LMFDB - Embedded newform 1200.4.f.r.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -4.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -4.00000i q^{7} -9.00000 q^{9} +48.0000 q^{11} +2.00000i q^{13} +114.000i q^{17} +140.000 q^{19} -12.0000 q^{21} -72.0000i q^{23} +27.0000i q^{27} -210.000 q^{29} -272.000 q^{31} -144.000i q^{33} +334.000i q^{37} +6.00000 q^{39} -198.000 q^{41} +268.000i q^{43} +216.000i q^{47} +327.000 q^{49} +342.000 q^{51} -78.0000i q^{53} -420.000i q^{57} +240.000 q^{59} +302.000 q^{61} +36.0000i q^{63} +596.000i q^{67} -216.000 q^{69} +768.000 q^{71} -478.000i q^{73} -192.000i q^{77} -640.000 q^{79} +81.0000 q^{81} +348.000i q^{83} +630.000i q^{87} -210.000 q^{89} +8.00000 q^{91} +816.000i q^{93} +1534.00i q^{97} -432.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} + 96 q^{11} + 280 q^{19} - 24 q^{21} - 420 q^{29} - 544 q^{31} + 12 q^{39} - 396 q^{41} + 654 q^{49} + 684 q^{51} + 480 q^{59} + 604 q^{61} - 432 q^{69} + 1536 q^{71} - 1280 q^{79} + 162 q^{81} - 420 q^{89} + 16 q^{91} - 864 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 96 * q^11 + 280 * q^19 - 24 * q^21 - 420 * q^29 - 544 * q^31 + 12 * q^39 - 396 * q^41 + 654 * q^49 + 684 * q^51 + 480 * q^59 + 604 * q^61 - 432 * q^69 + 1536 * q^71 - 1280 * q^79 + 162 * q^81 - 420 * q^89 + 16 * q^91 - 864 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$	− 4.00000i	− 0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	48.0000	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	0	0
$13$	2.00000i	0.0426692i	0.999772	+	0.0213346i	$0.00679154\pi$
$13$	2.00000i	0.0426692i	−0.999772	+	0.0213346i	$0.993208\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	114.000i	1.62642i	0.581974	+	0.813208i	$0.302281\pi$
$17$	114.000i	1.62642i	−0.581974	+	0.813208i	$0.697719\pi$
$18$	0	0
$19$	140.000	1.69043	0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000	1.69043	0.845216	+	0.534425i	$0.179472\pi$
$20$	0	0
$21$	−12.0000	−0.124696
$22$	0	0
$23$	− 72.0000i	− 0.652741i	−0.945242	−	0.326370i	$-0.894174\pi$
$23$	− 72.0000i	− 0.652741i	0.945242	−	0.326370i	$-0.105826\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	−272.000	−1.57589	−0.787946	−	0.615745i	$-0.788855\pi$
$31$	−272.000	−1.57589	−0.787946	+	0.615745i	$0.788855\pi$
$32$	0	0
$33$	− 144.000i	− 0.759612i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	334.000i	1.48403i	0.670381	+	0.742017i	$0.266131\pi$
$37$	334.000i	1.48403i	−0.670381	+	0.742017i	$0.733869\pi$
$38$	0	0
$39$	6.00000	0.0246351
$40$	0	0
$41$	−198.000	−0.754205	−0.377102	−	0.926172i	$-0.623080\pi$
$41$	−198.000	−0.754205	−0.377102	+	0.926172i	$0.623080\pi$
$42$	0	0
$43$	268.000i	0.950456i	0.879863	+	0.475228i	$0.157634\pi$
$43$	268.000i	0.950456i	−0.879863	+	0.475228i	$0.842366\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	216.000i	0.670358i	0.942154	+	0.335179i	$0.108797\pi$
$47$	216.000i	0.670358i	−0.942154	+	0.335179i	$0.891203\pi$
$48$	0	0
$49$	327.000	0.953353
$50$	0	0
$51$	342.000	0.939011
$52$	0	0
$53$	− 78.0000i	− 0.202153i	−0.994879	−	0.101077i	$-0.967771\pi$
$53$	− 78.0000i	− 0.202153i	0.994879	−	0.101077i	$-0.0322287\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 420.000i	− 0.975971i
$58$	0	0
$59$	240.000	0.529582	0.264791	−	0.964306i	$-0.414697\pi$
$59$	240.000	0.529582	0.264791	+	0.964306i	$0.414697\pi$
$60$	0	0
$61$	302.000	0.633888	0.316944	−	0.948444i	$-0.397343\pi$
$61$	302.000	0.633888	0.316944	+	0.948444i	$0.397343\pi$
$62$	0	0
$63$	36.0000i	0.0719932i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	596.000i	1.08676i	0.839487	+	0.543381i	$0.182856\pi$
$67$	596.000i	1.08676i	−0.839487	+	0.543381i	$0.817144\pi$
$68$	0	0
$69$	−216.000	−0.376860
$70$	0	0
$71$	768.000	1.28373	0.641865	−	0.766818i	$-0.278161\pi$
$71$	768.000	1.28373	0.641865	+	0.766818i	$0.278161\pi$
$72$	0	0
$73$	− 478.000i	− 0.766379i	−0.923670	−	0.383190i	$-0.874826\pi$
$73$	− 478.000i	− 0.766379i	0.923670	−	0.383190i	$-0.125174\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 192.000i	− 0.284161i
$78$	0	0
$79$	−640.000	−0.911464	−0.455732	−	0.890117i	$-0.650622\pi$
$79$	−640.000	−0.911464	−0.455732	+	0.890117i	$0.650622\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	348.000i	0.460216i	0.973165	+	0.230108i	$0.0739080\pi$
$83$	348.000i	0.460216i	−0.973165	+	0.230108i	$0.926092\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	630.000i	0.776357i
$88$	0	0
$89$	−210.000	−0.250112	−0.125056	−	0.992150i	$-0.539911\pi$
$89$	−210.000	−0.250112	−0.125056	+	0.992150i	$0.539911\pi$
$90$	0	0
$91$	8.00000	0.00921569
$92$	0	0
$93$	816.000i	0.909841i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1534.00i	1.60571i	0.596173	+	0.802856i	$0.296687\pi$
$97$	1534.00i	1.60571i	−0.596173	+	0.802856i	$0.703313\pi$
$98$	0	0
$99$	−432.000	−0.438562
$100$	0	0
$101$	1722.00	1.69649	0.848245	−	0.529605i	$-0.177660\pi$
$101$	1722.00	1.69649	0.848245	+	0.529605i	$0.177660\pi$
$102$	0	0
$103$	− 1052.00i	− 1.00638i	−0.864177	−	0.503188i	$-0.832160\pi$
$103$	− 1052.00i	− 1.00638i	0.864177	−	0.503188i	$-0.167840\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 564.000i	− 0.509570i	−0.966998	−	0.254785i	$-0.917995\pi$
$107$	− 564.000i	− 0.509570i	0.966998	−	0.254785i	$-0.0820046\pi$
$108$	0	0
$109$	610.000	0.536031	0.268016	−	0.963415i	$-0.413632\pi$
$109$	610.000	0.536031	0.268016	+	0.963415i	$0.413632\pi$
$110$	0	0
$111$	1002.00	0.856807
$112$	0	0
$113$	1302.00i	1.08391i	0.840407	+	0.541955i	$0.182316\pi$
$113$	1302.00i	1.08391i	−0.840407	+	0.541955i	$0.817684\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 18.0000i	− 0.0142231i
$118$	0	0
$119$	456.000	0.351273
$120$	0	0
$121$	973.000	0.731029
$122$	0	0
$123$	594.000i	0.435440i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 124.000i	− 0.0866395i	−0.999061	−	0.0433198i	$-0.986207\pi$
$127$	− 124.000i	− 0.0866395i	0.999061	−	0.0433198i	$-0.0137934\pi$
$128$	0	0
$129$	804.000	0.548746
$130$	0	0
$131$	−192.000	−0.128054	−0.0640272	−	0.997948i	$-0.520394\pi$
$131$	−192.000	−0.128054	−0.0640272	+	0.997948i	$0.520394\pi$
$132$	0	0
$133$	− 560.000i	− 0.365099i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2514.00i	1.56778i	0.620901	+	0.783889i	$0.286767\pi$
$137$	2514.00i	1.56778i	−0.620901	+	0.783889i	$0.713233\pi$
$138$	0	0
$139$	1340.00	0.817679	0.408839	−	0.912606i	$-0.365934\pi$
$139$	1340.00	0.817679	0.408839	+	0.912606i	$0.365934\pi$
$140$	0	0
$141$	648.000	0.387032
$142$	0	0
$143$	96.0000i	0.0561393i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 981.000i	− 0.550418i
$148$	0	0
$149$	−1410.00	−0.775246	−0.387623	−	0.921818i	$-0.626704\pi$
$149$	−1410.00	−0.775246	−0.387623	+	0.921818i	$0.626704\pi$
$150$	0	0
$151$	2128.00	1.14685	0.573424	−	0.819258i	$-0.305615\pi$
$151$	2128.00	1.14685	0.573424	+	0.819258i	$0.305615\pi$
$152$	0	0
$153$	− 1026.00i	− 0.542138i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3026.00i	− 1.53822i	−0.639114	−	0.769112i	$-0.720699\pi$
$157$	− 3026.00i	− 1.53822i	0.639114	−	0.769112i	$-0.279301\pi$
$158$	0	0
$159$	−234.000	−0.116713
$160$	0	0
$161$	−288.000	−0.140979
$162$	0	0
$163$	− 2612.00i	− 1.25514i	−0.778561	−	0.627569i	$-0.784050\pi$
$163$	− 2612.00i	− 1.25514i	0.778561	−	0.627569i	$-0.215950\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 24.0000i	− 0.0111208i	−0.999985	−	0.00556041i	$-0.998230\pi$
$167$	− 24.0000i	− 0.0111208i	0.999985	−	0.00556041i	$-0.00176994\pi$
$168$	0	0
$169$	2193.00	0.998179
$170$	0	0
$171$	−1260.00	−0.563477
$172$	0	0
$173$	1962.00i	0.862243i	0.902294	+	0.431122i	$0.141882\pi$
$173$	1962.00i	0.862243i	−0.902294	+	0.431122i	$0.858118\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 720.000i	− 0.305754i
$178$	0	0
$179$	−120.000	−0.0501074	−0.0250537	−	0.999686i	$-0.507976\pi$
$179$	−120.000	−0.0501074	−0.0250537	+	0.999686i	$0.507976\pi$
$180$	0	0
$181$	902.000	0.370415	0.185208	−	0.982699i	$-0.440704\pi$
$181$	902.000	0.370415	0.185208	+	0.982699i	$0.440704\pi$
$182$	0	0
$183$	− 906.000i	− 0.365975i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5472.00i	2.13985i
$188$	0	0
$189$	108.000	0.0415653
$190$	0	0
$191$	168.000	0.0636443	0.0318221	−	0.999494i	$-0.489869\pi$
$191$	168.000	0.0636443	0.0318221	+	0.999494i	$0.489869\pi$
$192$	0	0
$193$	− 1318.00i	− 0.491563i	−0.969325	−	0.245782i	$-0.920955\pi$
$193$	− 1318.00i	− 0.491563i	0.969325	−	0.245782i	$-0.0790446\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4014.00i	1.45170i	0.687851	+	0.725852i	$0.258554\pi$
$197$	4014.00i	1.45170i	−0.687851	+	0.725852i	$0.741446\pi$
$198$	0	0
$199$	2000.00	0.712443	0.356222	−	0.934401i	$-0.384065\pi$
$199$	2000.00	0.712443	0.356222	+	0.934401i	$0.384065\pi$
$200$	0	0
$201$	1788.00	0.627442
$202$	0	0
$203$	840.000i	0.290426i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	648.000i	0.217580i
$208$	0	0
$209$	6720.00	2.22408
$210$	0	0
$211$	3868.00	1.26201	0.631005	−	0.775779i	$-0.282643\pi$
$211$	3868.00	1.26201	0.631005	+	0.775779i	$0.282643\pi$
$212$	0	0
$213$	− 2304.00i	− 0.741162i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1088.00i	0.340361i
$218$	0	0
$219$	−1434.00	−0.442469
$220$	0	0
$221$	−228.000	−0.0693979
$222$	0	0
$223$	3148.00i	0.945317i	0.881246	+	0.472658i	$0.156706\pi$
$223$	3148.00i	0.945317i	−0.881246	+	0.472658i	$0.843294\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2556.00i	0.747347i	0.927560	+	0.373673i	$0.121902\pi$
$227$	2556.00i	0.747347i	−0.927560	+	0.373673i	$0.878098\pi$
$228$	0	0
$229$	610.000	0.176026	0.0880130	−	0.996119i	$-0.471948\pi$
$229$	610.000	0.176026	0.0880130	+	0.996119i	$0.471948\pi$
$230$	0	0
$231$	−576.000	−0.164061
$232$	0	0
$233$	− 2058.00i	− 0.578644i	−0.957232	−	0.289322i	$-0.906570\pi$
$233$	− 2058.00i	− 0.578644i	0.957232	−	0.289322i	$-0.0934298\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1920.00i	0.526234i
$238$	0	0
$239$	4920.00	1.33158	0.665792	−	0.746138i	$-0.268094\pi$
$239$	4920.00	1.33158	0.665792	+	0.746138i	$0.268094\pi$
$240$	0	0
$241$	−1438.00	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$241$	−1438.00	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	280.000i	0.0721294i
$248$	0	0
$249$	1044.00	0.265706
$250$	0	0
$251$	−792.000	−0.199166	−0.0995829	−	0.995029i	$-0.531751\pi$
$251$	−792.000	−0.199166	−0.0995829	+	0.995029i	$0.531751\pi$
$252$	0	0
$253$	− 3456.00i	− 0.858802i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2166.00i	− 0.525725i	−0.964833	−	0.262863i	$-0.915333\pi$
$257$	− 2166.00i	− 0.525725i	0.964833	−	0.262863i	$-0.0846666\pi$
$258$	0	0
$259$	1336.00	0.320521
$260$	0	0
$261$	1890.00	0.448230
$262$	0	0
$263$	− 3192.00i	− 0.748392i	−0.927350	−	0.374196i	$-0.877919\pi$
$263$	− 3192.00i	− 0.748392i	0.927350	−	0.374196i	$-0.122081\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	630.000i	0.144402i
$268$	0	0
$269$	−5490.00	−1.24435	−0.622177	−	0.782877i	$-0.713752\pi$
$269$	−5490.00	−1.24435	−0.622177	+	0.782877i	$0.713752\pi$
$270$	0	0
$271$	6328.00	1.41845	0.709223	−	0.704985i	$-0.249046\pi$
$271$	6328.00	1.41845	0.709223	+	0.704985i	$0.249046\pi$
$272$	0	0
$273$	− 24.0000i	− 0.00532068i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	574.000i	0.124507i	0.998060	+	0.0622533i	$0.0198287\pi$
$277$	574.000i	0.124507i	−0.998060	+	0.0622533i	$0.980171\pi$
$278$	0	0
$279$	2448.00	0.525297
$280$	0	0
$281$	4242.00	0.900557	0.450278	−	0.892888i	$-0.351325\pi$
$281$	4242.00	0.900557	0.450278	+	0.892888i	$0.351325\pi$
$282$	0	0
$283$	628.000i	0.131911i	0.997823	+	0.0659553i	$0.0210095\pi$
$283$	628.000i	0.131911i	−0.997823	+	0.0659553i	$0.978991\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	792.000i	0.162893i
$288$	0	0
$289$	−8083.00	−1.64523
$290$	0	0
$291$	4602.00	0.927058
$292$	0	0
$293$	− 558.000i	− 0.111258i	−0.998451	−	0.0556292i	$-0.982284\pi$
$293$	− 558.000i	− 0.111258i	0.998451	−	0.0556292i	$-0.0177165\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1296.00i	0.253204i
$298$	0	0
$299$	144.000	0.0278520
$300$	0	0
$301$	1072.00	0.205279
$302$	0	0
$303$	− 5166.00i	− 0.979468i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6964.00i	− 1.29465i	−0.762216	−	0.647323i	$-0.775888\pi$
$307$	− 6964.00i	− 1.29465i	0.762216	−	0.647323i	$-0.224112\pi$
$308$	0	0
$309$	−3156.00	−0.581031
$310$	0	0
$311$	−2832.00	−0.516360	−0.258180	−	0.966097i	$-0.583123\pi$
$311$	−2832.00	−0.516360	−0.258180	+	0.966097i	$0.583123\pi$
$312$	0	0
$313$	8642.00i	1.56062i	0.625392	+	0.780311i	$0.284939\pi$
$313$	8642.00i	1.56062i	−0.625392	+	0.780311i	$0.715061\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2214.00i	0.392273i	0.980577	+	0.196137i	$0.0628396\pi$
$317$	2214.00i	0.392273i	−0.980577	+	0.196137i	$0.937160\pi$
$318$	0	0
$319$	−10080.0	−1.76919
$320$	0	0
$321$	−1692.00	−0.294200
$322$	0	0
$323$	15960.0i	2.74934i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1830.00i	− 0.309478i
$328$	0	0
$329$	864.000	0.144784
$330$	0	0
$331$	−10772.0	−1.78877	−0.894385	−	0.447299i	$-0.852386\pi$
$331$	−10772.0	−1.78877	−0.894385	+	0.447299i	$0.852386\pi$
$332$	0	0
$333$	− 3006.00i	− 0.494678i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1654.00i	0.267356i	0.991025	+	0.133678i	$0.0426789\pi$
$337$	1654.00i	0.267356i	−0.991025	+	0.133678i	$0.957321\pi$
$338$	0	0
$339$	3906.00	0.625796
$340$	0	0
$341$	−13056.0	−2.07338
$342$	0	0
$343$	− 2680.00i	− 0.421885i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2196.00i	0.339733i	0.985467	+	0.169867i	$0.0543337\pi$
$347$	2196.00i	0.339733i	−0.985467	+	0.169867i	$0.945666\pi$
$348$	0	0
$349$	−8270.00	−1.26843	−0.634216	−	0.773156i	$-0.718677\pi$
$349$	−8270.00	−1.26843	−0.634216	+	0.773156i	$0.718677\pi$
$350$	0	0
$351$	−54.0000	−0.00821170
$352$	0	0
$353$	10302.0i	1.55331i	0.629923	+	0.776657i	$0.283086\pi$
$353$	10302.0i	1.55331i	−0.629923	+	0.776657i	$0.716914\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1368.00i	− 0.202807i
$358$	0	0
$359$	−2280.00	−0.335192	−0.167596	−	0.985856i	$-0.553600\pi$
$359$	−2280.00	−0.335192	−0.167596	+	0.985856i	$0.553600\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	0	0
$363$	− 2919.00i	− 0.422060i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8764.00i	− 1.24653i	−0.782010	−	0.623266i	$-0.785805\pi$
$367$	− 8764.00i	− 1.24653i	0.782010	−	0.623266i	$-0.214195\pi$
$368$	0	0
$369$	1782.00	0.251402
$370$	0	0
$371$	−312.000	−0.0436610
$372$	0	0
$373$	− 1318.00i	− 0.182958i	−0.995807	−	0.0914792i	$-0.970841\pi$
$373$	− 1318.00i	− 0.182958i	0.995807	−	0.0914792i	$-0.0291595\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 420.000i	− 0.0573769i
$378$	0	0
$379$	1100.00	0.149085	0.0745425	−	0.997218i	$-0.476250\pi$
$379$	1100.00	0.149085	0.0745425	+	0.997218i	$0.476250\pi$
$380$	0	0
$381$	−372.000	−0.0500214
$382$	0	0
$383$	3528.00i	0.470685i	0.971912	+	0.235343i	$0.0756212\pi$
$383$	3528.00i	0.470685i	−0.971912	+	0.235343i	$0.924379\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2412.00i	− 0.316819i
$388$	0	0
$389$	9630.00	1.25517	0.627584	−	0.778549i	$-0.284044\pi$
$389$	9630.00	1.25517	0.627584	+	0.778549i	$0.284044\pi$
$390$	0	0
$391$	8208.00	1.06163
$392$	0	0
$393$	576.000i	0.0739322i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3094.00i	0.391142i	0.980690	+	0.195571i	$0.0626560\pi$
$397$	3094.00i	0.391142i	−0.980690	+	0.195571i	$0.937344\pi$
$398$	0	0
$399$	−1680.00	−0.210790
$400$	0	0
$401$	−1638.00	−0.203985	−0.101992	−	0.994785i	$-0.532522\pi$
$401$	−1638.00	−0.203985	−0.101992	+	0.994785i	$0.532522\pi$
$402$	0	0
$403$	− 544.000i	− 0.0672421i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16032.0i	1.95252i
$408$	0	0
$409$	13750.0	1.66233	0.831166	−	0.556024i	$-0.187674\pi$
$409$	13750.0	1.66233	0.831166	+	0.556024i	$0.187674\pi$
$410$	0	0
$411$	7542.00	0.905157
$412$	0	0
$413$	− 960.000i	− 0.114379i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4020.00i	− 0.472087i
$418$	0	0
$419$	−12480.0	−1.45510	−0.727551	−	0.686053i	$-0.759342\pi$
$419$	−12480.0	−1.45510	−0.727551	+	0.686053i	$0.759342\pi$
$420$	0	0
$421$	7262.00	0.840685	0.420342	−	0.907366i	$-0.361910\pi$
$421$	7262.00	0.840685	0.420342	+	0.907366i	$0.361910\pi$
$422$	0	0
$423$	− 1944.00i	− 0.223453i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1208.00i	− 0.136907i
$428$	0	0
$429$	288.000	0.0324121
$430$	0	0
$431$	−9792.00	−1.09435	−0.547174	−	0.837019i	$-0.684296\pi$
$431$	−9792.00	−1.09435	−0.547174	+	0.837019i	$0.684296\pi$
$432$	0	0
$433$	1802.00i	0.199997i	0.994988	+	0.0999984i	$0.0318838\pi$
$433$	1802.00i	0.199997i	−0.994988	+	0.0999984i	$0.968116\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 10080.0i	− 1.10341i
$438$	0	0
$439$	−2320.00	−0.252227	−0.126113	−	0.992016i	$-0.540250\pi$
$439$	−2320.00	−0.252227	−0.126113	+	0.992016i	$0.540250\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	0	0
$443$	− 11172.0i	− 1.19819i	−0.800678	−	0.599095i	$-0.795527\pi$
$443$	− 11172.0i	− 1.19819i	0.800678	−	0.599095i	$-0.204473\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4230.00i	0.447589i
$448$	0	0
$449$	−6810.00	−0.715777	−0.357888	−	0.933764i	$-0.616503\pi$
$449$	−6810.00	−0.715777	−0.357888	+	0.933764i	$0.616503\pi$
$450$	0	0
$451$	−9504.00	−0.992297
$452$	0	0
$453$	− 6384.00i	− 0.662134i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17066.0i	− 1.74686i	−0.486952	−	0.873429i	$-0.661891\pi$
$457$	− 17066.0i	− 1.74686i	0.486952	−	0.873429i	$-0.338109\pi$
$458$	0	0
$459$	−3078.00	−0.313004
$460$	0	0
$461$	−18918.0	−1.91128	−0.955639	−	0.294541i	$-0.904833\pi$
$461$	−18918.0	−1.91128	−0.955639	+	0.294541i	$0.904833\pi$
$462$	0	0
$463$	− 1052.00i	− 0.105595i	−0.998605	−	0.0527976i	$-0.983186\pi$
$463$	− 1052.00i	− 0.105595i	0.998605	−	0.0527976i	$-0.0168138\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11076.0i	1.09751i	0.835984	+	0.548754i	$0.184898\pi$
$467$	11076.0i	1.09751i	−0.835984	+	0.548754i	$0.815102\pi$
$468$	0	0
$469$	2384.00	0.234718
$470$	0	0
$471$	−9078.00	−0.888094
$472$	0	0
$473$	12864.0i	1.25050i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	702.000i	0.0673844i
$478$	0	0
$479$	−9000.00	−0.858498	−0.429249	−	0.903186i	$-0.641222\pi$
$479$	−9000.00	−0.858498	−0.429249	+	0.903186i	$0.641222\pi$
$480$	0	0
$481$	−668.000	−0.0633226
$482$	0	0
$483$	864.000i	0.0813941i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8764.00i	− 0.815472i	−0.913100	−	0.407736i	$-0.866318\pi$
$487$	− 8764.00i	− 0.815472i	0.913100	−	0.407736i	$-0.133682\pi$
$488$	0	0
$489$	−7836.00	−0.724655
$490$	0	0
$491$	−5592.00	−0.513978	−0.256989	−	0.966414i	$-0.582730\pi$
$491$	−5592.00	−0.513978	−0.256989	+	0.966414i	$0.582730\pi$
$492$	0	0
$493$	− 23940.0i	− 2.18703i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3072.00i	− 0.277260i
$498$	0	0
$499$	4700.00	0.421645	0.210823	−	0.977524i	$-0.432386\pi$
$499$	4700.00	0.421645	0.210823	+	0.977524i	$0.432386\pi$
$500$	0	0
$501$	−72.0000	−0.00642060
$502$	0	0
$503$	11808.0i	1.04671i	0.852116	+	0.523353i	$0.175319\pi$
$503$	11808.0i	1.04671i	−0.852116	+	0.523353i	$0.824681\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6579.00i	− 0.576299i
$508$	0	0
$509$	−1170.00	−0.101885	−0.0509424	−	0.998702i	$-0.516222\pi$
$509$	−1170.00	−0.101885	−0.0509424	+	0.998702i	$0.516222\pi$
$510$	0	0
$511$	−1912.00	−0.165522
$512$	0	0
$513$	3780.00i	0.325324i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10368.0i	0.881981i
$518$	0	0
$519$	5886.00	0.497816
$520$	0	0
$521$	−16638.0	−1.39909	−0.699543	−	0.714590i	$-0.746613\pi$
$521$	−16638.0	−1.39909	−0.699543	+	0.714590i	$0.746613\pi$
$522$	0	0
$523$	− 15692.0i	− 1.31198i	−0.754771	−	0.655988i	$-0.772252\pi$
$523$	− 15692.0i	− 1.31198i	0.754771	−	0.655988i	$-0.227748\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 31008.0i	− 2.56305i
$528$	0	0
$529$	6983.00	0.573929
$530$	0	0
$531$	−2160.00	−0.176527
$532$	0	0
$533$	− 396.000i	− 0.0321814i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	360.000i	0.0289295i
$538$	0	0
$539$	15696.0	1.25431
$540$	0	0
$541$	−22018.0	−1.74977	−0.874887	−	0.484327i	$-0.839064\pi$
$541$	−22018.0	−1.74977	−0.874887	+	0.484327i	$0.839064\pi$
$542$	0	0
$543$	− 2706.00i	− 0.213859i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4564.00i	− 0.356751i	−0.983963	−	0.178375i	$-0.942916\pi$
$547$	− 4564.00i	− 0.356751i	0.983963	−	0.178375i	$-0.0570841\pi$
$548$	0	0
$549$	−2718.00	−0.211296
$550$	0	0
$551$	−29400.0	−2.27311
$552$	0	0
$553$	2560.00i	0.196858i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7734.00i	0.588331i	0.955755	+	0.294165i	$0.0950416\pi$
$557$	7734.00i	0.588331i	−0.955755	+	0.294165i	$0.904958\pi$
$558$	0	0
$559$	−536.000	−0.0405552
$560$	0	0
$561$	16416.0	1.23544
$562$	0	0
$563$	20148.0i	1.50824i	0.656739	+	0.754118i	$0.271935\pi$
$563$	20148.0i	1.50824i	−0.656739	+	0.754118i	$0.728065\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 324.000i	− 0.0239977i
$568$	0	0
$569$	24030.0	1.77046	0.885228	−	0.465156i	$-0.154002\pi$
$569$	24030.0	1.77046	0.885228	+	0.465156i	$0.154002\pi$
$570$	0	0
$571$	−2372.00	−0.173844	−0.0869222	−	0.996215i	$-0.527703\pi$
$571$	−2372.00	−0.173844	−0.0869222	+	0.996215i	$0.527703\pi$
$572$	0	0
$573$	− 504.000i	− 0.0367450i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8546.00i	− 0.616594i	−0.951290	−	0.308297i	$-0.900241\pi$
$577$	− 8546.00i	− 0.616594i	0.951290	−	0.308297i	$-0.0997590\pi$
$578$	0	0
$579$	−3954.00	−0.283804
$580$	0	0
$581$	1392.00	0.0993974
$582$	0	0
$583$	− 3744.00i	− 0.265970i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15444.0i	− 1.08593i	−0.839755	−	0.542966i	$-0.817301\pi$
$587$	− 15444.0i	− 1.08593i	0.839755	−	0.542966i	$-0.182699\pi$
$588$	0	0
$589$	−38080.0	−2.66394
$590$	0	0
$591$	12042.0	0.838142
$592$	0	0
$593$	18342.0i	1.27018i	0.772439	+	0.635089i	$0.219037\pi$
$593$	18342.0i	1.27018i	−0.772439	+	0.635089i	$0.780963\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6000.00i	− 0.411329i
$598$	0	0
$599$	24600.0	1.67801	0.839006	−	0.544123i	$-0.183137\pi$
$599$	24600.0	1.67801	0.839006	+	0.544123i	$0.183137\pi$
$600$	0	0
$601$	−8998.00	−0.610709	−0.305354	−	0.952239i	$-0.598775\pi$
$601$	−8998.00	−0.610709	−0.305354	+	0.952239i	$0.598775\pi$
$602$	0	0
$603$	− 5364.00i	− 0.362254i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4076.00i	0.272553i	0.990671	+	0.136277i	$0.0435136\pi$
$607$	4076.00i	0.272553i	−0.990671	+	0.136277i	$0.956486\pi$
$608$	0	0
$609$	2520.00	0.167677
$610$	0	0
$611$	−432.000	−0.0286037
$612$	0	0
$613$	− 4078.00i	− 0.268693i	−0.990934	−	0.134347i	$-0.957106\pi$
$613$	− 4078.00i	− 0.268693i	0.990934	−	0.134347i	$-0.0428935\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10086.0i	− 0.658099i	−0.944313	−	0.329049i	$-0.893272\pi$
$617$	− 10086.0i	− 0.658099i	0.944313	−	0.329049i	$-0.106728\pi$
$618$	0	0
$619$	8780.00	0.570110	0.285055	−	0.958511i	$-0.407988\pi$
$619$	8780.00	0.570110	0.285055	+	0.958511i	$0.407988\pi$
$620$	0	0
$621$	1944.00	0.125620
$622$	0	0
$623$	840.000i	0.0540191i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 20160.0i	− 1.28407i
$628$	0	0
$629$	−38076.0	−2.41366
$630$	0	0
$631$	−2792.00	−0.176145	−0.0880727	−	0.996114i	$-0.528071\pi$
$631$	−2792.00	−0.176145	−0.0880727	+	0.996114i	$0.528071\pi$
$632$	0	0
$633$	− 11604.0i	− 0.728622i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	654.000i	0.0406788i
$638$	0	0
$639$	−6912.00	−0.427910
$640$	0	0
$641$	7602.00	0.468426	0.234213	−	0.972185i	$-0.424749\pi$
$641$	7602.00	0.468426	0.234213	+	0.972185i	$0.424749\pi$
$642$	0	0
$643$	− 24212.0i	− 1.48496i	−0.669869	−	0.742479i	$-0.733650\pi$
$643$	− 24212.0i	− 1.48496i	0.669869	−	0.742479i	$-0.266350\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9456.00i	0.574581i	0.957844	+	0.287290i	$0.0927545\pi$
$647$	9456.00i	0.574581i	−0.957844	+	0.287290i	$0.907246\pi$
$648$	0	0
$649$	11520.0	0.696764
$650$	0	0
$651$	3264.00	0.196507
$652$	0	0
$653$	− 9558.00i	− 0.572792i	−0.958111	−	0.286396i	$-0.907543\pi$
$653$	− 9558.00i	− 0.572792i	0.958111	−	0.286396i	$-0.0924574\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4302.00i	0.255460i
$658$	0	0
$659$	−29280.0	−1.73078	−0.865392	−	0.501095i	$-0.832931\pi$
$659$	−29280.0	−1.73078	−0.865392	+	0.501095i	$0.832931\pi$
$660$	0	0
$661$	−29098.0	−1.71223	−0.856113	−	0.516789i	$-0.827127\pi$
$661$	−29098.0	−1.71223	−0.856113	+	0.516789i	$0.827127\pi$
$662$	0	0
$663$	684.000i	0.0400669i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15120.0i	0.877734i
$668$	0	0
$669$	9444.00	0.545779
$670$	0	0
$671$	14496.0	0.833997
$672$	0	0
$673$	− 11638.0i	− 0.666585i	−0.942823	−	0.333293i	$-0.891840\pi$
$673$	− 11638.0i	− 0.666585i	0.942823	−	0.333293i	$-0.108160\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3426.00i	− 0.194493i	−0.995260	−	0.0972466i	$-0.968996\pi$
$677$	− 3426.00i	− 0.194493i	0.995260	−	0.0972466i	$-0.0310035\pi$
$678$	0	0
$679$	6136.00	0.346801
$680$	0	0
$681$	7668.00	0.431481
$682$	0	0
$683$	20148.0i	1.12876i	0.825516	+	0.564379i	$0.190884\pi$
$683$	20148.0i	1.12876i	−0.825516	+	0.564379i	$0.809116\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1830.00i	− 0.101629i
$688$	0	0
$689$	156.000	0.00862573
$690$	0	0
$691$	29428.0	1.62011	0.810053	−	0.586356i	$-0.199438\pi$
$691$	29428.0	1.62011	0.810053	+	0.586356i	$0.199438\pi$
$692$	0	0
$693$	1728.00i	0.0947205i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 22572.0i	− 1.22665i
$698$	0	0
$699$	−6174.00	−0.334080
$700$	0	0
$701$	16242.0	0.875110	0.437555	−	0.899192i	$-0.355845\pi$
$701$	16242.0	0.875110	0.437555	+	0.899192i	$0.355845\pi$
$702$	0	0
$703$	46760.0i	2.50866i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6888.00i	− 0.366407i
$708$	0	0
$709$	−2030.00	−0.107529	−0.0537646	−	0.998554i	$-0.517122\pi$
$709$	−2030.00	−0.107529	−0.0537646	+	0.998554i	$0.517122\pi$
$710$	0	0
$711$	5760.00	0.303821
$712$	0	0
$713$	19584.0i	1.02865i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 14760.0i	− 0.768790i
$718$	0	0
$719$	6960.00	0.361007	0.180504	−	0.983574i	$-0.442227\pi$
$719$	6960.00	0.361007	0.180504	+	0.983574i	$0.442227\pi$
$720$	0	0
$721$	−4208.00	−0.217357
$722$	0	0
$723$	4314.00i	0.221908i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18596.0i	0.948676i	0.880343	+	0.474338i	$0.157313\pi$
$727$	18596.0i	0.948676i	−0.880343	+	0.474338i	$0.842687\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−30552.0	−1.54584
$732$	0	0
$733$	21242.0i	1.07038i	0.844731	+	0.535192i	$0.179761\pi$
$733$	21242.0i	1.07038i	−0.844731	+	0.535192i	$0.820239\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28608.0i	1.42984i
$738$	0	0
$739$	−340.000	−0.0169244	−0.00846218	−	0.999964i	$-0.502694\pi$
$739$	−340.000	−0.0169244	−0.00846218	+	0.999964i	$0.502694\pi$
$740$	0	0
$741$	840.000	0.0416440
$742$	0	0
$743$	21888.0i	1.08074i	0.841426	+	0.540372i	$0.181716\pi$
$743$	21888.0i	1.08074i	−0.841426	+	0.540372i	$0.818284\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3132.00i	− 0.153405i
$748$	0	0
$749$	−2256.00	−0.110057
$750$	0	0
$751$	−17792.0	−0.864500	−0.432250	−	0.901754i	$-0.642280\pi$
$751$	−17792.0	−0.864500	−0.432250	+	0.901754i	$0.642280\pi$
$752$	0	0
$753$	2376.00i	0.114988i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 37346.0i	− 1.79308i	−0.442960	−	0.896541i	$-0.646072\pi$
$757$	− 37346.0i	− 1.79308i	0.442960	−	0.896541i	$-0.353928\pi$
$758$	0	0
$759$	−10368.0	−0.495829
$760$	0	0
$761$	−11358.0	−0.541034	−0.270517	−	0.962715i	$-0.587195\pi$
$761$	−11358.0	−0.541034	−0.270517	+	0.962715i	$0.587195\pi$
$762$	0	0
$763$	− 2440.00i	− 0.115772i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	480.000i	0.0225969i
$768$	0	0
$769$	34270.0	1.60703	0.803516	−	0.595283i	$-0.202960\pi$
$769$	34270.0	1.60703	0.803516	+	0.595283i	$0.202960\pi$
$770$	0	0
$771$	−6498.00	−0.303528
$772$	0	0
$773$	− 13278.0i	− 0.617822i	−0.951091	−	0.308911i	$-0.900035\pi$
$773$	− 13278.0i	− 0.617822i	0.951091	−	0.308911i	$-0.0999645\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4008.00i	− 0.185053i
$778$	0	0
$779$	−27720.0	−1.27493
$780$	0	0
$781$	36864.0	1.68899
$782$	0	0
$783$	− 5670.00i	− 0.258786i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 11164.0i	− 0.505659i	−0.967511	−	0.252829i	$-0.918639\pi$
$787$	− 11164.0i	− 0.505659i	0.967511	−	0.252829i	$-0.0813612\pi$
$788$	0	0
$789$	−9576.00	−0.432084
$790$	0	0
$791$	5208.00	0.234103
$792$	0	0
$793$	604.000i	0.0270475i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5094.00i	0.226397i	0.993572	+	0.113199i	$0.0361097\pi$
$797$	5094.00i	0.226397i	−0.993572	+	0.113199i	$0.963890\pi$
$798$	0	0
$799$	−24624.0	−1.09028
$800$	0	0
$801$	1890.00	0.0833706
$802$	0	0
$803$	− 22944.0i	− 1.00831i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16470.0i	0.718428i
$808$	0	0
$809$	8790.00	0.382002	0.191001	−	0.981590i	$-0.438827\pi$
$809$	8790.00	0.382002	0.191001	+	0.981590i	$0.438827\pi$
$810$	0	0
$811$	−5852.00	−0.253380	−0.126690	−	0.991942i	$-0.540435\pi$
$811$	−5852.00	−0.253380	−0.126690	+	0.991942i	$0.540435\pi$
$812$	0	0
$813$	− 18984.0i	− 0.818940i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	37520.0i	1.60668i
$818$	0	0
$819$	−72.0000	−0.00307190
$820$	0	0
$821$	−29478.0	−1.25309	−0.626546	−	0.779384i	$-0.715532\pi$
$821$	−29478.0	−1.25309	−0.626546	+	0.779384i	$0.715532\pi$
$822$	0	0
$823$	− 39332.0i	− 1.66589i	−0.553356	−	0.832945i	$-0.686653\pi$
$823$	− 39332.0i	− 1.66589i	0.553356	−	0.832945i	$-0.313347\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6756.00i	0.284074i	0.989861	+	0.142037i	$0.0453652\pi$
$827$	6756.00i	0.284074i	−0.989861	+	0.142037i	$0.954635\pi$
$828$	0	0
$829$	−3950.00	−0.165488	−0.0827438	−	0.996571i	$-0.526368\pi$
$829$	−3950.00	−0.165488	−0.0827438	+	0.996571i	$0.526368\pi$
$830$	0	0
$831$	1722.00	0.0718839
$832$	0	0
$833$	37278.0i	1.55055i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 7344.00i	− 0.303280i
$838$	0	0
$839$	12360.0	0.508599	0.254300	−	0.967126i	$-0.418155\pi$
$839$	12360.0	0.508599	0.254300	+	0.967126i	$0.418155\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	0	0
$843$	− 12726.0i	− 0.519937i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3892.00i	− 0.157887i
$848$	0	0
$849$	1884.00	0.0761587
$850$	0	0
$851$	24048.0	0.968690
$852$	0	0
$853$	− 35998.0i	− 1.44496i	−0.691394	−	0.722478i	$-0.743003\pi$
$853$	− 35998.0i	− 1.44496i	0.691394	−	0.722478i	$-0.256997\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21594.0i	0.860720i	0.902657	+	0.430360i	$0.141613\pi$
$857$	21594.0i	0.860720i	−0.902657	+	0.430360i	$0.858387\pi$
$858$	0	0
$859$	9260.00	0.367808	0.183904	−	0.982944i	$-0.441126\pi$
$859$	9260.00	0.367808	0.183904	+	0.982944i	$0.441126\pi$
$860$	0	0
$861$	2376.00	0.0940463
$862$	0	0
$863$	− 31632.0i	− 1.24770i	−0.781544	−	0.623850i	$-0.785567\pi$
$863$	− 31632.0i	− 1.24770i	0.781544	−	0.623850i	$-0.214433\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	24249.0i	0.949872i
$868$	0	0
$869$	−30720.0	−1.19920
$870$	0	0
$871$	−1192.00	−0.0463713
$872$	0	0
$873$	− 13806.0i	− 0.535237i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39694.0i	1.52836i	0.645003	+	0.764180i	$0.276856\pi$
$877$	39694.0i	1.52836i	−0.645003	+	0.764180i	$0.723144\pi$
$878$	0	0
$879$	−1674.00	−0.0642351
$880$	0	0
$881$	1242.00	0.0474961	0.0237480	−	0.999718i	$-0.492440\pi$
$881$	1242.00	0.0474961	0.0237480	+	0.999718i	$0.492440\pi$
$882$	0	0
$883$	2668.00i	0.101682i	0.998707	+	0.0508411i	$0.0161902\pi$
$883$	2668.00i	0.101682i	−0.998707	+	0.0508411i	$0.983810\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 4344.00i	− 0.164439i	−0.996614	−	0.0822194i	$-0.973799\pi$
$887$	− 4344.00i	− 0.164439i	0.996614	−	0.0822194i	$-0.0262008\pi$
$888$	0	0
$889$	−496.000	−0.0187124
$890$	0	0
$891$	3888.00	0.146187
$892$	0	0
$893$	30240.0i	1.13319i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 432.000i	− 0.0160803i
$898$	0	0
$899$	57120.0	2.11909
$900$	0	0
$901$	8892.00	0.328785
$902$	0	0
$903$	− 3216.00i	− 0.118518i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4436.00i	0.162398i	0.996698	+	0.0811990i	$0.0258749\pi$
$907$	4436.00i	0.162398i	−0.996698	+	0.0811990i	$0.974125\pi$
$908$	0	0
$909$	−15498.0	−0.565496
$910$	0	0
$911$	−22752.0	−0.827450	−0.413725	−	0.910402i	$-0.635773\pi$
$911$	−22752.0	−0.827450	−0.413725	+	0.910402i	$0.635773\pi$
$912$	0	0
$913$	16704.0i	0.605500i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	768.000i	0.0276571i
$918$	0	0
$919$	−27160.0	−0.974892	−0.487446	−	0.873153i	$-0.662071\pi$
$919$	−27160.0	−0.974892	−0.487446	+	0.873153i	$0.662071\pi$
$920$	0	0
$921$	−20892.0	−0.747465
$922$	0	0
$923$	1536.00i	0.0547758i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9468.00i	0.335458i
$928$	0	0
$929$	33030.0	1.16650	0.583250	−	0.812292i	$-0.301781\pi$
$929$	33030.0	1.16650	0.583250	+	0.812292i	$0.301781\pi$
$930$	0	0
$931$	45780.0	1.61158
$932$	0	0
$933$	8496.00i	0.298121i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29974.0i	1.04505i	0.852625	+	0.522523i	$0.175009\pi$
$937$	29974.0i	1.04505i	−0.852625	+	0.522523i	$0.824991\pi$
$938$	0	0
$939$	25926.0	0.901026
$940$	0	0
$941$	13962.0	0.483686	0.241843	−	0.970315i	$-0.422248\pi$
$941$	13962.0	0.483686	0.241843	+	0.970315i	$0.422248\pi$
$942$	0	0
$943$	14256.0i	0.492300i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35196.0i	1.20773i	0.797088	+	0.603863i	$0.206373\pi$
$947$	35196.0i	1.20773i	−0.797088	+	0.603863i	$0.793627\pi$
$948$	0	0
$949$	956.000	0.0327008
$950$	0	0
$951$	6642.00	0.226479
$952$	0	0
$953$	− 28338.0i	− 0.963230i	−0.876383	−	0.481615i	$-0.840050\pi$
$953$	− 28338.0i	− 0.963230i	0.876383	−	0.481615i	$-0.159950\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	30240.0i	1.02144i
$958$	0	0
$959$	10056.0	0.338608
$960$	0	0
$961$	44193.0	1.48343
$962$	0	0
$963$	5076.00i	0.169857i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 17524.0i	− 0.582765i	−0.956607	−	0.291383i	$-0.905885\pi$
$967$	− 17524.0i	− 0.582765i	0.956607	−	0.291383i	$-0.0941153\pi$
$968$	0	0
$969$	47880.0	1.58733
$970$	0	0
$971$	26808.0	0.886004	0.443002	−	0.896521i	$-0.353913\pi$
$971$	26808.0	0.886004	0.443002	+	0.896521i	$0.353913\pi$
$972$	0	0
$973$	− 5360.00i	− 0.176602i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10914.0i	0.357390i	0.983905	+	0.178695i	$0.0571875\pi$
$977$	10914.0i	0.357390i	−0.983905	+	0.178695i	$0.942813\pi$
$978$	0	0
$979$	−10080.0	−0.329069
$980$	0	0
$981$	−5490.00	−0.178677
$982$	0	0
$983$	− 22272.0i	− 0.722652i	−0.932440	−	0.361326i	$-0.882324\pi$
$983$	− 22272.0i	− 0.722652i	0.932440	−	0.361326i	$-0.117676\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2592.00i	− 0.0835910i
$988$	0	0
$989$	19296.0	0.620402
$990$	0	0
$991$	−14072.0	−0.451071	−0.225536	−	0.974235i	$-0.572413\pi$
$991$	−14072.0	−0.451071	−0.225536	+	0.974235i	$0.572413\pi$
$992$	0	0
$993$	32316.0i	1.03275i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4826.00i	− 0.153301i	−0.997058	−	0.0766504i	$-0.975577\pi$
$997$	− 4826.00i	− 0.153301i	0.997058	−	0.0766504i	$-0.0244225\pi$
$998$	0	0
$999$	−9018.00	−0.285602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.r.49.1		2
4.3	odd	2		150.4.c.c.49.1		2
5.2	odd	4		240.4.a.b.1.1		1
5.3	odd	4		1200.4.a.ba.1.1		1
5.4	even	2	inner	1200.4.f.r.49.2		2
12.11	even	2		450.4.c.j.199.2		2
15.2	even	4		720.4.a.y.1.1		1
20.3	even	4		150.4.a.b.1.1		1
20.7	even	4		30.4.a.b.1.1	✓	1
20.19	odd	2		150.4.c.c.49.2		2
40.27	even	4		960.4.a.n.1.1		1
40.37	odd	4		960.4.a.bg.1.1		1
60.23	odd	4		450.4.a.r.1.1		1
60.47	odd	4		90.4.a.c.1.1		1
60.59	even	2		450.4.c.j.199.1		2
140.27	odd	4		1470.4.a.r.1.1		1
180.7	even	12		810.4.e.i.271.1		2
180.47	odd	12		810.4.e.p.271.1		2
180.67	even	12		810.4.e.i.541.1		2
180.167	odd	12		810.4.e.p.541.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.b.1.1	✓	1	20.7	even	4
90.4.a.c.1.1		1	60.47	odd	4
150.4.a.b.1.1		1	20.3	even	4
150.4.c.c.49.1		2	4.3	odd	2
150.4.c.c.49.2		2	20.19	odd	2
240.4.a.b.1.1		1	5.2	odd	4
450.4.a.r.1.1		1	60.23	odd	4
450.4.c.j.199.1		2	60.59	even	2
450.4.c.j.199.2		2	12.11	even	2
720.4.a.y.1.1		1	15.2	even	4
810.4.e.i.271.1		2	180.7	even	12
810.4.e.i.541.1		2	180.67	even	12
810.4.e.p.271.1		2	180.47	odd	12
810.4.e.p.541.1		2	180.167	odd	12
960.4.a.n.1.1		1	40.27	even	4
960.4.a.bg.1.1		1	40.37	odd	4
1200.4.a.ba.1.1		1	5.3	odd	4
1200.4.f.r.49.1		2	1.1	even	1	trivial
1200.4.f.r.49.2		2	5.4	even	2	inner
1470.4.a.r.1.1		1	140.27	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -4.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -4.00000i q^{7} -9.00000 q^{9} +48.0000 q^{11} +2.00000i q^{13} +114.000i q^{17} +140.000 q^{19} -12.0000 q^{21} -72.0000i q^{23} +27.0000i q^{27} -210.000 q^{29} -272.000 q^{31} -144.000i q^{33} +334.000i q^{37} +6.00000 q^{39} -198.000 q^{41} +268.000i q^{43} +216.000i q^{47} +327.000 q^{49} +342.000 q^{51} -78.0000i q^{53} -420.000i q^{57} +240.000 q^{59} +302.000 q^{61} +36.0000i q^{63} +596.000i q^{67} -216.000 q^{69} +768.000 q^{71} -478.000i q^{73} -192.000i q^{77} -640.000 q^{79} +81.0000 q^{81} +348.000i q^{83} +630.000i q^{87} -210.000 q^{89} +8.00000 q^{91} +816.000i q^{93} +1534.00i q^{97} -432.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} + 96 q^{11} + 280 q^{19} - 24 q^{21} - 420 q^{29} - 544 q^{31} + 12 q^{39} - 396 q^{41} + 654 q^{49} + 684 q^{51} + 480 q^{59} + 604 q^{61} - 432 q^{69} + 1536 q^{71} - 1280 q^{79} + 162 q^{81} - 420 q^{89} + 16 q^{91} - 864 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 96 * q^11 + 280 * q^19 - 24 * q^21 - 420 * q^29 - 544 * q^31 + 12 * q^39 - 396 * q^41 + 654 * q^49 + 684 * q^51 + 480 * q^59 + 604 * q^61 - 432 * q^69 + 1536 * q^71 - 1280 * q^79 + 162 * q^81 - 420 * q^89 + 16 * q^91 - 864 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more