LMFDB - Embedded newform 300.6.d.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,6,Mod(49,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("300.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	300.d (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:	$48.1151459439$
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 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	300.49
Dual form			300.6.d.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} -16.0000i q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} -16.0000i q^{7} -81.0000 q^{9} -564.000 q^{11} +370.000i q^{13} -1086.00i q^{17} +2860.00 q^{19} +144.000 q^{21} -1584.00i q^{23} -729.000i q^{27} -1134.00 q^{29} -6016.00 q^{31} -5076.00i q^{33} -538.000i q^{37} -3330.00 q^{39} +11370.0 q^{41} -5444.00i q^{43} +10296.0i q^{47} +16551.0 q^{49} +9774.00 q^{51} -34758.0i q^{53} +25740.0i q^{57} +26196.0 q^{59} +9422.00 q^{61} +1296.00i q^{63} -51124.0i q^{67} +14256.0 q^{69} +14520.0 q^{71} +22678.0i q^{73} +9024.00i q^{77} +97312.0 q^{79} +6561.00 q^{81} +7956.00i q^{83} -10206.0i q^{87} +47910.0 q^{89} +5920.00 q^{91} -54144.0i q^{93} +140738. i q^{97} +45684.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 162 q^{9}+O(q^{10}) $ 2 * q - 162 * q^9	$ 2 q - 162 q^{9} - 1128 q^{11} + 5720 q^{19} + 288 q^{21} - 2268 q^{29} - 12032 q^{31} - 6660 q^{39} + 22740 q^{41} + 33102 q^{49} + 19548 q^{51} + 52392 q^{59} + 18844 q^{61} + 28512 q^{69} + 29040 q^{71} + 194624 q^{79} + 13122 q^{81} + 95820 q^{89} + 11840 q^{91} + 91368 q^{99}+O(q^{100}) $ 2 * q - 162 * q^9 - 1128 * q^11 + 5720 * q^19 + 288 * q^21 - 2268 * q^29 - 12032 * q^31 - 6660 * q^39 + 22740 * q^41 + 33102 * q^49 + 19548 * q^51 + 52392 * q^59 + 18844 * q^61 + 28512 * q^69 + 29040 * q^71 + 194624 * q^79 + 13122 * q^81 + 95820 * q^89 + 11840 * q^91 + 91368 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	− 16.0000i	− 0.123417i	−0.998094	−	0.0617085i	$-0.980345\pi$
$7$	− 16.0000i	− 0.123417i	0.998094	−	0.0617085i	$-0.0196549\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−564.000	−1.40539	−0.702696	−	0.711490i	$-0.748021\pi$
$11$	−564.000	−1.40539	−0.702696	+	0.711490i	$0.748021\pi$
$12$	0	0
$13$	370.000i	0.607216i	0.952797	+	0.303608i	$0.0981914\pi$
$13$	370.000i	0.607216i	−0.952797	+	0.303608i	$0.901809\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1086.00i	− 0.911397i	−0.890134	−	0.455698i	$-0.849390\pi$
$17$	− 1086.00i	− 0.911397i	0.890134	−	0.455698i	$-0.150610\pi$
$18$	0	0
$19$	2860.00	1.81753	0.908766	−	0.417306i	$-0.137026\pi$
$19$	2860.00	1.81753	0.908766	+	0.417306i	$0.137026\pi$
$20$	0	0
$21$	144.000	0.0712548
$22$	0	0
$23$	− 1584.00i	− 0.624361i	−0.950023	−	0.312180i	$-0.898941\pi$
$23$	− 1584.00i	− 0.624361i	0.950023	−	0.312180i	$-0.101059\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	−1134.00	−0.250391	−0.125195	−	0.992132i	$-0.539956\pi$
$29$	−1134.00	−0.250391	−0.125195	+	0.992132i	$0.539956\pi$
$30$	0	0
$31$	−6016.00	−1.12436	−0.562178	−	0.827016i	$-0.690036\pi$
$31$	−6016.00	−1.12436	−0.562178	+	0.827016i	$0.690036\pi$
$32$	0	0
$33$	− 5076.00i	− 0.811403i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 538.000i	− 0.0646068i	−0.999478	−	0.0323034i	$-0.989716\pi$
$37$	− 538.000i	− 0.0646068i	0.999478	−	0.0323034i	$-0.0102843\pi$
$38$	0	0
$39$	−3330.00	−0.350576
$40$	0	0
$41$	11370.0	1.05633	0.528166	−	0.849141i	$-0.322880\pi$
$41$	11370.0	1.05633	0.528166	+	0.849141i	$0.322880\pi$
$42$	0	0
$43$	− 5444.00i	− 0.449001i	−0.974474	−	0.224500i	$-0.927925\pi$
$43$	− 5444.00i	− 0.449001i	0.974474	−	0.224500i	$-0.0720750\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10296.0i	0.679867i	0.940449	+	0.339933i	$0.110405\pi$
$47$	10296.0i	0.679867i	−0.940449	+	0.339933i	$0.889595\pi$
$48$	0	0
$49$	16551.0	0.984768
$50$	0	0
$51$	9774.00	0.526195
$52$	0	0
$53$	− 34758.0i	− 1.69967i	−0.527047	−	0.849836i	$-0.676701\pi$
$53$	− 34758.0i	− 1.69967i	0.527047	−	0.849836i	$-0.323299\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	25740.0i	1.04935i
$58$	0	0
$59$	26196.0	0.979727	0.489863	−	0.871799i	$-0.337047\pi$
$59$	26196.0	0.979727	0.489863	+	0.871799i	$0.337047\pi$
$60$	0	0
$61$	9422.00	0.324204	0.162102	−	0.986774i	$-0.448173\pi$
$61$	9422.00	0.324204	0.162102	+	0.986774i	$0.448173\pi$
$62$	0	0
$63$	1296.00i	0.0411390i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 51124.0i	− 1.39135i	−0.718354	−	0.695677i	$-0.755104\pi$
$67$	− 51124.0i	− 1.39135i	0.718354	−	0.695677i	$-0.244896\pi$
$68$	0	0
$69$	14256.0	0.360475
$70$	0	0
$71$	14520.0	0.341838	0.170919	−	0.985285i	$-0.445326\pi$
$71$	14520.0	0.341838	0.170919	+	0.985285i	$0.445326\pi$
$72$	0	0
$73$	22678.0i	0.498078i	0.968493	+	0.249039i	$0.0801148\pi$
$73$	22678.0i	0.498078i	−0.968493	+	0.249039i	$0.919885\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9024.00i	0.173449i
$78$	0	0
$79$	97312.0	1.75428	0.877140	−	0.480236i	$-0.159449\pi$
$79$	97312.0	1.75428	0.877140	+	0.480236i	$0.159449\pi$
$80$	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
$87$	− 10206.0i	− 0.144563i
$88$	0	0
$89$	47910.0	0.641137	0.320569	−	0.947225i	$-0.396126\pi$
$89$	47910.0	0.641137	0.320569	+	0.947225i	$0.396126\pi$
$90$	0	0
$91$	5920.00	0.0749408
$92$	0	0
$93$	− 54144.0i	− 0.649147i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	140738.i	1.51874i	0.650662	+	0.759368i	$0.274492\pi$
$97$	140738.i	1.51874i	−0.650662	+	0.759368i	$0.725508\pi$
$98$	0	0
$99$	45684.0	0.468464
$100$	0	0
$101$	85398.0	0.832999	0.416499	−	0.909136i	$-0.363257\pi$
$101$	85398.0	0.832999	0.416499	+	0.909136i	$0.363257\pi$
$102$	0	0
$103$	− 198656.i	− 1.84505i	−0.385935	−	0.922526i	$-0.626121\pi$
$103$	− 198656.i	− 1.84505i	0.385935	−	0.922526i	$-0.373879\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 116268.i	− 0.981750i	−0.871230	−	0.490875i	$-0.836677\pi$
$107$	− 116268.i	− 0.981750i	0.871230	−	0.490875i	$-0.163323\pi$
$108$	0	0
$109$	146722.	1.18285	0.591424	−	0.806361i	$-0.298566\pi$
$109$	146722.	1.18285	0.591424	+	0.806361i	$0.298566\pi$
$110$	0	0
$111$	4842.00	0.0373007
$112$	0	0
$113$	− 210882.i	− 1.55361i	−0.629738	−	0.776807i	$-0.716838\pi$
$113$	− 210882.i	− 1.55361i	0.629738	−	0.776807i	$-0.283162\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 29970.0i	− 0.202405i
$118$	0	0
$119$	−17376.0	−0.112482
$120$	0	0
$121$	157045.	0.975126
$122$	0	0
$123$	102330.i	0.609874i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	244424.i	1.34473i	0.740221	+	0.672364i	$0.234721\pi$
$127$	244424.i	1.34473i	−0.740221	+	0.672364i	$0.765279\pi$
$128$	0	0
$129$	48996.0	0.259231
$130$	0	0
$131$	−145308.	−0.739795	−0.369897	−	0.929073i	$-0.620607\pi$
$131$	−145308.	−0.739795	−0.369897	+	0.929073i	$0.620607\pi$
$132$	0	0
$133$	− 45760.0i	− 0.224314i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 125478.i	− 0.571171i	−0.958353	−	0.285586i	$-0.907812\pi$
$137$	− 125478.i	− 0.571171i	0.958353	−	0.285586i	$-0.0921881\pi$
$138$	0	0
$139$	−251756.	−1.10520	−0.552602	−	0.833445i	$-0.686365\pi$
$139$	−251756.	−1.10520	−0.552602	+	0.833445i	$0.686365\pi$
$140$	0	0
$141$	−92664.0	−0.392521
$142$	0	0
$143$	− 208680.i	− 0.853377i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	148959.i	0.568556i
$148$	0	0
$149$	167322.	0.617430	0.308715	−	0.951155i	$-0.400101\pi$
$149$	167322.	0.617430	0.308715	+	0.951155i	$0.400101\pi$
$150$	0	0
$151$	68120.0	0.243126	0.121563	−	0.992584i	$-0.461209\pi$
$151$	68120.0	0.243126	0.121563	+	0.992584i	$0.461209\pi$
$152$	0	0
$153$	87966.0i	0.303799i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 361762.i	− 1.17132i	−0.810559	−	0.585658i	$-0.800836\pi$
$157$	− 361762.i	− 1.17132i	0.810559	−	0.585658i	$-0.199164\pi$
$158$	0	0
$159$	312822.	0.981306
$160$	0	0
$161$	−25344.0	−0.0770567
$162$	0	0
$163$	− 121820.i	− 0.359128i	−0.983746	−	0.179564i	$-0.942531\pi$
$163$	− 121820.i	− 0.359128i	0.983746	−	0.179564i	$-0.0574687\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 409008.i	− 1.13486i	−0.823423	−	0.567428i	$-0.807939\pi$
$167$	− 409008.i	− 1.13486i	0.823423	−	0.567428i	$-0.192061\pi$
$168$	0	0
$169$	234393.	0.631288
$170$	0	0
$171$	−231660.	−0.605844
$172$	0	0
$173$	729954.i	1.85430i	0.374689	+	0.927151i	$0.377749\pi$
$173$	729954.i	1.85430i	−0.374689	+	0.927151i	$0.622251\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	235764.i	0.565645i
$178$	0	0
$179$	−704484.	−1.64338	−0.821691	−	0.569933i	$-0.806969\pi$
$179$	−704484.	−1.64338	−0.821691	+	0.569933i	$0.806969\pi$
$180$	0	0
$181$	−405850.	−0.920808	−0.460404	−	0.887709i	$-0.652295\pi$
$181$	−405850.	−0.920808	−0.460404	+	0.887709i	$0.652295\pi$
$182$	0	0
$183$	84798.0i	0.187179i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	612504.i	1.28087i
$188$	0	0
$189$	−11664.0	−0.0237516
$190$	0	0
$191$	183024.	0.363015	0.181508	−	0.983390i	$-0.441902\pi$
$191$	183024.	0.363015	0.181508	+	0.983390i	$0.441902\pi$
$192$	0	0
$193$	853054.i	1.64848i	0.566242	+	0.824239i	$0.308397\pi$
$193$	853054.i	1.64848i	−0.566242	+	0.824239i	$0.691603\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 476394.i	− 0.874582i	−0.899320	−	0.437291i	$-0.855938\pi$
$197$	− 476394.i	− 0.874582i	0.899320	−	0.437291i	$-0.144062\pi$
$198$	0	0
$199$	−470648.	−0.842488	−0.421244	−	0.906947i	$-0.638406\pi$
$199$	−470648.	−0.842488	−0.421244	+	0.906947i	$0.638406\pi$
$200$	0	0
$201$	460116.	0.803299
$202$	0	0
$203$	18144.0i	0.0309025i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	128304.i	0.208120i
$208$	0	0
$209$	−1.61304e6	−2.55434
$210$	0	0
$211$	−14140.0	−0.0218647	−0.0109323	−	0.999940i	$-0.503480\pi$
$211$	−14140.0	−0.0218647	−0.0109323	+	0.999940i	$0.503480\pi$
$212$	0	0
$213$	130680.i	0.197360i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	96256.0i	0.138765i
$218$	0	0
$219$	−204102.	−0.287566
$220$	0	0
$221$	401820.	0.553415
$222$	0	0
$223$	1.08052e6i	1.45503i	0.686094	+	0.727513i	$0.259324\pi$
$223$	1.08052e6i	1.45503i	−0.686094	+	0.727513i	$0.740676\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 340500.i	− 0.438584i	−0.975659	−	0.219292i	$-0.929625\pi$
$227$	− 340500.i	− 0.438584i	0.975659	−	0.219292i	$-0.0703747\pi$
$228$	0	0
$229$	787594.	0.992462	0.496231	−	0.868191i	$-0.334717\pi$
$229$	787594.	0.992462	0.496231	+	0.868191i	$0.334717\pi$
$230$	0	0
$231$	−81216.0	−0.100141
$232$	0	0
$233$	− 1.51881e6i	− 1.83279i	−0.400271	−	0.916397i	$-0.631084\pi$
$233$	− 1.51881e6i	− 1.83279i	0.400271	−	0.916397i	$-0.368916\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	875808.i	1.01283i
$238$	0	0
$239$	−1.18181e6	−1.33830	−0.669148	−	0.743129i	$-0.733341\pi$
$239$	−1.18181e6	−1.33830	−0.669148	+	0.743129i	$0.733341\pi$
$240$	0	0
$241$	−523342.	−0.580421	−0.290210	−	0.956963i	$-0.593725\pi$
$241$	−523342.	−0.580421	−0.290210	+	0.956963i	$0.593725\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.05820e6i	1.10363i
$248$	0	0
$249$	−71604.0	−0.0731878
$250$	0	0
$251$	−1.27741e6	−1.27981	−0.639907	−	0.768453i	$-0.721027\pi$
$251$	−1.27741e6	−1.27981	−0.639907	+	0.768453i	$0.721027\pi$
$252$	0	0
$253$	893376.i	0.877471i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.93112e6i	− 1.82379i	−0.410418	−	0.911897i	$-0.634617\pi$
$257$	− 1.93112e6i	− 1.82379i	0.410418	−	0.911897i	$-0.365383\pi$
$258$	0	0
$259$	−8608.00	−0.00797357
$260$	0	0
$261$	91854.0	0.0834635
$262$	0	0
$263$	822336.i	0.733094i	0.930399	+	0.366547i	$0.119460\pi$
$263$	822336.i	0.733094i	−0.930399	+	0.366547i	$0.880540\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	431190.i	0.370161i
$268$	0	0
$269$	−1.59181e6	−1.34125	−0.670625	−	0.741797i	$-0.733974\pi$
$269$	−1.59181e6	−1.34125	−0.670625	+	0.741797i	$0.733974\pi$
$270$	0	0
$271$	−1.21106e6	−1.00171	−0.500854	−	0.865532i	$-0.666981\pi$
$271$	−1.21106e6	−1.00171	−0.500854	+	0.865532i	$0.666981\pi$
$272$	0	0
$273$	53280.0i	0.0432671i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.34356e6i	− 1.83517i	−0.397536	−	0.917587i	$-0.630135\pi$
$277$	− 2.34356e6i	− 1.83517i	0.397536	−	0.917587i	$-0.369865\pi$
$278$	0	0
$279$	487296.	0.374785
$280$	0	0
$281$	1.86211e6	1.40682	0.703410	−	0.710784i	$-0.251660\pi$
$281$	1.86211e6	1.40682	0.703410	+	0.710784i	$0.251660\pi$
$282$	0	0
$283$	− 108212.i	− 0.0803173i	−0.999193	−	0.0401587i	$-0.987214\pi$
$283$	− 108212.i	− 0.0803173i	0.999193	−	0.0401587i	$-0.0127863\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 181920.i	− 0.130369i
$288$	0	0
$289$	240461.	0.169356
$290$	0	0
$291$	−1.26664e6	−0.876842
$292$	0	0
$293$	959658.i	0.653052i	0.945188	+	0.326526i	$0.105878\pi$
$293$	959658.i	0.653052i	−0.945188	+	0.326526i	$0.894122\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	411156.i	0.270468i
$298$	0	0
$299$	586080.	0.379122
$300$	0	0
$301$	−87104.0	−0.0554143
$302$	0	0
$303$	768582.i	0.480932i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.62729e6i	0.985416i	0.870195	+	0.492708i	$0.163993\pi$
$307$	1.62729e6i	0.985416i	−0.870195	+	0.492708i	$0.836007\pi$
$308$	0	0
$309$	1.78790e6	1.06524
$310$	0	0
$311$	−2.84086e6	−1.66551	−0.832757	−	0.553639i	$-0.813239\pi$
$311$	−2.84086e6	−1.66551	−0.832757	+	0.553639i	$0.813239\pi$
$312$	0	0
$313$	− 1.61715e6i	− 0.933014i	−0.884517	−	0.466507i	$-0.845512\pi$
$313$	− 1.61715e6i	− 0.933014i	0.884517	−	0.466507i	$-0.154488\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.04367e6i	− 1.14225i	−0.820863	−	0.571126i	$-0.806507\pi$
$317$	− 2.04367e6i	− 1.14225i	0.820863	−	0.571126i	$-0.193493\pi$
$318$	0	0
$319$	639576.	0.351897
$320$	0	0
$321$	1.04641e6	0.566813
$322$	0	0
$323$	− 3.10596e6i	− 1.65649i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.32050e6i	0.682918i
$328$	0	0
$329$	164736.	0.0839071
$330$	0	0
$331$	1.00425e6	0.503817	0.251908	−	0.967751i	$-0.418942\pi$
$331$	1.00425e6	0.503817	0.251908	+	0.967751i	$0.418942\pi$
$332$	0	0
$333$	43578.0i	0.0215356i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.91960e6i	1.40039i	0.713952	+	0.700195i	$0.246904\pi$
$337$	2.91960e6i	1.40039i	−0.713952	+	0.700195i	$0.753096\pi$
$338$	0	0
$339$	1.89794e6	0.896980
$340$	0	0
$341$	3.39302e6	1.58016
$342$	0	0
$343$	− 533728.i	− 0.244954i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.21075e6i	− 0.985634i	−0.870133	−	0.492817i	$-0.835967\pi$
$347$	− 2.21075e6i	− 0.985634i	0.870133	−	0.492817i	$-0.164033\pi$
$348$	0	0
$349$	−375566.	−0.165053	−0.0825264	−	0.996589i	$-0.526299\pi$
$349$	−375566.	−0.165053	−0.0825264	+	0.996589i	$0.526299\pi$
$350$	0	0
$351$	269730.	0.116859
$352$	0	0
$353$	− 1.49992e6i	− 0.640666i	−0.947305	−	0.320333i	$-0.896205\pi$
$353$	− 1.49992e6i	− 0.640666i	0.947305	−	0.320333i	$-0.103795\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 156384.i	− 0.0649414i
$358$	0	0
$359$	−2.49626e6	−1.02224	−0.511122	−	0.859508i	$-0.670770\pi$
$359$	−2.49626e6	−1.02224	−0.511122	+	0.859508i	$0.670770\pi$
$360$	0	0
$361$	5.70350e6	2.30342
$362$	0	0
$363$	1.41340e6i	0.562989i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.09714e6i	1.58787i	0.608000	+	0.793937i	$0.291972\pi$
$367$	4.09714e6i	1.58787i	−0.608000	+	0.793937i	$0.708028\pi$
$368$	0	0
$369$	−920970.	−0.352111
$370$	0	0
$371$	−556128.	−0.209768
$372$	0	0
$373$	− 1.24213e6i	− 0.462271i	−0.972922	−	0.231135i	$-0.925756\pi$
$373$	− 1.24213e6i	− 0.462271i	0.972922	−	0.231135i	$-0.0742440\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 419580.i	− 0.152041i
$378$	0	0
$379$	2.28413e6	0.816814	0.408407	−	0.912800i	$-0.366084\pi$
$379$	2.28413e6	0.816814	0.408407	+	0.912800i	$0.366084\pi$
$380$	0	0
$381$	−2.19982e6	−0.776379
$382$	0	0
$383$	565128.i	0.196857i	0.995144	+	0.0984283i	$0.0313815\pi$
$383$	565128.i	0.196857i	−0.995144	+	0.0984283i	$0.968618\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	440964.i	0.149667i
$388$	0	0
$389$	3.37367e6	1.13039	0.565196	−	0.824957i	$-0.308801\pi$
$389$	3.37367e6	1.13039	0.565196	+	0.824957i	$0.308801\pi$
$390$	0	0
$391$	−1.72022e6	−0.569040
$392$	0	0
$393$	− 1.30777e6i	− 0.427121i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 745138.i	− 0.237280i	−0.992937	−	0.118640i	$-0.962147\pi$
$397$	− 745138.i	− 0.237280i	0.992937	−	0.118640i	$-0.0378534\pi$
$398$	0	0
$399$	411840.	0.129508
$400$	0	0
$401$	864786.	0.268564	0.134282	−	0.990943i	$-0.457127\pi$
$401$	864786.	0.268564	0.134282	+	0.990943i	$0.457127\pi$
$402$	0	0
$403$	− 2.22592e6i	− 0.682727i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	303432.i	0.0907978i
$408$	0	0
$409$	5.70189e6	1.68543	0.842715	−	0.538359i	$-0.180956\pi$
$409$	5.70189e6	1.68543	0.842715	+	0.538359i	$0.180956\pi$
$410$	0	0
$411$	1.12930e6	0.329766
$412$	0	0
$413$	− 419136.i	− 0.120915i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.26580e6i	− 0.638090i
$418$	0	0
$419$	2.91947e6	0.812398	0.406199	−	0.913785i	$-0.366854\pi$
$419$	2.91947e6	0.812398	0.406199	+	0.913785i	$0.366854\pi$
$420$	0	0
$421$	1.08815e6	0.299215	0.149608	−	0.988745i	$-0.452199\pi$
$421$	1.08815e6	0.299215	0.149608	+	0.988745i	$0.452199\pi$
$422$	0	0
$423$	− 833976.i	− 0.226622i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 150752.i	− 0.0400123i
$428$	0	0
$429$	1.87812e6	0.492697
$430$	0	0
$431$	4.64573e6	1.20465	0.602325	−	0.798251i	$-0.294241\pi$
$431$	4.64573e6	1.20465	0.602325	+	0.798251i	$0.294241\pi$
$432$	0	0
$433$	702094.i	0.179960i	0.995944	+	0.0899799i	$0.0286803\pi$
$433$	702094.i	0.179960i	−0.995944	+	0.0899799i	$0.971320\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.53024e6i	− 1.13480i
$438$	0	0
$439$	1.63343e6	0.404520	0.202260	−	0.979332i	$-0.435171\pi$
$439$	1.63343e6	0.404520	0.202260	+	0.979332i	$0.435171\pi$
$440$	0	0
$441$	−1.34063e6	−0.328256
$442$	0	0
$443$	− 5.39500e6i	− 1.30612i	−0.757308	−	0.653058i	$-0.773486\pi$
$443$	− 5.39500e6i	− 1.30612i	0.757308	−	0.653058i	$-0.226514\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.50590e6i	0.356473i
$448$	0	0
$449$	918462.	0.215003	0.107502	−	0.994205i	$-0.465715\pi$
$449$	918462.	0.215003	0.107502	+	0.994205i	$0.465715\pi$
$450$	0	0
$451$	−6.41268e6	−1.48456
$452$	0	0
$453$	613080.i	0.140369i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.55695e6i	− 0.348726i	−0.984681	−	0.174363i	$-0.944213\pi$
$457$	− 1.55695e6i	− 0.348726i	0.984681	−	0.174363i	$-0.0557866\pi$
$458$	0	0
$459$	−791694.	−0.175398
$460$	0	0
$461$	−5.19299e6	−1.13806	−0.569030	−	0.822316i	$-0.692681\pi$
$461$	−5.19299e6	−1.13806	−0.569030	+	0.822316i	$0.692681\pi$
$462$	0	0
$463$	665848.i	0.144352i	0.997392	+	0.0721760i	$0.0229943\pi$
$463$	665848.i	0.144352i	−0.997392	+	0.0721760i	$0.977006\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.84073e6i	1.45148i	0.687970	+	0.725739i	$0.258502\pi$
$467$	6.84073e6i	1.45148i	−0.687970	+	0.725739i	$0.741498\pi$
$468$	0	0
$469$	−817984.	−0.171717
$470$	0	0
$471$	3.25586e6	0.676259
$472$	0	0
$473$	3.07042e6i	0.631022i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.81540e6i	0.566557i
$478$	0	0
$479$	−2.26214e6	−0.450486	−0.225243	−	0.974303i	$-0.572318\pi$
$479$	−2.26214e6	−0.450486	−0.225243	+	0.974303i	$0.572318\pi$
$480$	0	0
$481$	199060.	0.0392303
$482$	0	0
$483$	− 228096.i	− 0.0444887i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.78443e6i	− 1.29626i	−0.761531	−	0.648128i	$-0.775552\pi$
$487$	− 6.78443e6i	− 1.29626i	0.761531	−	0.648128i	$-0.224448\pi$
$488$	0	0
$489$	1.09638e6	0.207343
$490$	0	0
$491$	−6.75964e6	−1.26538	−0.632688	−	0.774407i	$-0.718048\pi$
$491$	−6.75964e6	−1.26538	−0.632688	+	0.774407i	$0.718048\pi$
$492$	0	0
$493$	1.23152e6i	0.228205i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 232320.i	− 0.0421886i
$498$	0	0
$499$	−8.73256e6	−1.56997	−0.784983	−	0.619517i	$-0.787329\pi$
$499$	−8.73256e6	−1.56997	−0.784983	+	0.619517i	$0.787329\pi$
$500$	0	0
$501$	3.68107e6	0.655209
$502$	0	0
$503$	− 4.75426e6i	− 0.837843i	−0.908022	−	0.418921i	$-0.862408\pi$
$503$	− 4.75426e6i	− 0.837843i	0.908022	−	0.418921i	$-0.137592\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.10954e6i	0.364475i
$508$	0	0
$509$	2.13074e6	0.364532	0.182266	−	0.983249i	$-0.441657\pi$
$509$	2.13074e6	0.364532	0.182266	+	0.983249i	$0.441657\pi$
$510$	0	0
$511$	362848.	0.0614713
$512$	0	0
$513$	− 2.08494e6i	− 0.349784i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.80694e6i	− 0.955479i
$518$	0	0
$519$	−6.56959e6	−1.07058
$520$	0	0
$521$	1.10967e7	1.79101	0.895507	−	0.445048i	$-0.146813\pi$
$521$	1.10967e7	1.79101	0.895507	+	0.445048i	$0.146813\pi$
$522$	0	0
$523$	− 941252.i	− 0.150471i	−0.997166	−	0.0752353i	$-0.976029\pi$
$523$	− 941252.i	− 0.150471i	0.997166	−	0.0752353i	$-0.0239708\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.53338e6i	1.02473i
$528$	0	0
$529$	3.92729e6	0.610174
$530$	0	0
$531$	−2.12188e6	−0.326576
$532$	0	0
$533$	4.20690e6i	0.641422i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 6.34036e6i	− 0.948807i
$538$	0	0
$539$	−9.33476e6	−1.38399
$540$	0	0
$541$	2.39896e6	0.352395	0.176197	−	0.984355i	$-0.443620\pi$
$541$	2.39896e6	0.352395	0.176197	+	0.984355i	$0.443620\pi$
$542$	0	0
$543$	− 3.65265e6i	− 0.531629i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.55851e6i	0.222711i	0.993781	+	0.111355i	$0.0355191\pi$
$547$	1.55851e6i	0.222711i	−0.993781	+	0.111355i	$0.964481\pi$
$548$	0	0
$549$	−763182.	−0.108068
$550$	0	0
$551$	−3.24324e6	−0.455093
$552$	0	0
$553$	− 1.55699e6i	− 0.216508i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.19451e7i	1.63137i	0.578496	+	0.815685i	$0.303640\pi$
$557$	1.19451e7i	1.63137i	−0.578496	+	0.815685i	$0.696360\pi$
$558$	0	0
$559$	2.01428e6	0.272640
$560$	0	0
$561$	−5.51254e6	−0.739510
$562$	0	0
$563$	9.63906e6i	1.28163i	0.767694	+	0.640817i	$0.221404\pi$
$563$	9.63906e6i	1.28163i	−0.767694	+	0.640817i	$0.778596\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 104976.i	− 0.0137130i
$568$	0	0
$569$	−830106.	−0.107486	−0.0537431	−	0.998555i	$-0.517115\pi$
$569$	−830106.	−0.107486	−0.0537431	+	0.998555i	$0.517115\pi$
$570$	0	0
$571$	−3.32914e6	−0.427309	−0.213654	−	0.976909i	$-0.568537\pi$
$571$	−3.32914e6	−0.427309	−0.213654	+	0.976909i	$0.568537\pi$
$572$	0	0
$573$	1.64722e6i	0.209587i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.40026e6i	0.550223i	0.961412	+	0.275111i	$0.0887147\pi$
$577$	4.40026e6i	0.550223i	−0.961412	+	0.275111i	$0.911285\pi$
$578$	0	0
$579$	−7.67749e6	−0.951749
$580$	0	0
$581$	127296.	0.0156450
$582$	0	0
$583$	1.96035e7i	2.38870i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.25848e6i	− 0.869461i	−0.900561	−	0.434731i	$-0.856844\pi$
$587$	− 7.25848e6i	− 0.869461i	0.900561	−	0.434731i	$-0.143156\pi$
$588$	0	0
$589$	−1.72058e7	−2.04355
$590$	0	0
$591$	4.28755e6	0.504940
$592$	0	0
$593$	1.55171e7i	1.81206i	0.423210	+	0.906032i	$0.360903\pi$
$593$	1.55171e7i	1.81206i	−0.423210	+	0.906032i	$0.639097\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.23583e6i	− 0.486410i
$598$	0	0
$599$	1.02670e7	1.16917	0.584583	−	0.811334i	$-0.301258\pi$
$599$	1.02670e7	1.16917	0.584583	+	0.811334i	$0.301258\pi$
$600$	0	0
$601$	−9.42362e6	−1.06422	−0.532110	−	0.846675i	$-0.678601\pi$
$601$	−9.42362e6	−1.06422	−0.532110	+	0.846675i	$0.678601\pi$
$602$	0	0
$603$	4.14104e6i	0.463785i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7.67289e6i	− 0.845254i	−0.906304	−	0.422627i	$-0.861108\pi$
$607$	− 7.67289e6i	− 0.845254i	0.906304	−	0.422627i	$-0.138892\pi$
$608$	0	0
$609$	−163296.	−0.0178415
$610$	0	0
$611$	−3.80952e6	−0.412826
$612$	0	0
$613$	2.30598e6i	0.247859i	0.992291	+	0.123929i	$0.0395496\pi$
$613$	2.30598e6i	0.247859i	−0.992291	+	0.123929i	$0.960450\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 505254.i	− 0.0534314i	−0.999643	−	0.0267157i	$-0.991495\pi$
$617$	− 505254.i	− 0.0534314i	0.999643	−	0.0267157i	$-0.00850489\pi$
$618$	0	0
$619$	−4.61380e6	−0.483986	−0.241993	−	0.970278i	$-0.577801\pi$
$619$	−4.61380e6	−0.483986	−0.241993	+	0.970278i	$0.577801\pi$
$620$	0	0
$621$	−1.15474e6	−0.120158
$622$	0	0
$623$	− 766560.i	− 0.0791272i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.45174e7i	− 1.47475i
$628$	0	0
$629$	−584268.	−0.0588824
$630$	0	0
$631$	4.77327e6	0.477247	0.238623	−	0.971112i	$-0.423304\pi$
$631$	4.77327e6	0.477247	0.238623	+	0.971112i	$0.423304\pi$
$632$	0	0
$633$	− 127260.i	− 0.0126236i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.12387e6i	0.597967i
$638$	0	0
$639$	−1.17612e6	−0.113946
$640$	0	0
$641$	−1.09254e7	−1.05025	−0.525125	−	0.851025i	$-0.675981\pi$
$641$	−1.09254e7	−1.05025	−0.525125	+	0.851025i	$0.675981\pi$
$642$	0	0
$643$	− 1.13952e6i	− 0.108691i	−0.998522	−	0.0543454i	$-0.982693\pi$
$643$	− 1.13952e6i	− 0.108691i	0.998522	−	0.0543454i	$-0.0173072\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.10955e6i	0.479868i	0.970789	+	0.239934i	$0.0771259\pi$
$647$	5.10955e6i	0.479868i	−0.970789	+	0.239934i	$0.922874\pi$
$648$	0	0
$649$	−1.47745e7	−1.37690
$650$	0	0
$651$	−866304.	−0.0801157
$652$	0	0
$653$	7.81591e6i	0.717293i	0.933474	+	0.358646i	$0.116762\pi$
$653$	7.81591e6i	0.717293i	−0.933474	+	0.358646i	$0.883238\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.83692e6i	− 0.166026i
$658$	0	0
$659$	1.05905e7	0.949954	0.474977	−	0.879998i	$-0.342456\pi$
$659$	1.05905e7	0.949954	0.474977	+	0.879998i	$0.342456\pi$
$660$	0	0
$661$	4.67092e6	0.415814	0.207907	−	0.978149i	$-0.433335\pi$
$661$	4.67092e6	0.415814	0.207907	+	0.978149i	$0.433335\pi$
$662$	0	0
$663$	3.61638e6i	0.319514i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.79626e6i	0.156334i
$668$	0	0
$669$	−9.72468e6	−0.840059
$670$	0	0
$671$	−5.31401e6	−0.455634
$672$	0	0
$673$	− 2.07728e7i	− 1.76790i	−0.467585	−	0.883948i	$-0.654876\pi$
$673$	− 2.07728e7i	− 1.76790i	0.467585	−	0.883948i	$-0.345124\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.67497e7i	1.40454i	0.711911	+	0.702270i	$0.247830\pi$
$677$	1.67497e7i	1.40454i	−0.711911	+	0.702270i	$0.752170\pi$
$678$	0	0
$679$	2.25181e6	0.187438
$680$	0	0
$681$	3.06450e6	0.253216
$682$	0	0
$683$	− 4.41700e6i	− 0.362306i	−0.983455	−	0.181153i	$-0.942017\pi$
$683$	− 4.41700e6i	− 0.362306i	0.983455	−	0.181153i	$-0.0579829\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.08835e6i	0.572998i
$688$	0	0
$689$	1.28605e7	1.03207
$690$	0	0
$691$	1.86481e7	1.48573	0.742863	−	0.669443i	$-0.233467\pi$
$691$	1.86481e7	1.48573	0.742863	+	0.669443i	$0.233467\pi$
$692$	0	0
$693$	− 730944.i	− 0.0578164i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.23478e7i	− 0.962739i
$698$	0	0
$699$	1.36693e7	1.05816
$700$	0	0
$701$	7.28389e6	0.559845	0.279923	−	0.960023i	$-0.409691\pi$
$701$	7.28389e6	0.559845	0.279923	+	0.960023i	$0.409691\pi$
$702$	0	0
$703$	− 1.53868e6i	− 0.117425i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.36637e6i	− 0.102806i
$708$	0	0
$709$	−1.82063e7	−1.36021	−0.680105	−	0.733115i	$-0.738066\pi$
$709$	−1.82063e7	−1.36021	−0.680105	+	0.733115i	$0.738066\pi$
$710$	0	0
$711$	−7.88227e6	−0.584760
$712$	0	0
$713$	9.52934e6i	0.702003i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.06363e7i	− 0.772666i
$718$	0	0
$719$	−6.46901e6	−0.466676	−0.233338	−	0.972396i	$-0.574965\pi$
$719$	−6.46901e6	−0.466676	−0.233338	+	0.972396i	$0.574965\pi$
$720$	0	0
$721$	−3.17850e6	−0.227711
$722$	0	0
$723$	− 4.71008e6i	− 0.335106i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.39991e7i	− 0.982343i	−0.871063	−	0.491172i	$-0.836569\pi$
$727$	− 1.39991e7i	− 0.982343i	0.871063	−	0.491172i	$-0.163431\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−5.91218e6	−0.409218
$732$	0	0
$733$	− 2.51019e7i	− 1.72562i	−0.505526	−	0.862811i	$-0.668702\pi$
$733$	− 2.51019e7i	− 1.72562i	0.505526	−	0.862811i	$-0.331298\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.88339e7i	1.95540i
$738$	0	0
$739$	−9.73514e6	−0.655739	−0.327870	−	0.944723i	$-0.606331\pi$
$739$	−9.73514e6	−0.655739	−0.327870	+	0.944723i	$0.606331\pi$
$740$	0	0
$741$	−9.52380e6	−0.637184
$742$	0	0
$743$	− 3.03370e6i	− 0.201604i	−0.994906	−	0.100802i	$-0.967859\pi$
$743$	− 3.03370e6i	− 0.201604i	0.994906	−	0.100802i	$-0.0321409\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 644436.i	− 0.0422550i
$748$	0	0
$749$	−1.86029e6	−0.121165
$750$	0	0
$751$	−6.07557e6	−0.393086	−0.196543	−	0.980495i	$-0.562971\pi$
$751$	−6.07557e6	−0.393086	−0.196543	+	0.980495i	$0.562971\pi$
$752$	0	0
$753$	− 1.14967e7i	− 0.738901i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 4.82103e6i	− 0.305774i	−0.988244	−	0.152887i	$-0.951143\pi$
$757$	− 4.82103e6i	− 0.305774i	0.988244	−	0.152887i	$-0.0488570\pi$
$758$	0	0
$759$	−8.04038e6	−0.506608
$760$	0	0
$761$	5.82748e6	0.364770	0.182385	−	0.983227i	$-0.441618\pi$
$761$	5.82748e6	0.364770	0.182385	+	0.983227i	$0.441618\pi$
$762$	0	0
$763$	− 2.34755e6i	− 0.145984i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.69252e6i	0.594906i
$768$	0	0
$769$	2.71546e7	1.65587	0.827936	−	0.560822i	$-0.189515\pi$
$769$	2.71546e7	1.65587	0.827936	+	0.560822i	$0.189515\pi$
$770$	0	0
$771$	1.73801e7	1.05297
$772$	0	0
$773$	− 6678.00i	0 0.000401974i	−1.00000		0.000200987i	$-0.999936\pi$
$773$	− 6678.00i	0 0.000401974i	1.00000		0.000200987i	$-6.39761e-5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 77472.0i	− 0.00460354i
$778$	0	0
$779$	3.25182e7	1.91992
$780$	0	0
$781$	−8.18928e6	−0.480417
$782$	0	0
$783$	826686.i	0.0481877i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.44352e7i	0.830781i	0.909643	+	0.415391i	$0.136355\pi$
$787$	1.44352e7i	0.830781i	−0.909643	+	0.415391i	$0.863645\pi$
$788$	0	0
$789$	−7.40102e6	−0.423252
$790$	0	0
$791$	−3.37411e6	−0.191742
$792$	0	0
$793$	3.48614e6i	0.196862i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.90792e6i	− 0.106393i	−0.998584	−	0.0531967i	$-0.983059\pi$
$797$	− 1.90792e6i	− 0.106393i	0.998584	−	0.0531967i	$-0.0169410\pi$
$798$	0	0
$799$	1.11815e7	0.619629
$800$	0	0
$801$	−3.88071e6	−0.213712
$802$	0	0
$803$	− 1.27904e7i	− 0.699995i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.43263e7i	− 0.774371i
$808$	0	0
$809$	3.48543e7	1.87234	0.936171	−	0.351546i	$-0.114344\pi$
$809$	3.48543e7	1.87234	0.936171	+	0.351546i	$0.114344\pi$
$810$	0	0
$811$	1.41263e7	0.754182	0.377091	−	0.926176i	$-0.376924\pi$
$811$	1.41263e7	0.754182	0.377091	+	0.926176i	$0.376924\pi$
$812$	0	0
$813$	− 1.08995e7i	− 0.578336i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.55698e7i	− 0.816073i
$818$	0	0
$819$	−479520.	−0.0249803
$820$	0	0
$821$	1.78858e7	0.926082	0.463041	−	0.886337i	$-0.346758\pi$
$821$	1.78858e7	0.926082	0.463041	+	0.886337i	$0.346758\pi$
$822$	0	0
$823$	1.73695e7i	0.893897i	0.894560	+	0.446948i	$0.147489\pi$
$823$	1.73695e7i	0.893897i	−0.894560	+	0.446948i	$0.852511\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.84745e7i	− 0.939310i	−0.882850	−	0.469655i	$-0.844378\pi$
$827$	− 1.84745e7i	− 0.939310i	0.882850	−	0.469655i	$-0.155622\pi$
$828$	0	0
$829$	3.02647e7	1.52950	0.764750	−	0.644327i	$-0.222862\pi$
$829$	3.02647e7	1.52950	0.764750	+	0.644327i	$0.222862\pi$
$830$	0	0
$831$	2.10921e7	1.05954
$832$	0	0
$833$	− 1.79744e7i	− 0.897515i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.38566e6i	0.216382i
$838$	0	0
$839$	2.10237e7	1.03111	0.515555	−	0.856856i	$-0.327586\pi$
$839$	2.10237e7	1.03111	0.515555	+	0.856856i	$0.327586\pi$
$840$	0	0
$841$	−1.92252e7	−0.937305
$842$	0	0
$843$	1.67590e7i	0.812228i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.51272e6i	− 0.120347i
$848$	0	0
$849$	973908.	0.0463712
$850$	0	0
$851$	−852192.	−0.0403379
$852$	0	0
$853$	2.60690e7i	1.22674i	0.789796	+	0.613369i	$0.210186\pi$
$853$	2.60690e7i	1.22674i	−0.789796	+	0.613369i	$0.789814\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	489738.i	0.0227778i	0.999935	+	0.0113889i	$0.00362528\pi$
$857$	489738.i	0.0227778i	−0.999935	+	0.0113889i	$0.996375\pi$
$858$	0	0
$859$	2.17067e7	1.00371	0.501857	−	0.864951i	$-0.332650\pi$
$859$	2.17067e7	1.00371	0.501857	+	0.864951i	$0.332650\pi$
$860$	0	0
$861$	1.63728e6	0.0752688
$862$	0	0
$863$	− 2.91038e7i	− 1.33022i	−0.746747	−	0.665108i	$-0.768385\pi$
$863$	− 2.91038e7i	− 1.33022i	0.746747	−	0.665108i	$-0.231615\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.16415e6i	0.0977776i
$868$	0	0
$869$	−5.48840e7	−2.46545
$870$	0	0
$871$	1.89159e7	0.844853
$872$	0	0
$873$	− 1.13998e7i	− 0.506245i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.11648e7i	0.929215i	0.885517	+	0.464607i	$0.153805\pi$
$877$	2.11648e7i	0.929215i	−0.885517	+	0.464607i	$0.846195\pi$
$878$	0	0
$879$	−8.63692e6	−0.377039
$880$	0	0
$881$	−9.60651e6	−0.416990	−0.208495	−	0.978023i	$-0.566857\pi$
$881$	−9.60651e6	−0.416990	−0.208495	+	0.978023i	$0.566857\pi$
$882$	0	0
$883$	− 2.58825e7i	− 1.11713i	−0.829460	−	0.558566i	$-0.811352\pi$
$883$	− 2.58825e7i	− 1.11713i	0.829460	−	0.558566i	$-0.188648\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.43249e7i	1.03811i	0.854742	+	0.519053i	$0.173715\pi$
$887$	2.43249e7i	1.03811i	−0.854742	+	0.519053i	$0.826285\pi$
$888$	0	0
$889$	3.91078e6	0.165962
$890$	0	0
$891$	−3.70040e6	−0.156155
$892$	0	0
$893$	2.94466e7i	1.23568i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.27472e6i	0.218886i
$898$	0	0
$899$	6.82214e6	0.281528
$900$	0	0
$901$	−3.77472e7	−1.54908
$902$	0	0
$903$	− 783936.i	− 0.0319935i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 7.87371e6i	− 0.317805i	−0.987294	−	0.158903i	$-0.949204\pi$
$907$	− 7.87371e6i	− 0.317805i	0.987294	−	0.158903i	$-0.0507956\pi$
$908$	0	0
$909$	−6.91724e6	−0.277666
$910$	0	0
$911$	4.74478e7	1.89418	0.947088	−	0.320975i	$-0.104010\pi$
$911$	4.74478e7	1.89418	0.947088	+	0.320975i	$0.104010\pi$
$912$	0	0
$913$	− 4.48718e6i	− 0.178155i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.32493e6i	0.0913032i
$918$	0	0
$919$	−605288.	−0.0236414	−0.0118207	−	0.999930i	$-0.503763\pi$
$919$	−605288.	−0.0236414	−0.0118207	+	0.999930i	$0.503763\pi$
$920$	0	0
$921$	−1.46456e7	−0.568930
$922$	0	0
$923$	5.37240e6i	0.207570i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.60911e7i	0.615017i
$928$	0	0
$929$	9.52438e6	0.362074	0.181037	−	0.983476i	$-0.442055\pi$
$929$	9.52438e6	0.362074	0.181037	+	0.983476i	$0.442055\pi$
$930$	0	0
$931$	4.73359e7	1.78985
$932$	0	0
$933$	− 2.55677e7i	− 0.961585i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 9.15708e6i	− 0.340728i	−0.985381	−	0.170364i	$-0.945506\pi$
$937$	− 9.15708e6i	− 0.340728i	0.985381	−	0.170364i	$-0.0544944\pi$
$938$	0	0
$939$	1.45543e7	0.538676
$940$	0	0
$941$	2.46291e6	0.0906723	0.0453361	−	0.998972i	$-0.485564\pi$
$941$	2.46291e6	0.0906723	0.0453361	+	0.998972i	$0.485564\pi$
$942$	0	0
$943$	− 1.80101e7i	− 0.659533i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.13961e6i	0.294937i	0.989067	+	0.147468i	$0.0471124\pi$
$947$	8.13961e6i	0.294937i	−0.989067	+	0.147468i	$0.952888\pi$
$948$	0	0
$949$	−8.39086e6	−0.302441
$950$	0	0
$951$	1.83930e7	0.659479
$952$	0	0
$953$	− 1.65014e7i	− 0.588557i	−0.955720	−	0.294278i	$-0.904921\pi$
$953$	− 1.65014e7i	− 0.588557i	0.955720	−	0.294278i	$-0.0950792\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.75618e6i	0.203168i
$958$	0	0
$959$	−2.00765e6	−0.0704922
$960$	0	0
$961$	7.56310e6	0.264175
$962$	0	0
$963$	9.41771e6i	0.327250i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.72505e7i	− 0.937148i	−0.883424	−	0.468574i	$-0.844768\pi$
$967$	− 2.72505e7i	− 0.937148i	0.883424	−	0.468574i	$-0.155232\pi$
$968$	0	0
$969$	2.79536e7	0.956377
$970$	0	0
$971$	−1.77172e7	−0.603040	−0.301520	−	0.953460i	$-0.597494\pi$
$971$	−1.77172e7	−0.603040	−0.301520	+	0.953460i	$0.597494\pi$
$972$	0	0
$973$	4.02810e6i	0.136401i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.57062e7i	0.861591i	0.902450	+	0.430795i	$0.141767\pi$
$977$	2.57062e7i	0.861591i	−0.902450	+	0.430795i	$0.858233\pi$
$978$	0	0
$979$	−2.70212e7	−0.901049
$980$	0	0
$981$	−1.18845e7	−0.394283
$982$	0	0
$983$	− 4.05067e6i	− 0.133704i	−0.997763	−	0.0668518i	$-0.978705\pi$
$983$	− 4.05067e6i	− 0.133704i	0.997763	−	0.0668518i	$-0.0212955\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.48262e6i	0.0484438i
$988$	0	0
$989$	−8.62330e6	−0.280338
$990$	0	0
$991$	−1.71299e6	−0.0554078	−0.0277039	−	0.999616i	$-0.508820\pi$
$991$	−1.71299e6	−0.0554078	−0.0277039	+	0.999616i	$0.508820\pi$
$992$	0	0
$993$	9.03827e6i	0.290879i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.09874e7i	0.668683i	0.942452	+	0.334341i	$0.108514\pi$
$997$	2.09874e7i	0.668683i	−0.942452	+	0.334341i	$0.891486\pi$
$998$	0	0
$999$	−392202.	−0.0124336

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.6.d.a.49.2		2
3.2	odd	2		900.6.d.i.649.1		2
5.2	odd	4		300.6.a.e.1.1		1
5.3	odd	4		60.6.a.b.1.1	✓	1
5.4	even	2	inner	300.6.d.a.49.1		2
15.2	even	4		900.6.a.g.1.1		1
15.8	even	4		180.6.a.a.1.1		1
15.14	odd	2		900.6.d.i.649.2		2
20.3	even	4		240.6.a.m.1.1		1
40.3	even	4		960.6.a.c.1.1		1
40.13	odd	4		960.6.a.r.1.1		1
60.23	odd	4		720.6.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.a.b.1.1	✓	1	5.3	odd	4
180.6.a.a.1.1		1	15.8	even	4
240.6.a.m.1.1		1	20.3	even	4
300.6.a.e.1.1		1	5.2	odd	4
300.6.d.a.49.1		2	5.4	even	2	inner
300.6.d.a.49.2		2	1.1	even	1	trivial
720.6.a.f.1.1		1	60.23	odd	4
900.6.a.g.1.1		1	15.2	even	4
900.6.d.i.649.1		2	3.2	odd	2
900.6.d.i.649.2		2	15.14	odd	2
960.6.a.c.1.1		1	40.3	even	4
960.6.a.r.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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	300.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} -16.0000i q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} -16.0000i q^{7} -81.0000 q^{9} -564.000 q^{11} +370.000i q^{13} -1086.00i q^{17} +2860.00 q^{19} +144.000 q^{21} -1584.00i q^{23} -729.000i q^{27} -1134.00 q^{29} -6016.00 q^{31} -5076.00i q^{33} -538.000i q^{37} -3330.00 q^{39} +11370.0 q^{41} -5444.00i q^{43} +10296.0i q^{47} +16551.0 q^{49} +9774.00 q^{51} -34758.0i q^{53} +25740.0i q^{57} +26196.0 q^{59} +9422.00 q^{61} +1296.00i q^{63} -51124.0i q^{67} +14256.0 q^{69} +14520.0 q^{71} +22678.0i q^{73} +9024.00i q^{77} +97312.0 q^{79} +6561.00 q^{81} +7956.00i q^{83} -10206.0i q^{87} +47910.0 q^{89} +5920.00 q^{91} -54144.0i q^{93} +140738. i q^{97} +45684.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 162 q^{9}+O(q^{10}) \) 2 * q - 162 * q^9	\( 2 q - 162 q^{9} - 1128 q^{11} + 5720 q^{19} + 288 q^{21} - 2268 q^{29} - 12032 q^{31} - 6660 q^{39} + 22740 q^{41} + 33102 q^{49} + 19548 q^{51} + 52392 q^{59} + 18844 q^{61} + 28512 q^{69} + 29040 q^{71} + 194624 q^{79} + 13122 q^{81} + 95820 q^{89} + 11840 q^{91} + 91368 q^{99}+O(q^{100}) \) 2 * q - 162 * q^9 - 1128 * q^11 + 5720 * q^19 + 288 * q^21 - 2268 * q^29 - 12032 * q^31 - 6660 * q^39 + 22740 * q^41 + 33102 * q^49 + 19548 * q^51 + 52392 * q^59 + 18844 * q^61 + 28512 * q^69 + 29040 * q^71 + 194624 * q^79 + 13122 * q^81 + 95820 * q^89 + 11840 * q^91 + 91368 * q^99

Introduction

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