LMFDB - Embedded newform 600.6.f.i.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,6,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	600.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:	$96.2302918878$
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 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.6.f.i.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} -80.0000i q^{7} -81.0000 q^{9} +684.000 q^{11} +978.000i q^{13} -862.000i q^{17} -916.000 q^{19} +720.000 q^{21} +1552.00i q^{23} -729.000i q^{27} +7314.00 q^{29} -9312.00 q^{31} +6156.00i q^{33} -8826.00i q^{37} -8802.00 q^{39} -3286.00 q^{41} -7556.00i q^{43} -5960.00i q^{47} +10407.0 q^{49} +7758.00 q^{51} +8698.00i q^{53} -8244.00i q^{57} +42036.0 q^{59} +37518.0 q^{61} +6480.00i q^{63} +29324.0i q^{67} -13968.0 q^{69} +84408.0 q^{71} +46550.0i q^{73} -54720.0i q^{77} -26752.0 q^{79} +6561.00 q^{81} +7956.00i q^{83} +65826.0i q^{87} -59674.0 q^{89} +78240.0 q^{91} -83808.0i q^{93} +136898. i q^{97} -55404.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 162 q^{9} + 1368 q^{11} - 1832 q^{19} + 1440 q^{21} + 14628 q^{29} - 18624 q^{31} - 17604 q^{39} - 6572 q^{41} + 20814 q^{49} + 15516 q^{51} + 84072 q^{59} + 75036 q^{61} - 27936 q^{69} + 168816 q^{71}+ \cdots - 110808 q^{99}+O(q^{100}) $ 2 * q - 162 * q^9 + 1368 * q^11 - 1832 * q^19 + 1440 * q^21 + 14628 * q^29 - 18624 * q^31 - 17604 * q^39 - 6572 * q^41 + 20814 * q^49 + 15516 * q^51 + 84072 * q^59 + 75036 * q^61 - 27936 * q^69 + 168816 * q^71 - 53504 * q^79 + 13122 * q^81 - 119348 * q^89 + 156480 * q^91 - 110808 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 80.0000i	− 0.617085i	−0.951211	−	0.308542i	$-0.900159\pi$
$7$	− 80.0000i	− 0.617085i	0.951211	−	0.308542i	$-0.0998412\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	684.000	1.70441	0.852206	−	0.523207i	$-0.175265\pi$
$11$	684.000	1.70441	0.852206	+	0.523207i	$0.175265\pi$
$12$	0	0
$13$	978.000i	1.60502i	0.596639	+	0.802510i	$0.296503\pi$
$13$	978.000i	1.60502i	−0.596639	+	0.802510i	$0.703497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 862.000i	− 0.723411i	−0.932292	−	0.361705i	$-0.882195\pi$
$17$	− 862.000i	− 0.723411i	0.932292	−	0.361705i	$-0.117805\pi$
$18$	0	0
$19$	−916.000	−0.582119	−0.291059	−	0.956705i	$-0.594008\pi$
$19$	−916.000	−0.582119	−0.291059	+	0.956705i	$0.594008\pi$
$20$	0	0
$21$	720.000	0.356274
$22$	0	0
$23$	1552.00i	0.611747i	0.952072	+	0.305874i	$0.0989485\pi$
$23$	1552.00i	0.611747i	−0.952072	+	0.305874i	$0.901051\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	7314.00	1.61495	0.807477	−	0.589900i	$-0.200833\pi$
$29$	7314.00	1.61495	0.807477	+	0.589900i	$0.200833\pi$
$30$	0	0
$31$	−9312.00	−1.74036	−0.870179	−	0.492735i	$-0.835997\pi$
$31$	−9312.00	−1.74036	−0.870179	+	0.492735i	$0.835997\pi$
$32$	0	0
$33$	6156.00i	0.984042i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8826.00i	− 1.05989i	−0.848033	−	0.529944i	$-0.822213\pi$
$37$	− 8826.00i	− 1.05989i	0.848033	−	0.529944i	$-0.177787\pi$
$38$	0	0
$39$	−8802.00	−0.926659
$40$	0	0
$41$	−3286.00	−0.305287	−0.152643	−	0.988281i	$-0.548779\pi$
$41$	−3286.00	−0.305287	−0.152643	+	0.988281i	$0.548779\pi$
$42$	0	0
$43$	− 7556.00i	− 0.623190i	−0.950215	−	0.311595i	$-0.899137\pi$
$43$	− 7556.00i	− 0.623190i	0.950215	−	0.311595i	$-0.100863\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5960.00i	− 0.393552i	−0.980449	−	0.196776i	$-0.936953\pi$
$47$	− 5960.00i	− 0.393552i	0.980449	−	0.196776i	$-0.0630471\pi$
$48$	0	0
$49$	10407.0	0.619206
$50$	0	0
$51$	7758.00	0.417661
$52$	0	0
$53$	8698.00i	0.425334i	0.977125	+	0.212667i	$0.0682149\pi$
$53$	8698.00i	0.425334i	−0.977125	+	0.212667i	$0.931785\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8244.00i	− 0.336086i
$58$	0	0
$59$	42036.0	1.57214	0.786070	−	0.618137i	$-0.212112\pi$
$59$	42036.0	1.57214	0.786070	+	0.618137i	$0.212112\pi$
$60$	0	0
$61$	37518.0	1.29097	0.645483	−	0.763774i	$-0.276656\pi$
$61$	37518.0	1.29097	0.645483	+	0.763774i	$0.276656\pi$
$62$	0	0
$63$	6480.00i	0.205695i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	29324.0i	0.798061i	0.916938	+	0.399031i	$0.130653\pi$
$67$	29324.0i	0.798061i	−0.916938	+	0.399031i	$0.869347\pi$
$68$	0	0
$69$	−13968.0	−0.353193
$70$	0	0
$71$	84408.0	1.98718	0.993591	−	0.113033i	$-0.0360566\pi$
$71$	84408.0	1.98718	0.993591	+	0.113033i	$0.0360566\pi$
$72$	0	0
$73$	46550.0i	1.02238i	0.859468	+	0.511190i	$0.170795\pi$
$73$	46550.0i	1.02238i	−0.859468	+	0.511190i	$0.829205\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 54720.0i	− 1.05177i
$78$	0	0
$79$	−26752.0	−0.482268	−0.241134	−	0.970492i	$-0.577519\pi$
$79$	−26752.0	−0.482268	−0.241134	+	0.970492i	$0.577519\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	7956.00i	0.126765i	0.997989	+	0.0633825i	$0.0201888\pi$
$83$	7956.00i	0.126765i	−0.997989	+	0.0633825i	$0.979811\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	65826.0i	0.932394i
$88$	0	0
$89$	−59674.0	−0.798565	−0.399282	−	0.916828i	$-0.630741\pi$
$89$	−59674.0	−0.798565	−0.399282	+	0.916828i	$0.630741\pi$
$90$	0	0
$91$	78240.0	0.990434
$92$	0	0
$93$	− 83808.0i	− 1.00480i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	136898.i	1.47730i	0.674091	+	0.738648i	$0.264536\pi$
$97$	136898.i	1.47730i	−0.674091	+	0.738648i	$0.735464\pi$
$98$	0	0
$99$	−55404.0	−0.568137
$100$	0	0
$101$	−202858.	−1.97874	−0.989370	−	0.145421i	$-0.953546\pi$
$101$	−202858.	−1.97874	−0.989370	+	0.145421i	$0.953546\pi$
$102$	0	0
$103$	8576.00i	0.0796511i	0.999207	+	0.0398255i	$0.0126802\pi$
$103$	8576.00i	0.0796511i	−0.999207	+	0.0398255i	$0.987320\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19948.0i	− 0.168438i	−0.996447	−	0.0842190i	$-0.973160\pi$
$107$	− 19948.0i	− 0.168438i	0.996447	−	0.0842190i	$-0.0268395\pi$
$108$	0	0
$109$	−37598.0	−0.303109	−0.151554	−	0.988449i	$-0.548428\pi$
$109$	−37598.0	−0.303109	−0.151554	+	0.988449i	$0.548428\pi$
$110$	0	0
$111$	79434.0	0.611926
$112$	0	0
$113$	191838.i	1.41331i	0.707556	+	0.706657i	$0.249797\pi$
$113$	191838.i	1.41331i	−0.707556	+	0.706657i	$0.750203\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 79218.0i	− 0.535007i
$118$	0	0
$119$	−68960.0	−0.446406
$120$	0	0
$121$	306805.	1.90502
$122$	0	0
$123$	− 29574.0i	− 0.176257i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	98888.0i	0.544044i	0.962291	+	0.272022i	$0.0876924\pi$
$127$	98888.0i	0.544044i	−0.962291	+	0.272022i	$0.912308\pi$
$128$	0	0
$129$	68004.0	0.359799
$130$	0	0
$131$	29636.0	0.150883	0.0754417	−	0.997150i	$-0.475963\pi$
$131$	29636.0	0.150883	0.0754417	+	0.997150i	$0.475963\pi$
$132$	0	0
$133$	73280.0i	0.359217i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4614.00i	− 0.0210028i	−0.999945	−	0.0105014i	$-0.996657\pi$
$137$	− 4614.00i	− 0.0210028i	0.999945	−	0.0105014i	$-0.00334275\pi$
$138$	0	0
$139$	254292.	1.11634	0.558169	−	0.829727i	$-0.311504\pi$
$139$	254292.	1.11634	0.558169	+	0.829727i	$0.311504\pi$
$140$	0	0
$141$	53640.0	0.227217
$142$	0	0
$143$	668952.i	2.73561i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	93663.0i	0.357499i
$148$	0	0
$149$	83226.0	0.307110	0.153555	−	0.988140i	$-0.450928\pi$
$149$	83226.0	0.307110	0.153555	+	0.988140i	$0.450928\pi$
$150$	0	0
$151$	−212616.	−0.758846	−0.379423	−	0.925223i	$-0.623877\pi$
$151$	−212616.	−0.758846	−0.379423	+	0.925223i	$0.623877\pi$
$152$	0	0
$153$	69822.0i	0.241137i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	112702.i	0.364907i	0.983214	+	0.182454i	$0.0584039\pi$
$157$	112702.i	0.364907i	−0.983214	+	0.182454i	$0.941596\pi$
$158$	0	0
$159$	−78282.0	−0.245566
$160$	0	0
$161$	124160.	0.377500
$162$	0	0
$163$	411172.i	1.21214i	0.795409	+	0.606072i	$0.207256\pi$
$163$	411172.i	1.21214i	−0.795409	+	0.606072i	$0.792744\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	258896.i	0.718347i	0.933271	+	0.359173i	$0.116941\pi$
$167$	258896.i	0.718347i	−0.933271	+	0.359173i	$0.883059\pi$
$168$	0	0
$169$	−585191.	−1.57609
$170$	0	0
$171$	74196.0	0.194040
$172$	0	0
$173$	397026.i	1.00856i	0.863539	+	0.504282i	$0.168243\pi$
$173$	397026.i	1.00856i	−0.863539	+	0.504282i	$0.831757\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	378324.i	0.907676i
$178$	0	0
$179$	473468.	1.10448	0.552240	−	0.833685i	$-0.313773\pi$
$179$	473468.	1.10448	0.552240	+	0.833685i	$0.313773\pi$
$180$	0	0
$181$	−79834.0	−0.181130	−0.0905652	−	0.995891i	$-0.528867\pi$
$181$	−79834.0	−0.181130	−0.0905652	+	0.995891i	$0.528867\pi$
$182$	0	0
$183$	337662.i	0.745340i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 589608.i	− 1.23299i
$188$	0	0
$189$	−58320.0	−0.118758
$190$	0	0
$191$	397360.	0.788135	0.394068	−	0.919081i	$-0.371068\pi$
$191$	397360.	0.788135	0.394068	+	0.919081i	$0.371068\pi$
$192$	0	0
$193$	− 777858.i	− 1.50317i	−0.659638	−	0.751583i	$-0.729291\pi$
$193$	− 777858.i	− 1.50317i	0.659638	−	0.751583i	$-0.270709\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	254678.i	0.467548i	0.972291	+	0.233774i	$0.0751075\pi$
$197$	254678.i	0.467548i	−0.972291	+	0.233774i	$0.924892\pi$
$198$	0	0
$199$	540264.	0.967104	0.483552	−	0.875316i	$-0.339346\pi$
$199$	540264.	0.967104	0.483552	+	0.875316i	$0.339346\pi$
$200$	0	0
$201$	−263916.	−0.460761
$202$	0	0
$203$	− 585120.i	− 0.996563i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 125712.i	− 0.203916i
$208$	0	0
$209$	−626544.	−0.992169
$210$	0	0
$211$	1.05690e6	1.63428	0.817142	−	0.576436i	$-0.195557\pi$
$211$	1.05690e6	1.63428	0.817142	+	0.576436i	$0.195557\pi$
$212$	0	0
$213$	759672.i	1.14730i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	744960.i	1.07395i
$218$	0	0
$219$	−418950.	−0.590272
$220$	0	0
$221$	843036.	1.16109
$222$	0	0
$223$	1.09063e6i	1.46864i	0.678802	+	0.734321i	$0.262499\pi$
$223$	1.09063e6i	1.46864i	−0.678802	+	0.734321i	$0.737501\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2772.00i	− 0.00357050i	−0.999998	−	0.00178525i	$-0.999432\pi$
$227$	− 2772.00i	− 0.00357050i	0.999998	−	0.00178525i	$-0.000568262\pi$
$228$	0	0
$229$	304458.	0.383653	0.191827	−	0.981429i	$-0.438559\pi$
$229$	304458.	0.383653	0.191827	+	0.981429i	$0.438559\pi$
$230$	0	0
$231$	492480.	0.607238
$232$	0	0
$233$	329990.i	0.398209i	0.979978	+	0.199104i	$0.0638033\pi$
$233$	329990.i	0.398209i	−0.979978	+	0.199104i	$0.936197\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 240768.i	− 0.278438i
$238$	0	0
$239$	−721584.	−0.817132	−0.408566	−	0.912729i	$-0.633971\pi$
$239$	−721584.	−0.817132	−0.408566	+	0.912729i	$0.633971\pi$
$240$	0	0
$241$	271538.	0.301154	0.150577	−	0.988598i	$-0.451887\pi$
$241$	271538.	0.301154	0.150577	+	0.988598i	$0.451887\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 895848.i	− 0.934312i
$248$	0	0
$249$	−71604.0	−0.0731878
$250$	0	0
$251$	1.34534e6	1.34787	0.673935	−	0.738791i	$-0.264603\pi$
$251$	1.34534e6	1.34787	0.673935	+	0.738791i	$0.264603\pi$
$252$	0	0
$253$	1.06157e6i	1.04267i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.62290e6i	1.53270i	0.642421	+	0.766352i	$0.277930\pi$
$257$	1.62290e6i	1.53270i	−0.642421	+	0.766352i	$0.722070\pi$
$258$	0	0
$259$	−706080.	−0.654040
$260$	0	0
$261$	−592434.	−0.538318
$262$	0	0
$263$	− 472128.i	− 0.420892i	−0.977606	−	0.210446i	$-0.932508\pi$
$263$	− 472128.i	− 0.420892i	0.977606	−	0.210446i	$-0.0674916\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 537066.i	− 0.461052i
$268$	0	0
$269$	−2430.00	−0.00204751	−0.00102375	−	0.999999i	$-0.500326\pi$
$269$	−2430.00	−0.00204751	−0.00102375	+	0.999999i	$0.500326\pi$
$270$	0	0
$271$	1.65157e6	1.36607	0.683035	−	0.730385i	$-0.260659\pi$
$271$	1.65157e6	1.36607	0.683035	+	0.730385i	$0.260659\pi$
$272$	0	0
$273$	704160.i	0.571827i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.21129e6i	1.73159i	0.500397	+	0.865796i	$0.333187\pi$
$277$	2.21129e6i	1.73159i	−0.500397	+	0.865796i	$0.666813\pi$
$278$	0	0
$279$	754272.	0.580120
$280$	0	0
$281$	−423014.	−0.319587	−0.159793	−	0.987150i	$-0.551083\pi$
$281$	−423014.	−0.319587	−0.159793	+	0.987150i	$0.551083\pi$
$282$	0	0
$283$	487052.i	0.361501i	0.983529	+	0.180750i	$0.0578526\pi$
$283$	487052.i	0.361501i	−0.983529	+	0.180750i	$0.942147\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	262880.i	0.188388i
$288$	0	0
$289$	676813.	0.476677
$290$	0	0
$291$	−1.23208e6	−0.852918
$292$	0	0
$293$	1.02692e6i	0.698825i	0.936969	+	0.349412i	$0.113619\pi$
$293$	1.02692e6i	0.698825i	−0.936969	+	0.349412i	$0.886381\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 498636.i	− 0.328014i
$298$	0	0
$299$	−1.51786e6	−0.981867
$300$	0	0
$301$	−604480.	−0.384561
$302$	0	0
$303$	− 1.82572e6i	− 1.14243i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.07133e6i	− 0.648751i	−0.945928	−	0.324376i	$-0.894846\pi$
$307$	− 1.07133e6i	− 0.648751i	0.945928	−	0.324376i	$-0.105154\pi$
$308$	0	0
$309$	−77184.0	−0.0459866
$310$	0	0
$311$	1.87422e6	1.09880	0.549400	−	0.835559i	$-0.314856\pi$
$311$	1.87422e6	1.09880	0.549400	+	0.835559i	$0.314856\pi$
$312$	0	0
$313$	− 2.92883e6i	− 1.68979i	−0.534932	−	0.844895i	$-0.679663\pi$
$313$	− 2.92883e6i	− 1.68979i	0.534932	−	0.844895i	$-0.320337\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.94831e6i	− 1.64788i	−0.566680	−	0.823938i	$-0.691773\pi$
$317$	− 2.94831e6i	− 1.64788i	0.566680	−	0.823938i	$-0.308227\pi$
$318$	0	0
$319$	5.00278e6	2.75254
$320$	0	0
$321$	179532.	0.0972477
$322$	0	0
$323$	789592.i	0.421111i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 338382.i	− 0.175000i
$328$	0	0
$329$	−476800.	−0.242855
$330$	0	0
$331$	−856100.	−0.429491	−0.214746	−	0.976670i	$-0.568892\pi$
$331$	−856100.	−0.429491	−0.214746	+	0.976670i	$0.568892\pi$
$332$	0	0
$333$	714906.i	0.353296i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.11272e6i	− 1.49302i	−0.665375	−	0.746509i	$-0.731728\pi$
$337$	− 3.11272e6i	− 1.49302i	0.665375	−	0.746509i	$-0.268272\pi$
$338$	0	0
$339$	−1.72654e6	−0.815977
$340$	0	0
$341$	−6.36941e6	−2.96629
$342$	0	0
$343$	− 2.17712e6i	− 0.999188i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.09651e6i	− 0.934701i	−0.884072	−	0.467351i	$-0.845209\pi$
$347$	− 2.09651e6i	− 0.934701i	0.884072	−	0.467351i	$-0.154791\pi$
$348$	0	0
$349$	4.44677e6	1.95425	0.977127	−	0.212656i	$-0.0682113\pi$
$349$	4.44677e6	1.95425	0.977127	+	0.212656i	$0.0682113\pi$
$350$	0	0
$351$	712962.	0.308886
$352$	0	0
$353$	− 1.80434e6i	− 0.770692i	−0.922772	−	0.385346i	$-0.874082\pi$
$353$	− 1.80434e6i	− 0.770692i	0.922772	−	0.385346i	$-0.125918\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 620640.i	− 0.257733i
$358$	0	0
$359$	−2.84270e6	−1.16411	−0.582055	−	0.813149i	$-0.697751\pi$
$359$	−2.84270e6	−1.16411	−0.582055	+	0.813149i	$0.697751\pi$
$360$	0	0
$361$	−1.63704e6	−0.661138
$362$	0	0
$363$	2.76124e6i	1.09986i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.65561e6i	0.641641i	0.947140	+	0.320821i	$0.103959\pi$
$367$	1.65561e6i	0.641641i	−0.947140	+	0.320821i	$0.896041\pi$
$368$	0	0
$369$	266166.	0.101762
$370$	0	0
$371$	695840.	0.262467
$372$	0	0
$373$	199690.i	0.0743163i	0.999309	+	0.0371582i	$0.0118305\pi$
$373$	199690.i	0.0743163i	−0.999309	+	0.0371582i	$0.988169\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.15309e6i	2.59203i
$378$	0	0
$379$	1.45610e6	0.520707	0.260353	−	0.965513i	$-0.416161\pi$
$379$	1.45610e6	0.520707	0.260353	+	0.965513i	$0.416161\pi$
$380$	0	0
$381$	−889992.	−0.314104
$382$	0	0
$383$	4.62548e6i	1.61124i	0.592434	+	0.805619i	$0.298167\pi$
$383$	4.62548e6i	1.61124i	−0.592434	+	0.805619i	$0.701833\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	612036.i	0.207730i
$388$	0	0
$389$	−3.51068e6	−1.17630	−0.588149	−	0.808753i	$-0.700143\pi$
$389$	−3.51068e6	−1.17630	−0.588149	+	0.808753i	$0.700143\pi$
$390$	0	0
$391$	1.33782e6	0.442545
$392$	0	0
$393$	266724.i	0.0871125i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.84773e6i	1.54370i	0.635807	+	0.771848i	$0.280667\pi$
$397$	4.84773e6i	1.54370i	−0.635807	+	0.771848i	$0.719333\pi$
$398$	0	0
$399$	−659520.	−0.207394
$400$	0	0
$401$	−4.21515e6	−1.30904	−0.654519	−	0.756046i	$-0.727129\pi$
$401$	−4.21515e6	−1.30904	−0.654519	+	0.756046i	$0.727129\pi$
$402$	0	0
$403$	− 9.10714e6i	− 2.79331i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.03698e6i	− 1.80648i
$408$	0	0
$409$	−4.49535e6	−1.32879	−0.664394	−	0.747383i	$-0.731310\pi$
$409$	−4.49535e6	−1.32879	−0.664394	+	0.747383i	$0.731310\pi$
$410$	0	0
$411$	41526.0	0.0121259
$412$	0	0
$413$	− 3.36288e6i	− 0.970144i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.28863e6i	0.644518i
$418$	0	0
$419$	4.44571e6	1.23710	0.618552	−	0.785744i	$-0.287720\pi$
$419$	4.44571e6	1.23710	0.618552	+	0.785744i	$0.287720\pi$
$420$	0	0
$421$	−4.87185e6	−1.33964	−0.669821	−	0.742523i	$-0.733629\pi$
$421$	−4.87185e6	−1.33964	−0.669821	+	0.742523i	$0.733629\pi$
$422$	0	0
$423$	482760.i	0.131184i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.00144e6i	− 0.796636i
$428$	0	0
$429$	−6.02057e6	−1.57941
$430$	0	0
$431$	549152.	0.142397	0.0711983	−	0.997462i	$-0.477318\pi$
$431$	549152.	0.142397	0.0711983	+	0.997462i	$0.477318\pi$
$432$	0	0
$433$	− 2.37675e6i	− 0.609206i	−0.952479	−	0.304603i	$-0.901476\pi$
$433$	− 2.37675e6i	− 0.609206i	0.952479	−	0.304603i	$-0.0985239\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.42163e6i	− 0.356110i
$438$	0	0
$439$	−1.31188e6	−0.324887	−0.162444	−	0.986718i	$-0.551938\pi$
$439$	−1.31188e6	−0.324887	−0.162444	+	0.986718i	$0.551938\pi$
$440$	0	0
$441$	−842967.	−0.206402
$442$	0	0
$443$	− 2.92914e6i	− 0.709138i	−0.935030	−	0.354569i	$-0.884628\pi$
$443$	− 2.92914e6i	− 0.709138i	0.935030	−	0.354569i	$-0.115372\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	749034.i	0.177310i
$448$	0	0
$449$	4.30777e6	1.00841	0.504205	−	0.863584i	$-0.331786\pi$
$449$	4.30777e6	1.00841	0.504205	+	0.863584i	$0.331786\pi$
$450$	0	0
$451$	−2.24762e6	−0.520334
$452$	0	0
$453$	− 1.91354e6i	− 0.438120i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.48196e6i	1.00387i	0.864905	+	0.501935i	$0.167378\pi$
$457$	4.48196e6i	1.00387i	−0.864905	+	0.501935i	$0.832622\pi$
$458$	0	0
$459$	−628398.	−0.139220
$460$	0	0
$461$	−95906.0	−0.0210181	−0.0105091	−	0.999945i	$-0.503345\pi$
$461$	−95906.0	−0.0210181	−0.0105091	+	0.999945i	$0.503345\pi$
$462$	0	0
$463$	− 7.24487e6i	− 1.57065i	−0.619086	−	0.785323i	$-0.712497\pi$
$463$	− 7.24487e6i	− 1.57065i	0.619086	−	0.785323i	$-0.287503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.80528e6i	− 0.383048i	−0.981488	−	0.191524i	$-0.938657\pi$
$467$	− 1.80528e6i	− 0.383048i	0.981488	−	0.191524i	$-0.0613430\pi$
$468$	0	0
$469$	2.34592e6	0.492471
$470$	0	0
$471$	−1.01432e6	−0.210679
$472$	0	0
$473$	− 5.16830e6i	− 1.06217i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 704538.i	− 0.141778i
$478$	0	0
$479$	1.02682e6	0.204481	0.102241	−	0.994760i	$-0.467399\pi$
$479$	1.02682e6	0.204481	0.102241	+	0.994760i	$0.467399\pi$
$480$	0	0
$481$	8.63183e6	1.70114
$482$	0	0
$483$	1.11744e6i	0.217950i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.43013e6i	− 1.22856i	−0.789087	−	0.614281i	$-0.789446\pi$
$487$	− 6.43013e6i	− 1.22856i	0.789087	−	0.614281i	$-0.210554\pi$
$488$	0	0
$489$	−3.70055e6	−0.699832
$490$	0	0
$491$	−6.89637e6	−1.29097	−0.645486	−	0.763772i	$-0.723345\pi$
$491$	−6.89637e6	−1.29097	−0.645486	+	0.763772i	$0.723345\pi$
$492$	0	0
$493$	− 6.30467e6i	− 1.16827i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.75264e6i	− 1.22626i
$498$	0	0
$499$	8.27403e6	1.48753	0.743765	−	0.668441i	$-0.233038\pi$
$499$	8.27403e6	1.48753	0.743765	+	0.668441i	$0.233038\pi$
$500$	0	0
$501$	−2.33006e6	−0.414738
$502$	0	0
$503$	5.83070e6i	1.02755i	0.857926	+	0.513773i	$0.171753\pi$
$503$	5.83070e6i	1.02755i	−0.857926	+	0.513773i	$0.828247\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5.26672e6i	− 0.909956i
$508$	0	0
$509$	−3.66421e6	−0.626881	−0.313441	−	0.949608i	$-0.601482\pi$
$509$	−3.66421e6	−0.626881	−0.313441	+	0.949608i	$0.601482\pi$
$510$	0	0
$511$	3.72400e6	0.630895
$512$	0	0
$513$	667764.i	0.112029i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.07664e6i	− 0.670774i
$518$	0	0
$519$	−3.57323e6	−0.582295
$520$	0	0
$521$	−5.79381e6	−0.935126	−0.467563	−	0.883960i	$-0.654868\pi$
$521$	−5.79381e6	−0.935126	−0.467563	+	0.883960i	$0.654868\pi$
$522$	0	0
$523$	− 6.06676e6i	− 0.969845i	−0.874557	−	0.484922i	$-0.838848\pi$
$523$	− 6.06676e6i	− 0.969845i	0.874557	−	0.484922i	$-0.161152\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.02694e6i	1.25899i
$528$	0	0
$529$	4.02764e6	0.625765
$530$	0	0
$531$	−3.40492e6	−0.524047
$532$	0	0
$533$	− 3.21371e6i	− 0.489991i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.26121e6i	0.637672i
$538$	0	0
$539$	7.11839e6	1.05538
$540$	0	0
$541$	−2.19330e6	−0.322184	−0.161092	−	0.986939i	$-0.551502\pi$
$541$	−2.19330e6	−0.322184	−0.161092	+	0.986939i	$0.551502\pi$
$542$	0	0
$543$	− 718506.i	− 0.104576i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.03263e7i	1.47563i	0.675004	+	0.737814i	$0.264142\pi$
$547$	1.03263e7i	1.47563i	−0.675004	+	0.737814i	$0.735858\pi$
$548$	0	0
$549$	−3.03896e6	−0.430322
$550$	0	0
$551$	−6.69962e6	−0.940094
$552$	0	0
$553$	2.14016e6i	0.297600i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.01187e6i	− 0.547910i	−0.961742	−	0.273955i	$-0.911668\pi$
$557$	− 4.01187e6i	− 0.547910i	0.961742	−	0.273955i	$-0.0883320\pi$
$558$	0	0
$559$	7.38977e6	1.00023
$560$	0	0
$561$	5.30647e6	0.711867
$562$	0	0
$563$	9.59663e6i	1.27599i	0.770040	+	0.637996i	$0.220236\pi$
$563$	9.59663e6i	1.27599i	−0.770040	+	0.637996i	$0.779764\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 524880.i	− 0.0685650i
$568$	0	0
$569$	−6.76649e6	−0.876159	−0.438079	−	0.898936i	$-0.644341\pi$
$569$	−6.76649e6	−0.876159	−0.438079	+	0.898936i	$0.644341\pi$
$570$	0	0
$571$	1.42954e7	1.83488	0.917439	−	0.397877i	$-0.130253\pi$
$571$	1.42954e7	1.83488	0.917439	+	0.397877i	$0.130253\pi$
$572$	0	0
$573$	3.57624e6i	0.455030i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.38116e7i	− 1.72705i	−0.504309	−	0.863523i	$-0.668253\pi$
$577$	− 1.38116e7i	− 1.72705i	0.504309	−	0.863523i	$-0.331747\pi$
$578$	0	0
$579$	7.00072e6	0.867854
$580$	0	0
$581$	636480.	0.0782248
$582$	0	0
$583$	5.94943e6i	0.724943i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.82424e6i	− 0.338303i	−0.985590	−	0.169151i	$-0.945897\pi$
$587$	− 2.82424e6i	− 0.338303i	0.985590	−	0.169151i	$-0.0541027\pi$
$588$	0	0
$589$	8.52979e6	1.01310
$590$	0	0
$591$	−2.29210e6	−0.269939
$592$	0	0
$593$	6.72749e6i	0.785626i	0.919618	+	0.392813i	$0.128498\pi$
$593$	6.72749e6i	0.785626i	−0.919618	+	0.392813i	$0.871502\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.86238e6i	0.558358i
$598$	0	0
$599$	−1.02563e7	−1.16795	−0.583977	−	0.811770i	$-0.698504\pi$
$599$	−1.02563e7	−1.16795	−0.583977	+	0.811770i	$0.698504\pi$
$600$	0	0
$601$	−6.93684e6	−0.783385	−0.391693	−	0.920096i	$-0.628110\pi$
$601$	−6.93684e6	−0.783385	−0.391693	+	0.920096i	$0.628110\pi$
$602$	0	0
$603$	− 2.37524e6i	− 0.266020i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 6.04044e6i	− 0.665422i	−0.943029	−	0.332711i	$-0.892037\pi$
$607$	− 6.04044e6i	− 0.665422i	0.943029	−	0.332711i	$-0.107963\pi$
$608$	0	0
$609$	5.26608e6	0.575366
$610$	0	0
$611$	5.82888e6	0.631658
$612$	0	0
$613$	− 5.65002e6i	− 0.607294i	−0.952785	−	0.303647i	$-0.901796\pi$
$613$	− 5.65002e6i	− 0.607294i	0.952785	−	0.303647i	$-0.0982043\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.77818e6i	− 0.822555i	−0.911510	−	0.411278i	$-0.865083\pi$
$617$	− 7.77818e6i	− 0.822555i	0.911510	−	0.411278i	$-0.134917\pi$
$618$	0	0
$619$	−5.86584e6	−0.615323	−0.307662	−	0.951496i	$-0.599546\pi$
$619$	−5.86584e6	−0.615323	−0.307662	+	0.951496i	$0.599546\pi$
$620$	0	0
$621$	1.13141e6	0.117731
$622$	0	0
$623$	4.77392e6i	0.492782i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 5.63890e6i	− 0.572829i
$628$	0	0
$629$	−7.60801e6	−0.766734
$630$	0	0
$631$	−4.14394e6	−0.414324	−0.207162	−	0.978307i	$-0.566423\pi$
$631$	−4.14394e6	−0.414324	−0.207162	+	0.978307i	$0.566423\pi$
$632$	0	0
$633$	9.51210e6i	0.943555i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.01780e7i	0.993839i
$638$	0	0
$639$	−6.83705e6	−0.662394
$640$	0	0
$641$	−8.41769e6	−0.809185	−0.404593	−	0.914497i	$-0.632587\pi$
$641$	−8.41769e6	−0.809185	−0.404593	+	0.914497i	$0.632587\pi$
$642$	0	0
$643$	− 1.79931e7i	− 1.71625i	−0.513444	−	0.858123i	$-0.671630\pi$
$643$	− 1.79931e7i	− 1.71625i	0.513444	−	0.858123i	$-0.328370\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 7.92533e6i	− 0.744315i	−0.928170	−	0.372157i	$-0.878618\pi$
$647$	− 7.92533e6i	− 0.744315i	0.928170	−	0.372157i	$-0.121382\pi$
$648$	0	0
$649$	2.87526e7	2.67957
$650$	0	0
$651$	−6.70464e6	−0.620045
$652$	0	0
$653$	1.53749e7i	1.41101i	0.708704	+	0.705506i	$0.249280\pi$
$653$	1.53749e7i	1.41101i	−0.708704	+	0.705506i	$0.750720\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.77055e6i	− 0.340793i
$658$	0	0
$659$	−1.46609e7	−1.31507	−0.657534	−	0.753425i	$-0.728400\pi$
$659$	−1.46609e7	−1.31507	−0.657534	+	0.753425i	$0.728400\pi$
$660$	0	0
$661$	−3.32825e6	−0.296287	−0.148143	−	0.988966i	$-0.547330\pi$
$661$	−3.32825e6	−0.296287	−0.148143	+	0.988966i	$0.547330\pi$
$662$	0	0
$663$	7.58732e6i	0.670355i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.13513e7i	0.987943i
$668$	0	0
$669$	−9.81569e6	−0.847921
$670$	0	0
$671$	2.56623e7	2.20034
$672$	0	0
$673$	− 2.08463e7i	− 1.77415i	−0.461621	−	0.887077i	$-0.652732\pi$
$673$	− 2.08463e7i	− 1.77415i	0.461621	−	0.887077i	$-0.347268\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.60880e7i	1.34906i	0.738248	+	0.674530i	$0.235653\pi$
$677$	1.60880e7i	1.34906i	−0.738248	+	0.674530i	$0.764347\pi$
$678$	0	0
$679$	1.09518e7	0.911618
$680$	0	0
$681$	24948.0	0.00206143
$682$	0	0
$683$	− 3.93124e6i	− 0.322461i	−0.986917	−	0.161231i	$-0.948454\pi$
$683$	− 3.93124e6i	− 0.322461i	0.986917	−	0.161231i	$-0.0515463\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.74012e6i	0.221502i
$688$	0	0
$689$	−8.50664e6	−0.682669
$690$	0	0
$691$	9.05300e6	0.721269	0.360634	−	0.932707i	$-0.382560\pi$
$691$	9.05300e6	0.721269	0.360634	+	0.932707i	$0.382560\pi$
$692$	0	0
$693$	4.43232e6i	0.350589i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.83253e6i	0.220848i
$698$	0	0
$699$	−2.96991e6	−0.229906
$700$	0	0
$701$	1.82887e7	1.40568	0.702842	−	0.711346i	$-0.251914\pi$
$701$	1.82887e7	1.40568	0.702842	+	0.711346i	$0.251914\pi$
$702$	0	0
$703$	8.08462e6i	0.616980i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.62286e7i	1.22105i
$708$	0	0
$709$	−1.05416e7	−0.787572	−0.393786	−	0.919202i	$-0.628835\pi$
$709$	−1.05416e7	−0.787572	−0.393786	+	0.919202i	$0.628835\pi$
$710$	0	0
$711$	2.16691e6	0.160756
$712$	0	0
$713$	− 1.44522e7i	− 1.06466i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.49426e6i	− 0.471771i
$718$	0	0
$719$	−1.34280e7	−0.968703	−0.484352	−	0.874873i	$-0.660944\pi$
$719$	−1.34280e7	−0.968703	−0.484352	+	0.874873i	$0.660944\pi$
$720$	0	0
$721$	686080.	0.0491515
$722$	0	0
$723$	2.44384e6i	0.173871i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.97059e6i	0.699657i	0.936814	+	0.349828i	$0.113760\pi$
$727$	9.97059e6i	0.699657i	−0.936814	+	0.349828i	$0.886240\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−6.51327e6	−0.450823
$732$	0	0
$733$	2.70572e7i	1.86004i	0.367509	+	0.930020i	$0.380211\pi$
$733$	2.70572e7i	1.86004i	−0.367509	+	0.930020i	$0.619789\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.00576e7i	1.36022i
$738$	0	0
$739$	−4.87076e6	−0.328084	−0.164042	−	0.986453i	$-0.552453\pi$
$739$	−4.87076e6	−0.328084	−0.164042	+	0.986453i	$0.552453\pi$
$740$	0	0
$741$	8.06263e6	0.539425
$742$	0	0
$743$	− 144288.i	− 0.00958867i	−0.999989	−	0.00479433i	$-0.998474\pi$
$743$	− 144288.i	− 0.00958867i	0.999989	−	0.00479433i	$-0.00152609\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 644436.i	− 0.0422550i
$748$	0	0
$749$	−1.59584e6	−0.103941
$750$	0	0
$751$	−8.74882e6	−0.566043	−0.283022	−	0.959114i	$-0.591337\pi$
$751$	−8.74882e6	−0.566043	−0.283022	+	0.959114i	$0.591337\pi$
$752$	0	0
$753$	1.21081e7i	0.778193i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 5.58062e6i	− 0.353951i	−0.984215	−	0.176975i	$-0.943369\pi$
$757$	− 5.58062e6i	− 0.353951i	0.984215	−	0.176975i	$-0.0566313\pi$
$758$	0	0
$759$	−9.55411e6	−0.601985
$760$	0	0
$761$	273178.	0.0170995	0.00854976	−	0.999963i	$-0.497278\pi$
$761$	273178.	0.0170995	0.00854976	+	0.999963i	$0.497278\pi$
$762$	0	0
$763$	3.00784e6i	0.187044i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.11112e7i	2.52332i
$768$	0	0
$769$	2.16358e7	1.31934	0.659672	−	0.751554i	$-0.270695\pi$
$769$	2.16358e7	1.31934	0.659672	+	0.751554i	$0.270695\pi$
$770$	0	0
$771$	−1.46061e7	−0.884907
$772$	0	0
$773$	− 5.39836e6i	− 0.324947i	−0.986713	−	0.162474i	$-0.948053\pi$
$773$	− 5.39836e6i	− 0.324947i	0.986713	−	0.162474i	$-0.0519473\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.35472e6i	− 0.377610i
$778$	0	0
$779$	3.00998e6	0.177713
$780$	0	0
$781$	5.77351e7	3.38698
$782$	0	0
$783$	− 5.33191e6i	− 0.310798i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.56497e7i	0.900677i	0.892858	+	0.450338i	$0.148697\pi$
$787$	1.56497e7i	0.900677i	−0.892858	+	0.450338i	$0.851303\pi$
$788$	0	0
$789$	4.24915e6	0.243002
$790$	0	0
$791$	1.53470e7	0.872134
$792$	0	0
$793$	3.66926e7i	2.07203i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.43553e7i	− 1.35815i	−0.734069	−	0.679074i	$-0.762381\pi$
$797$	− 2.43553e7i	− 1.35815i	0.734069	−	0.679074i	$-0.237619\pi$
$798$	0	0
$799$	−5.13752e6	−0.284699
$800$	0	0
$801$	4.83359e6	0.266188
$802$	0	0
$803$	3.18402e7i	1.74256i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 21870.0i	− 0.00118213i
$808$	0	0
$809$	2.60329e7	1.39846	0.699231	−	0.714896i	$-0.253526\pi$
$809$	2.60329e7	1.39846	0.699231	+	0.714896i	$0.253526\pi$
$810$	0	0
$811$	−2.63808e7	−1.40843	−0.704217	−	0.709985i	$-0.748702\pi$
$811$	−2.63808e7	−1.40843	−0.704217	+	0.709985i	$0.748702\pi$
$812$	0	0
$813$	1.48641e7i	0.788701i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.92130e6i	0.362771i
$818$	0	0
$819$	−6.33744e6	−0.330145
$820$	0	0
$821$	−7.46105e6	−0.386315	−0.193158	−	0.981168i	$-0.561873\pi$
$821$	−7.46105e6	−0.386315	−0.193158	+	0.981168i	$0.561873\pi$
$822$	0	0
$823$	3.34734e6i	0.172266i	0.996284	+	0.0861332i	$0.0274511\pi$
$823$	3.34734e6i	0.172266i	−0.996284	+	0.0861332i	$0.972549\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.70089e7i	− 1.88167i	−0.338872	−	0.940833i	$-0.610045\pi$
$827$	− 3.70089e7i	− 1.88167i	0.338872	−	0.940833i	$-0.389955\pi$
$828$	0	0
$829$	−2.35921e7	−1.19229	−0.596143	−	0.802878i	$-0.703301\pi$
$829$	−2.35921e7	−1.19229	−0.596143	+	0.802878i	$0.703301\pi$
$830$	0	0
$831$	−1.99016e7	−0.999735
$832$	0	0
$833$	− 8.97083e6i	− 0.447940i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.78845e6i	0.334932i
$838$	0	0
$839$	3.54805e7	1.74014	0.870071	−	0.492926i	$-0.164073\pi$
$839$	3.54805e7	1.74014	0.870071	+	0.492926i	$0.164073\pi$
$840$	0	0
$841$	3.29834e7	1.60807
$842$	0	0
$843$	− 3.80713e6i	− 0.184514i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.45444e7i	− 1.17556i
$848$	0	0
$849$	−4.38347e6	−0.208713
$850$	0	0
$851$	1.36980e7	0.648383
$852$	0	0
$853$	5.54993e6i	0.261165i	0.991437	+	0.130582i	$0.0416847\pi$
$853$	5.54993e6i	0.261165i	−0.991437	+	0.130582i	$0.958315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.21344e7i	− 1.02948i	−0.857348	−	0.514738i	$-0.827889\pi$
$857$	− 2.21344e7i	− 1.02948i	0.857348	−	0.514738i	$-0.172111\pi$
$858$	0	0
$859$	9.65533e6	0.446462	0.223231	−	0.974766i	$-0.428340\pi$
$859$	9.65533e6	0.446462	0.223231	+	0.974766i	$0.428340\pi$
$860$	0	0
$861$	−2.36592e6	−0.108766
$862$	0	0
$863$	− 1.43771e7i	− 0.657120i	−0.944483	−	0.328560i	$-0.893437\pi$
$863$	− 1.43771e7i	− 0.657120i	0.944483	−	0.328560i	$-0.106563\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.09132e6i	0.275210i
$868$	0	0
$869$	−1.82984e7	−0.821983
$870$	0	0
$871$	−2.86789e7	−1.28090
$872$	0	0
$873$	− 1.10887e7i	− 0.492432i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.01032e6i	0.395586i	0.980244	+	0.197793i	$0.0633774\pi$
$877$	9.01032e6i	0.395586i	−0.980244	+	0.197793i	$0.936623\pi$
$878$	0	0
$879$	−9.24230e6	−0.403467
$880$	0	0
$881$	−1.64420e7	−0.713700	−0.356850	−	0.934162i	$-0.616149\pi$
$881$	−1.64420e7	−0.713700	−0.356850	+	0.934162i	$0.616149\pi$
$882$	0	0
$883$	− 6.06516e6i	− 0.261783i	−0.991397	−	0.130891i	$-0.958216\pi$
$883$	− 6.06516e6i	− 0.261783i	0.991397	−	0.130891i	$-0.0417839\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.85746e7i	1.21947i	0.792605	+	0.609735i	$0.208724\pi$
$887$	2.85746e7i	1.21947i	−0.792605	+	0.609735i	$0.791276\pi$
$888$	0	0
$889$	7.91104e6	0.335722
$890$	0	0
$891$	4.48772e6	0.189379
$892$	0	0
$893$	5.45936e6i	0.229094i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.36607e7i	− 0.566881i
$898$	0	0
$899$	−6.81080e7	−2.81060
$900$	0	0
$901$	7.49768e6	0.307691
$902$	0	0
$903$	− 5.44032e6i	− 0.222027i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.47466e7i	− 0.595215i	−0.954688	−	0.297607i	$-0.903811\pi$
$907$	− 1.47466e7i	− 0.595215i	0.954688	−	0.297607i	$-0.0961887\pi$
$908$	0	0
$909$	1.64315e7	0.659580
$910$	0	0
$911$	−2.61222e6	−0.104283	−0.0521416	−	0.998640i	$-0.516605\pi$
$911$	−2.61222e6	−0.104283	−0.0521416	+	0.998640i	$0.516605\pi$
$912$	0	0
$913$	5.44190e6i	0.216060i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.37088e6i	− 0.0931078i
$918$	0	0
$919$	4.66079e7	1.82042	0.910208	−	0.414152i	$-0.135922\pi$
$919$	4.66079e7	1.82042	0.910208	+	0.414152i	$0.135922\pi$
$920$	0	0
$921$	9.64199e6	0.374557
$922$	0	0
$923$	8.25510e7i	3.18947i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 694656.i	− 0.0265504i
$928$	0	0
$929$	3.16320e7	1.20251	0.601254	−	0.799058i	$-0.294668\pi$
$929$	3.16320e7	1.20251	0.601254	+	0.799058i	$0.294668\pi$
$930$	0	0
$931$	−9.53281e6	−0.360451
$932$	0	0
$933$	1.68679e7i	0.634393i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.57021e7i	0.956355i	0.878263	+	0.478177i	$0.158702\pi$
$937$	2.57021e7i	0.956355i	−0.878263	+	0.478177i	$0.841298\pi$
$938$	0	0
$939$	2.63594e7	0.975601
$940$	0	0
$941$	−7.01907e6	−0.258408	−0.129204	−	0.991618i	$-0.541242\pi$
$941$	−7.01907e6	−0.258408	−0.129204	+	0.991618i	$0.541242\pi$
$942$	0	0
$943$	− 5.09987e6i	− 0.186758i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.38467e6i	0.122643i	0.998118	+	0.0613213i	$0.0195314\pi$
$947$	3.38467e6i	0.122643i	−0.998118	+	0.0613213i	$0.980469\pi$
$948$	0	0
$949$	−4.55259e7	−1.64094
$950$	0	0
$951$	2.65348e7	0.951401
$952$	0	0
$953$	9.74473e6i	0.347566i	0.984784	+	0.173783i	$0.0555991\pi$
$953$	9.74473e6i	0.347566i	−0.984784	+	0.173783i	$0.944401\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.50250e7i	1.58918i
$958$	0	0
$959$	−369120.	−0.0129605
$960$	0	0
$961$	5.80842e7	2.02885
$962$	0	0
$963$	1.61579e6i	0.0561460i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.09405e7i	− 0.376244i	−0.982146	−	0.188122i	$-0.939760\pi$
$967$	− 1.09405e7i	− 0.376244i	0.982146	−	0.188122i	$-0.0602400\pi$
$968$	0	0
$969$	−7.10633e6	−0.243128
$970$	0	0
$971$	−2.25329e7	−0.766953	−0.383476	−	0.923551i	$-0.625273\pi$
$971$	−2.25329e7	−0.766953	−0.383476	+	0.923551i	$0.625273\pi$
$972$	0	0
$973$	− 2.03434e7i	− 0.688875i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.79146e7i	− 0.935610i	−0.883832	−	0.467805i	$-0.845045\pi$
$977$	− 2.79146e7i	− 0.935610i	0.883832	−	0.467805i	$-0.154955\pi$
$978$	0	0
$979$	−4.08170e7	−1.36108
$980$	0	0
$981$	3.04544e6	0.101036
$982$	0	0
$983$	3.07607e7i	1.01534i	0.861551	+	0.507670i	$0.169493\pi$
$983$	3.07607e7i	1.01534i	−0.861551	+	0.507670i	$0.830507\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.29120e6i	− 0.140212i
$988$	0	0
$989$	1.17269e7	0.381235
$990$	0	0
$991$	−4.31296e6	−0.139505	−0.0697527	−	0.997564i	$-0.522221\pi$
$991$	−4.31296e6	−0.139505	−0.0697527	+	0.997564i	$0.522221\pi$
$992$	0	0
$993$	− 7.70490e6i	− 0.247967i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.85667e7i	− 0.910170i	−0.890448	−	0.455085i	$-0.849609\pi$
$997$	− 2.85667e7i	− 0.910170i	0.890448	−	0.455085i	$-0.150391\pi$
$998$	0	0
$999$	−6.43415e6	−0.203975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.6.f.i.49.2		2
5.2	odd	4		600.6.a.g.1.1		1
5.3	odd	4		120.6.a.c.1.1	✓	1
5.4	even	2	inner	600.6.f.i.49.1		2
15.8	even	4		360.6.a.c.1.1		1
20.3	even	4		240.6.a.n.1.1		1
40.3	even	4		960.6.a.f.1.1		1
40.13	odd	4		960.6.a.o.1.1		1
60.23	odd	4		720.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.a.c.1.1	✓	1	5.3	odd	4
240.6.a.n.1.1		1	20.3	even	4
360.6.a.c.1.1		1	15.8	even	4
600.6.a.g.1.1		1	5.2	odd	4
600.6.f.i.49.1		2	5.4	even	2	inner
600.6.f.i.49.2		2	1.1	even	1	trivial
720.6.a.g.1.1		1	60.23	odd	4
960.6.a.f.1.1		1	40.3	even	4
960.6.a.o.1.1		1	40.13	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} -80.0000i q^{7} -81.0000 q^{9} +684.000 q^{11} +978.000i q^{13} -862.000i q^{17} -916.000 q^{19} +720.000 q^{21} +1552.00i q^{23} -729.000i q^{27} +7314.00 q^{29} -9312.00 q^{31} +6156.00i q^{33} -8826.00i q^{37} -8802.00 q^{39} -3286.00 q^{41} -7556.00i q^{43} -5960.00i q^{47} +10407.0 q^{49} +7758.00 q^{51} +8698.00i q^{53} -8244.00i q^{57} +42036.0 q^{59} +37518.0 q^{61} +6480.00i q^{63} +29324.0i q^{67} -13968.0 q^{69} +84408.0 q^{71} +46550.0i q^{73} -54720.0i q^{77} -26752.0 q^{79} +6561.00 q^{81} +7956.00i q^{83} +65826.0i q^{87} -59674.0 q^{89} +78240.0 q^{91} -83808.0i q^{93} +136898. i q^{97} -55404.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 162 q^{9} + 1368 q^{11} - 1832 q^{19} + 1440 q^{21} + 14628 q^{29} - 18624 q^{31} - 17604 q^{39} - 6572 q^{41} + 20814 q^{49} + 15516 q^{51} + 84072 q^{59} + 75036 q^{61} - 27936 q^{69} + 168816 q^{71}+ \cdots - 110808 q^{99}+O(q^{100}) \) 2 * q - 162 * q^9 + 1368 * q^11 - 1832 * q^19 + 1440 * q^21 + 14628 * q^29 - 18624 * q^31 - 17604 * q^39 - 6572 * q^41 + 20814 * q^49 + 15516 * q^51 + 84072 * q^59 + 75036 * q^61 - 27936 * q^69 + 168816 * q^71 - 53504 * q^79 + 13122 * q^81 - 119348 * q^89 + 156480 * q^91 - 110808 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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