	0
$535$	0	0
$536$	−2576.00	−0.207586
$537$	0	0
$538$	− 12396.0i	− 0.993363i
$539$	5022.00	0.401323
$540$	0	0
$541$	23030.0	1.83020	0.915099	−	0.403229i	$-0.132112\pi$
$541$	23030.0	1.83020	0.915099	+	0.403229i	$0.132112\pi$
$542$	− 1750.00i	− 0.138688i
$543$	0	0
$544$	−480.000	−0.0378306
$545$	0	0
$546$	0	0
$547$	3814.00i	0.298126i	0.988828	+	0.149063i	$0.0476257\pi$
$547$	3814.00i	0.298126i	−0.988828	+	0.149063i	$0.952374\pi$
$548$	− 588.000i	− 0.0458360i
$549$	0	0
$550$	0	0
$551$	3588.00	0.277412
$552$	0	0
$553$	− 8984.00i	− 0.690847i
$554$	−10972.0	−0.841436
$555$	0	0
$556$	6416.00	0.489387
$557$	− 22266.0i	− 1.69379i	−0.531761	−	0.846895i	$-0.678469\pi$
$557$	− 22266.0i	− 1.69379i	0.531761	−	0.846895i	$-0.321531\pi$
$558$	0	0
$559$	1520.00	0.115007
$560$	0	0
$561$	0	0
$562$	6408.00i	0.480970i
$563$	− 23844.0i	− 1.78491i	−0.451136	−	0.892455i	$-0.648981\pi$
$563$	− 23844.0i	− 1.78491i	0.451136	−	0.892455i	$-0.351019\pi$
$564$	0	0
$565$	0	0
$566$	14644.0	1.08751
$567$	0	0
$568$	− 2160.00i	− 0.159563i
$569$	−7488.00	−0.551693	−0.275846	−	0.961202i	$-0.588958\pi$
$569$	−7488.00	−0.551693	−0.275846	+	0.961202i	$0.588958\pi$
$570$	0	0
$571$	5111.00	0.374586	0.187293	−	0.982304i	$-0.440029\pi$
$571$	5111.00	0.374586	0.187293	+	0.982304i	$0.440029\pi$
$572$	− 576.000i	− 0.0421045i
$573$	0	0
$574$	−3936.00	−0.286212
$575$	0	0
$576$	0	0
$577$	− 6986.00i	− 0.504040i	−0.967722	−	0.252020i	$-0.918905\pi$
$577$	− 6986.00i	− 0.504040i	0.967722	−	0.252020i	$-0.0810948\pi$
$578$	− 9376.00i	− 0.674724i
$579$	0	0
$580$	0	0
$581$	−6168.00	−0.440433
$582$	0	0
$583$	− 3186.00i	− 0.226330i
$584$	−2032.00	−0.143981
$585$	0	0
$586$	2706.00	0.190757
$587$	− 20571.0i	− 1.44643i	−0.690622	−	0.723216i	$-0.742663\pi$
$587$	− 20571.0i	− 1.44643i	0.690622	−	0.723216i	$-0.257337\pi$
$588$	0	0
$589$	1955.00	0.136765
$590$	0	0
$591$	0	0
$592$	− 1184.00i	− 0.0821995i
$593$	23241.0i	1.60943i	0.593660	+	0.804716i	$0.297683\pi$
$593$	23241.0i	1.60943i	−0.593660	+	0.804716i	$0.702317\pi$
$594$	0	0
$595$	0	0
$596$	4872.00	0.334840
$597$	0	0
$598$	1008.00i	0.0689301i
$599$	−20208.0	−1.37842	−0.689212	−	0.724559i	$-0.742043\pi$
$599$	−20208.0	−1.37842	−0.689212	+	0.724559i	$0.742043\pi$
$600$	0	0
$601$	−9055.00	−0.614578	−0.307289	−	0.951616i	$-0.599422\pi$
$601$	−9055.00	−0.614578	−0.307289	+	0.951616i	$0.599422\pi$
$602$	3040.00i	0.205816i
$603$	0	0
$604$	8992.00	0.605760
$605$	0	0
$606$	0	0
$607$	− 15554.0i	− 1.04006i	−0.854148	−	0.520031i	$-0.825920\pi$
$607$	− 15554.0i	− 1.04006i	0.854148	−	0.520031i	$-0.174080\pi$
$608$	736.000i	0.0490933i
$609$	0	0
$610$	0	0
$611$	2304.00	0.152553
$612$	0	0
$613$	− 5632.00i	− 0.371084i	−0.982636	−	0.185542i	$-0.940596\pi$
$613$	− 5632.00i	− 0.371084i	0.982636	−	0.185542i	$-0.0594040\pi$
$614$	−3316.00	−0.217953
$615$	0	0
$616$	1152.00	0.0753497
$617$	9141.00i	0.596439i	0.954497	+	0.298219i	$0.0963927\pi$
$617$	9141.00i	0.596439i	−0.954497	+	0.298219i	$0.903607\pi$
$618$	0	0
$619$	13372.0	0.868281	0.434141	−	0.900845i	$-0.357052\pi$
$619$	13372.0	0.868281	0.434141	+	0.900845i	$0.357052\pi$
$620$	0	0
$621$	0	0
$622$	2088.00i	0.134600i
$623$	− 1584.00i	− 0.101865i
$624$	0	0
$625$	0	0
$626$	5176.00	0.330471
$627$	0	0
$628$	− 11992.0i	− 0.761995i
$629$	−1110.00	−0.0703634
$630$	0	0
$631$	11165.0	0.704392	0.352196	−	0.935926i	$-0.385435\pi$
$631$	11165.0	0.704392	0.352196	+	0.935926i	$0.385435\pi$
$632$	8984.00i	0.565450i
$633$	0	0
$634$	2898.00	0.181537
$635$	0	0
$636$	0	0
$637$	2232.00i	0.138831i
$638$	5616.00i	0.348495i
$639$	0	0
$640$	0	0
$641$	−912.000	−0.0561963	−0.0280982	−	0.999605i	$-0.508945\pi$
$641$	−912.000	−0.0561963	−0.0280982	+	0.999605i	$0.508945\pi$
$642$	0	0
$643$	− 27952.0i	− 1.71434i	−0.515035	−	0.857169i	$-0.672221\pi$
$643$	− 27952.0i	− 1.71434i	0.515035	−	0.857169i	$-0.327779\pi$
$644$	−2016.00	−0.123356
$645$	0	0
$646$	690.000	0.0420243
$647$	6285.00i	0.381899i	0.981600	+	0.190950i	$0.0611567\pi$
$647$	6285.00i	0.381899i	−0.981600	+	0.190950i	$0.938843\pi$
$648$	0	0
$649$	−14256.0	−0.862245
$650$	0	0
$651$	0	0
$652$	− 13880.0i	− 0.833716i
$653$	− 16497.0i	− 0.988633i	−0.869282	−	0.494317i	$-0.835418\pi$
$653$	− 16497.0i	− 0.988633i	0.869282	−	0.494317i	$-0.164582\pi$
$654$	0	0
$655$	0	0
$656$	3936.00	0.234261
$657$	0	0
$658$	4608.00i	0.273007i
$659$	−14844.0	−0.877451	−0.438725	−	0.898621i	$-0.644570\pi$
$659$	−14844.0	−0.877451	−0.438725	+	0.898621i	$0.644570\pi$
$660$	0	0
$661$	31934.0	1.87911	0.939553	−	0.342404i	$-0.111241\pi$
$661$	31934.0	1.87911	0.939553	+	0.342404i	$0.111241\pi$
$662$	− 9760.00i	− 0.573011i
$663$	0	0
$664$	6168.00	0.360489
$665$	0	0
$666$	0	0
$667$	− 9828.00i	− 0.570527i
$668$	1548.00i	0.0896616i
$669$	0	0
$670$	0	0
$671$	−16326.0	−0.939282
$672$	0	0
$673$	− 24352.0i	− 1.39480i	−0.716682	−	0.697400i	$-0.754340\pi$
$673$	− 24352.0i	− 1.39480i	0.716682	−	0.697400i	$-0.245660\pi$
$674$	15488.0	0.885127
$675$	0	0
$676$	−8532.00	−0.485435
$677$	10374.0i	0.588929i	0.955662	+	0.294465i	$0.0951413\pi$
$677$	10374.0i	0.588929i	−0.955662	+	0.294465i	$0.904859\pi$
$678$	0	0
$679$	9536.00	0.538966
$680$	0	0
$681$	0	0
$682$	3060.00i	0.171809i
$683$	− 7347.00i	− 0.411603i	−0.978594	−	0.205802i	$-0.934020\pi$
$683$	− 7347.00i	− 0.411603i	0.978594	−	0.205802i	$-0.0659802\pi$
$684$	0	0
$685$	0	0
$686$	−9952.00	−0.553891
$687$	0	0
$688$	− 3040.00i	− 0.168458i
$689$	1416.00	0.0782951
$690$	0	0
$691$	−5371.00	−0.295691	−0.147845	−	0.989010i	$-0.547234\pi$
$691$	−5371.00	−0.295691	−0.147845	+	0.989010i	$0.547234\pi$
$692$	3420.00i	0.187874i
$693$	0	0
$694$	−1608.00	−0.0879522
$695$	0	0
$696$	0	0
$697$	− 3690.00i	− 0.200529i
$698$	− 5630.00i	− 0.305299i
$699$	0	0
$700$	0	0
$701$	7086.00	0.381790	0.190895	−	0.981610i	$-0.438861\pi$
$701$	7086.00	0.381790	0.190895	+	0.981610i	$0.438861\pi$
$702$	0	0
$703$	1702.00i	0.0913117i
$704$	−1152.00	−0.0616728
$705$	0	0
$706$	−7476.00	−0.398531
$707$	13536.0i	0.720048i
$708$	0	0
$709$	−17186.0	−0.910344	−0.455172	−	0.890404i	$-0.650422\pi$
$709$	−17186.0	−0.910344	−0.455172	+	0.890404i	$0.650422\pi$
$710$	0	0
$711$	0	0
$712$	1584.00i	0.0833749i
$713$	− 5355.00i	− 0.281271i
$714$	0	0
$715$	0	0
$716$	−1056.00	−0.0551181
$717$	0	0
$718$	− 22044.0i	− 1.14579i
$719$	23814.0	1.23520	0.617602	−	0.786490i	$-0.288104\pi$
$719$	23814.0	1.23520	0.617602	+	0.786490i	$0.288104\pi$
$720$	0	0
$721$	13984.0	0.722318
$722$	12660.0i	0.652571i
$723$	0	0
$724$	10204.0	0.523797
$725$	0	0
$726$	0	0
$727$	22732.0i	1.15967i	0.814732	+	0.579837i	$0.196884\pi$
$727$	22732.0i	1.15967i	−0.814732	+	0.579837i	$0.803116\pi$
$728$	512.000i	0.0260659i
$729$	0	0
$730$	0	0
$731$	−2850.00	−0.144201
$732$	0	0
$733$	4664.00i	0.235019i	0.993072	+	0.117509i	$0.0374910\pi$
$733$	4664.00i	0.235019i	−0.993072	+	0.117509i	$0.962509\pi$
$734$	−15088.0	−0.758731
$735$	0	0
$736$	2016.00	0.100966
$737$	5796.00i	0.289686i
$738$	0	0
$739$	−5501.00	−0.273826	−0.136913	−	0.990583i	$-0.543718\pi$
$739$	−5501.00	−0.273826	−0.136913	+	0.990583i	$0.543718\pi$
$740$	0	0
$741$	0	0
$742$	2832.00i	0.140116i
$743$	− 27096.0i	− 1.33789i	−0.743310	−	0.668947i	$-0.766745\pi$
$743$	− 27096.0i	− 1.33789i	0.743310	−	0.668947i	$-0.233255\pi$
$744$	0	0
$745$	0	0
$746$	−10808.0	−0.530441
$747$	0	0
$748$	1080.00i	0.0527924i
$749$	7584.00	0.369978
$750$	0	0
$751$	−5659.00	−0.274967	−0.137483	−	0.990504i	$-0.543901\pi$
$751$	−5659.00	−0.274967	−0.137483	+	0.990504i	$0.543901\pi$
$752$	− 4608.00i	− 0.223453i
$753$	0	0
$754$	−2496.00	−0.120556
$755$	0	0
$756$	0	0
$757$	− 37694.0i	− 1.80979i	−0.425634	−	0.904895i	$-0.639949\pi$
$757$	− 37694.0i	− 1.80979i	0.425634	−	0.904895i	$-0.360051\pi$
$758$	− 4670.00i	− 0.223776i
$759$	0	0
$760$	0	0
$761$	−6588.00	−0.313817	−0.156909	−	0.987613i	$-0.550153\pi$
$761$	−6588.00	−0.313817	−0.156909	+	0.987613i	$0.550153\pi$
$762$	0	0
$763$	4744.00i	0.225091i
$764$	8952.00	0.423916
$765$	0	0
$766$	−13266.0	−0.625744
$767$	− 6336.00i	− 0.298279i
$768$	0	0
$769$	19.0000	0.000890972 0	0.000445486	−	1.00000i	$-0.499858\pi$
$769$	19.0000	0.000890972 0	0.000445486		1.00000i	$0.499858\pi$
$770$	0	0
$771$	0	0
$772$	− 8720.00i	− 0.406528i
$773$	33639.0i	1.56521i	0.622516	+	0.782607i	$0.286111\pi$
$773$	33639.0i	1.56521i	−0.622516	+	0.782607i	$0.713889\pi$
$774$	0	0
$775$	0	0
$776$	−9536.00	−0.441137
$777$	0	0
$778$	− 15132.0i	− 0.697312i
$779$	−5658.00	−0.260230
$780$	0	0
$781$	−4860.00	−0.222669
$782$	− 1890.00i	− 0.0864274i
$783$	0	0
$784$	4464.00	0.203353
$785$	0	0
$786$	0	0
$787$	− 23474.0i	− 1.06322i	−0.846988	−	0.531612i	$-0.821586\pi$
$787$	− 23474.0i	− 1.06322i	0.846988	−	0.531612i	$-0.178414\pi$
$788$	10308.0i	0.465999i
$789$	0	0
$790$	0	0
$791$	−8496.00	−0.381900
$792$	0	0
$793$	− 7256.00i	− 0.324928i
$794$	14840.0	0.663290
$795$	0	0
$796$	5648.00	0.251493
$797$	− 7917.00i	− 0.351863i	−0.984402	−	0.175931i	$-0.943706\pi$
$797$	− 7917.00i	− 0.351863i	0.984402	−	0.175931i	$-0.0562937\pi$
$798$	0	0
$799$	−4320.00	−0.191277
$800$	0	0
$801$	0	0
$802$	17004.0i	0.748668i
$803$	4572.00i	0.200925i
$804$	0	0
$805$	0	0
$806$	−1360.00	−0.0594342
$807$	0	0
$808$	− 13536.0i	− 0.589350i
$809$	41202.0	1.79059	0.895294	−	0.445476i	$-0.146966\pi$
$809$	41202.0	1.79059	0.895294	+	0.445476i	$0.146966\pi$
$810$	0	0
$811$	35492.0	1.53674	0.768368	−	0.640008i	$-0.221069\pi$
$811$	35492.0	1.53674	0.768368	+	0.640008i	$0.221069\pi$
$812$	− 4992.00i	− 0.215745i
$813$	0	0
$814$	−2664.00	−0.114709
$815$	0	0
$816$	0	0
$817$	4370.00i	0.187132i
$818$	− 3806.00i	− 0.162682i
$819$	0	0
$820$	0	0
$821$	−7146.00	−0.303772	−0.151886	−	0.988398i	$-0.548535\pi$
$821$	−7146.00	−0.303772	−0.151886	+	0.988398i	$0.548535\pi$
$822$	0	0
$823$	8882.00i	0.376193i	0.982151	+	0.188097i	$0.0602318\pi$
$823$	8882.00i	0.376193i	−0.982151	+	0.188097i	$0.939768\pi$
$824$	−13984.0	−0.591208
$825$	0	0
$826$	12672.0	0.533796
$827$	21705.0i	0.912644i	0.889815	+	0.456322i	$0.150834\pi$
$827$	21705.0i	0.912644i	−0.889815	+	0.456322i	$0.849166\pi$
$828$	0	0
$829$	−29018.0	−1.21573	−0.607863	−	0.794042i	$-0.707973\pi$
$829$	−29018.0	−1.21573	−0.607863	+	0.794042i	$0.707973\pi$
$830$	0	0
$831$	0	0
$832$	− 512.000i	− 0.0213346i
$833$	− 4185.00i	− 0.174072i
$834$	0	0
$835$	0	0
$836$	1656.00	0.0685095
$837$	0	0
$838$	26964.0i	1.11152i
$839$	−31164.0	−1.28236	−0.641180	−	0.767390i	$-0.721555\pi$
$839$	−31164.0	−1.28236	−0.641180	+	0.767390i	$0.721555\pi$
$840$	0	0
$841$	−53.0000	−0.00217311
$842$	3074.00i	0.125816i
$843$	0	0
$844$	1228.00	0.0500823
$845$	0	0
$846$	0	0
$847$	8056.00i	0.326809i
$848$	− 2832.00i	− 0.114683i
$849$	0	0
$850$	0	0
$851$	4662.00	0.187792
$852$	0	0
$853$	49160.0i	1.97328i	0.162921	+	0.986639i	$0.447908\pi$
$853$	49160.0i	1.97328i	−0.162921	+	0.986639i	$0.552092\pi$
$854$	14512.0	0.581488
$855$	0	0
$856$	−7584.00	−0.302822
$857$	2349.00i	0.0936293i	0.998904	+	0.0468147i	$0.0149070\pi$
$857$	2349.00i	0.0936293i	−0.998904	+	0.0468147i	$0.985093\pi$
$858$	0	0
$859$	28195.0	1.11991	0.559954	−	0.828524i	$-0.310819\pi$
$859$	28195.0	1.11991	0.559954	+	0.828524i	$0.310819\pi$
$860$	0	0
$861$	0	0
$862$	20736.0i	0.819340i
$863$	− 23997.0i	− 0.946544i	−0.880916	−	0.473272i	$-0.843073\pi$
$863$	− 23997.0i	− 0.946544i	0.880916	−	0.473272i	$-0.156927\pi$
$864$	0	0
$865$	0	0
$866$	−26336.0	−1.03341
$867$	0	0
$868$	− 2720.00i	− 0.106363i
$869$	20214.0	0.789083
$870$	0	0
$871$	−2576.00	−0.100212
$872$	− 4744.00i	− 0.184234i
$873$	0	0
$874$	−2898.00	−0.112158
$875$	0	0
$876$	0	0
$877$	− 46286.0i	− 1.78217i	−0.453832	−	0.891087i	$-0.649943\pi$
$877$	− 46286.0i	− 1.78217i	0.453832	−	0.891087i	$-0.350057\pi$
$878$	14638.0i	0.562652i
$879$	0	0
$880$	0	0
$881$	39636.0	1.51574	0.757872	−	0.652403i	$-0.226239\pi$
$881$	39636.0	1.51574	0.757872	+	0.652403i	$0.226239\pi$
$882$	0	0
$883$	− 16744.0i	− 0.638143i	−0.947731	−	0.319072i	$-0.896629\pi$
$883$	− 16744.0i	− 0.638143i	0.947731	−	0.319072i	$-0.103371\pi$
$884$	−480.000	−0.0182626
$885$	0	0
$886$	−8238.00	−0.312371
$887$	− 1251.00i	− 0.0473557i	−0.999720	−	0.0236778i	$-0.992462\pi$
$887$	− 1251.00i	− 0.0473557i	0.999720	−	0.0236778i	$-0.00753759\pi$
$888$	0	0
$889$	−2608.00	−0.0983909
$890$	0	0
$891$	0	0
$892$	− 20936.0i	− 0.785862i
$893$	6624.00i	0.248224i
$894$	0	0
$895$	0	0
$896$	1024.00	0.0381802
$897$	0	0
$898$	− 10776.0i	− 0.400445i
$899$	13260.0	0.491931
$900$	0	0
$901$	−2655.00	−0.0981697
$902$	− 8856.00i	− 0.326910i
$903$	0	0
$904$	8496.00	0.312580
$905$	0	0
$906$	0	0
$907$	36988.0i	1.35410i	0.735938	+	0.677049i	$0.236741\pi$
$907$	36988.0i	1.35410i	−0.735938	+	0.677049i	$0.763259\pi$
$908$	6036.00i	0.220608i
$909$	0	0
$910$	0	0
$911$	16404.0	0.596585	0.298292	−	0.954475i	$-0.403583\pi$
$911$	16404.0	0.596585	0.298292	+	0.954475i	$0.403583\pi$
$912$	0	0
$913$	− 13878.0i	− 0.503061i
$914$	5504.00	0.199186
$915$	0	0
$916$	4844.00	0.174727
$917$	7920.00i	0.285214i
$918$	0	0
$919$	664.000	0.0238339	0.0119169	−	0.999929i	$-0.496207\pi$
$919$	664.000	0.0238339	0.0119169	+	0.999929i	$0.496207\pi$
$920$	0	0
$921$	0	0
$922$	8628.00i	0.308187i
$923$	− 2160.00i	− 0.0770285i
$924$	0	0
$925$	0	0
$926$	−11588.0	−0.411237
$927$	0	0
$928$	4992.00i	0.176585i
$929$	39642.0	1.40001	0.700006	−	0.714137i	$-0.253180\pi$
$929$	39642.0	1.40001	0.700006	+	0.714137i	$0.253180\pi$
$930$	0	0
$931$	−6417.00	−0.225895
$932$	− 24984.0i	− 0.878088i
$933$	0	0
$934$	−12618.0	−0.442049
$935$	0	0
$936$	0	0
$937$	36028.0i	1.25612i	0.778165	+	0.628059i	$0.216151\pi$
$937$	36028.0i	1.25612i	−0.778165	+	0.628059i	$0.783849\pi$
$938$	− 5152.00i	− 0.179338i
$939$	0	0
$940$	0	0
$941$	23058.0	0.798798	0.399399	−	0.916777i	$-0.369219\pi$
$941$	23058.0	0.798798	0.399399	+	0.916777i	$0.369219\pi$
$942$	0	0
$943$	15498.0i	0.535190i
$944$	−12672.0	−0.436905
$945$	0	0
$946$	−6840.00	−0.235082
$947$	− 19953.0i	− 0.684673i	−0.939577	−	0.342337i	$-0.888782\pi$
$947$	− 19953.0i	− 0.684673i	0.939577	−	0.342337i	$-0.111218\pi$
$948$	0	0
$949$	−2032.00	−0.0695063
$950$	0	0
$951$	0	0
$952$	− 960.000i	− 0.0326825i
$953$	25638.0i	0.871455i	0.900079	+	0.435727i	$0.143509\pi$
$953$	25638.0i	0.871455i	−0.900079	+	0.435727i	$0.856491\pi$
$954$	0	0
$955$	0	0
$956$	18600.0	0.629254
$957$	0	0
$958$	29652.0i	1.00001i
$959$	1176.00	0.0395986
$960$	0	0
$961$	−22566.0	−0.757477
$962$	− 1184.00i	− 0.0396816i
$963$	0	0
$964$	12580.0	0.420306
$965$	0	0
$966$	0	0
$967$	27034.0i	0.899023i	0.893275	+	0.449511i	$0.148402\pi$
$967$	27034.0i	0.899023i	−0.893275	+	0.449511i	$0.851598\pi$
$968$	− 8056.00i	− 0.267489i
$969$	0	0
$970$	0	0
$971$	14802.0	0.489206	0.244603	−	0.969623i	$-0.421342\pi$
$971$	14802.0	0.489206	0.244603	+	0.969623i	$0.421342\pi$
$972$	0	0
$973$	12832.0i	0.422790i
$974$	−13516.0	−0.444641
$975$	0	0
$976$	−14512.0	−0.475940
$977$	9186.00i	0.300805i	0.988625	+	0.150402i	$0.0480569\pi$
$977$	9186.00i	0.300805i	−0.988625	+	0.150402i	$0.951943\pi$
$978$	0	0
$979$	3564.00	0.116349
$980$	0	0
$981$	0	0
$982$	29148.0i	0.947200i
$983$	31647.0i	1.02684i	0.858138	+	0.513419i	$0.171621\pi$
$983$	31647.0i	1.02684i	−0.858138	+	0.513419i	$0.828379\pi$
$984$	0	0
$985$	0	0
$986$	4680.00	0.151158
$987$	0	0
$988$	736.000i	0.0236997i
$989$	11970.0	0.384857
$990$	0	0
$991$	−48823.0	−1.56500	−0.782499	−	0.622651i	$-0.786055\pi$
$991$	−48823.0	−1.56500	−0.782499	+	0.622651i	$0.786055\pi$
$992$	2720.00i	0.0870565i
$993$	0	0
$994$	4320.00	0.137849
$995$	0	0
$996$	0	0
$997$	13066.0i	0.415050i	0.978230	+	0.207525i	$0.0665408\pi$
$997$	13066.0i	0.415050i	−0.978230	+	0.207525i	$0.933459\pi$
$998$	25222.0i	0.799988i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.n.649.1		2
3.2	odd	2		1350.4.c.g.649.2		2
5.2	odd	4		270.4.a.l.1.1	yes	1
5.3	odd	4		1350.4.a.f.1.1		1
5.4	even	2	inner	1350.4.c.n.649.2		2
15.2	even	4		270.4.a.b.1.1	✓	1
15.8	even	4		1350.4.a.t.1.1		1
15.14	odd	2		1350.4.c.g.649.1		2
20.7	even	4		2160.4.a.m.1.1		1
45.2	even	12		810.4.e.v.271.1		2
45.7	odd	12		810.4.e.b.271.1		2
45.22	odd	12		810.4.e.b.541.1		2
45.32	even	12		810.4.e.v.541.1		2
60.47	odd	4		2160.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.b.1.1	✓	1	15.2	even	4
270.4.a.l.1.1	yes	1	5.2	odd	4
810.4.e.b.271.1		2	45.7	odd	12
810.4.e.b.541.1		2	45.22	odd	12
810.4.e.v.271.1		2	45.2	even	12
810.4.e.v.541.1		2	45.32	even	12
1350.4.a.f.1.1		1	5.3	odd	4
1350.4.a.t.1.1		1	15.8	even	4
1350.4.c.g.649.1		2	15.14	odd	2
1350.4.c.g.649.2		2	3.2	odd	2
1350.4.c.n.649.1		2	1.1	even	1	trivial
1350.4.c.n.649.2		2	5.4	even	2	inner
2160.4.a.c.1.1		1	60.47	odd	4
2160.4.a.m.1.1		1	20.7	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} -8.00000i q^{7} +8.00000i q^{8} +18.0000 q^{11} +8.00000i q^{13} -16.0000 q^{14} +16.0000 q^{16} -15.0000i q^{17} -23.0000 q^{19} -36.0000i q^{22} +63.0000i q^{23} +16.0000 q^{26} +32.0000i q^{28} -156.000 q^{29} -85.0000 q^{31} -32.0000i q^{32} -30.0000 q^{34} -74.0000i q^{37} +46.0000i q^{38} +246.000 q^{41} -190.000i q^{43} -72.0000 q^{44} +126.000 q^{46} -288.000i q^{47} +279.000 q^{49} -32.0000i q^{52} -177.000i q^{53} +64.0000 q^{56} +312.000i q^{58} -792.000 q^{59} -907.000 q^{61} +170.000i q^{62} -64.0000 q^{64} +322.000i q^{67} +60.0000i q^{68} -270.000 q^{71} +254.000i q^{73} -148.000 q^{74} +92.0000 q^{76} -144.000i q^{77} +1123.00 q^{79} -492.000i q^{82} -771.000i q^{83} -380.000 q^{86} +144.000i q^{88} +198.000 q^{89} +64.0000 q^{91} -252.000i q^{92} -576.000 q^{94} +1192.00i q^{97} -558.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 36 q^{11} - 32 q^{14} + 32 q^{16} - 46 q^{19} + 32 q^{26} - 312 q^{29} - 170 q^{31} - 60 q^{34} + 492 q^{41} - 144 q^{44} + 252 q^{46} + 558 q^{49} + 128 q^{56} - 1584 q^{59} - 1814 q^{61}+ \cdots - 1152 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 + 36 * q^11 - 32 * q^14 + 32 * q^16 - 46 * q^19 + 32 * q^26 - 312 * q^29 - 170 * q^31 - 60 * q^34 + 492 * q^41 - 144 * q^44 + 252 * q^46 + 558 * q^49 + 128 * q^56 - 1584 * q^59 - 1814 * q^61 - 128 * q^64 - 540 * q^71 - 296 * q^74 + 184 * q^76 + 2246 * q^79 - 760 * q^86 + 396 * q^89 + 128 * q^91 - 1152 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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