LMFDB - Embedded newform 1350.2.c.c.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.7798042729$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.2.c.c.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} -3.00000 q^{11} -5.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +4.00000 q^{19} +3.00000i q^{22} -9.00000i q^{23} -5.00000 q^{26} -2.00000i q^{28} -3.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} -10.0000i q^{37} -4.00000i q^{38} +1.00000i q^{43} +3.00000 q^{44} -9.00000 q^{46} -9.00000i q^{47} +3.00000 q^{49} +5.00000i q^{52} -12.0000i q^{53} -2.00000 q^{56} +3.00000i q^{58} +12.0000 q^{59} +2.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} -3.00000i q^{68} -12.0000 q^{71} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -6.00000i q^{77} +13.0000 q^{79} +6.00000i q^{83} +1.00000 q^{86} -3.00000i q^{88} -12.0000 q^{89} +10.0000 q^{91} +9.00000i q^{92} -9.00000 q^{94} +2.00000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} - 10 q^{26} - 6 q^{29} + 10 q^{31} + 6 q^{34} + 6 q^{44} - 18 q^{46} + 6 q^{49} - 4 q^{56} + 24 q^{59} + 4 q^{61} - 2 q^{64} - 24 q^{71} - 20 q^{74} - 8 q^{76} + 26 q^{79} + 2 q^{86} - 24 q^{89} + 20 q^{91} - 18 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 - 10 * q^26 - 6 * q^29 + 10 * q^31 + 6 * q^34 + 6 * q^44 - 18 * q^46 + 6 * q^49 - 4 * q^56 + 24 * q^59 + 4 * q^61 - 2 * q^64 - 24 * q^71 - 20 * q^74 - 8 * q^76 + 26 * q^79 + 2 * q^86 - 24 * q^89 + 20 * q^91 - 18 * q^94

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	− 5.00000i	− 1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	3.00000i	0.639602i
$23$	− 9.00000i	− 1.87663i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$	− 9.00000i	− 1.87663i	0.345782	−	0.938315i	$-0.387614\pi$
$24$	0	0
$25$	0	0
$26$	−5.00000	−0.980581
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−9.00000	−1.32698
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	5.00000i	0.693375i
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	3.00000i	0.393919i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 5.00000i	− 0.635001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 3.00000i	− 0.363803i
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	1.00000	0.107833
$87$	0	0
$88$	− 3.00000i	− 0.319801i
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	9.00000i	0.938315i
$93$	0	0
$94$	−9.00000	−0.928279
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−12.0000	−1.16554
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	0	0
$112$	2.00000i	0.188982i
$113$	− 15.0000i	− 1.41108i	−0.708669	−	0.705541i	$-0.750704\pi$
$113$	− 15.0000i	− 1.41108i	0.708669	−	0.705541i	$-0.249296\pi$
$114$	0	0
$115$	0	0
$116$	3.00000	0.278543
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	−5.00000	−0.449013
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	12.0000i	1.00702i
$143$	15.0000i	1.25436i
$144$	0	0
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	10.0000i	0.821995i
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	4.00000i	0.324443i
$153$	0	0
$154$	−6.00000	−0.483494
$155$	0	0
$156$	0	0
$157$	− 13.0000i	− 1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$	− 13.0000i	− 1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	− 13.0000i	− 1.03422i
$159$	0	0
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	0	0
$165$	0	0
$166$	6.00000	0.465690
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	− 1.00000i	− 0.0762493i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	12.0000i	0.899438i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	− 10.0000i	− 0.741249i
$183$	0	0
$184$	9.00000	0.663489
$185$	0	0
$186$	0	0
$187$	− 9.00000i	− 0.658145i
$188$	9.00000i	0.656392i
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	0	0
$202$	15.0000i	1.05540i
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	0	0
$206$	−2.00000	−0.139347
$207$	0	0
$208$	− 5.00000i	− 0.346688i
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	12.0000i	0.824163i
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	0	0
$217$	10.0000i	0.678844i
$218$	20.0000i	1.35457i
$219$	0	0
$220$	0	0
$221$	15.0000	1.00901
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−15.0000	−0.997785
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	− 3.00000i	− 0.196960i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	6.00000i	0.388922i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	− 20.0000i	− 1.27257i
$248$	5.00000i	0.317500i
$249$	0	0
$250$	0	0
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	0	0
$253$	27.0000i	1.69748i
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0000i	0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$257$	15.0000i	0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$258$	0	0
$259$	20.0000	1.24274
$260$	0	0
$261$	0	0
$262$	− 3.00000i	− 0.185341i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	3.00000i	0.181902i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	2.00000i	0.119952i
$279$	0	0
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	15.0000	0.886969
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 10.0000i	− 0.585206i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	− 21.0000i	− 1.21650i
$299$	−45.0000	−2.60242
$300$	0	0
$301$	−2.00000	−0.115278
$302$	− 11.0000i	− 0.632979i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	11.0000i	0.627803i	0.949456	+	0.313902i	$0.101636\pi$
$307$	11.0000i	0.627803i	−0.949456	+	0.313902i	$0.898364\pi$
$308$	6.00000i	0.341882i
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	− 8.00000i	− 0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$	− 8.00000i	− 0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	−13.0000	−0.731307
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	0	0
$322$	− 18.0000i	− 1.00310i
$323$	12.0000i	0.667698i
$324$	0	0
$325$	0	0
$326$	−11.0000	−0.609234
$327$	0	0
$328$	0	0
$329$	18.0000	0.992372
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	− 6.00000i	− 0.329293i
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	32.0000i	1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$	32.0000i	1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$	12.0000i	0.652714i
$339$	0	0
$340$	0	0
$341$	−15.0000	−0.812296
$342$	0	0
$343$	20.0000i	1.07990i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	18.0000	0.967686
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	3.00000i	0.159901i
$353$	3.00000i	0.159674i	0.996808	+	0.0798369i	$0.0254400\pi$
$353$	3.00000i	0.159674i	−0.996808	+	0.0798369i	$0.974560\pi$
$354$	0	0
$355$	0	0
$356$	12.0000	0.635999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 20.0000i	− 1.05118i
$363$	0	0
$364$	−10.0000	−0.524142
$365$	0	0
$366$	0	0
$367$	26.0000i	1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$	26.0000i	1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$	− 9.00000i	− 0.469157i
$369$	0	0
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	1.00000i	0.0517780i	0.999665	+	0.0258890i	$0.00824165\pi$
$373$	1.00000i	0.0517780i	−0.999665	+	0.0258890i	$0.991758\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	9.00000	0.464140
$377$	15.0000i	0.772539i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	6.00000i	0.306987i
$383$	− 9.00000i	− 0.459879i	−0.973205	−	0.229939i	$-0.926147\pi$
$383$	− 9.00000i	− 0.459879i	0.973205	−	0.229939i	$-0.0738528\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	27.0000	1.36545
$392$	3.00000i	0.151523i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.00000i	− 0.351320i	−0.984451	−	0.175660i	$-0.943794\pi$
$397$	− 7.00000i	− 0.351320i	0.984451	−	0.175660i	$-0.0562059\pi$
$398$	− 7.00000i	− 0.350878i
$399$	0	0
$400$	0	0
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	− 25.0000i	− 1.24534i
$404$	15.0000	0.746278
$405$	0	0
$406$	−6.00000	−0.297775
$407$	30.0000i	1.48704i
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	0	0
$412$	2.00000i	0.0985329i
$413$	24.0000i	1.18096i
$414$	0	0
$415$	0	0
$416$	−5.00000	−0.245145
$417$	0	0
$418$	12.0000i	0.586939i
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	22.0000i	1.07094i
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	6.00000i	0.290021i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	20.0000	0.957826
$437$	− 36.0000i	− 1.72211i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	− 15.0000i	− 0.713477i
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	−8.00000	−0.378811
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	15.0000i	0.705541i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	− 4.00000i	− 0.186908i
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 38.0000i	− 1.76601i	−0.469364	−	0.883005i	$-0.655517\pi$
$463$	− 38.0000i	− 1.76601i	0.469364	−	0.883005i	$-0.344483\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	6.00000	0.277945
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	− 3.00000i	− 0.137940i
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	0	0
$478$	6.00000i	0.274434i
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−50.0000	−2.27980
$482$	19.0000i	0.865426i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	2.00000i	0.0905357i
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	− 9.00000i	− 0.405340i
$494$	−20.0000	−0.899843
$495$	0	0
$496$	5.00000	0.224507
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	− 9.00000i	− 0.401690i
$503$	− 9.00000i	− 0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$	− 9.00000i	− 0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$	0	0
$505$	0	0
$506$	27.0000	1.20030
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	15.0000	0.661622
$515$	0	0
$516$	0	0
$517$	27.0000i	1.18746i
$518$	− 20.0000i	− 0.878750i
$519$	0	0
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	31.0000i	1.35554i	0.735276	+	0.677768i	$0.237052\pi$
$523$	31.0000i	1.35554i	−0.735276	+	0.677768i	$0.762948\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	12.0000	0.523225
$527$	15.0000i	0.653410i
$528$	0	0
$529$	−58.0000	−2.52174
$530$	0	0
$531$	0	0
$532$	− 8.00000i	− 0.346844i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	9.00000i	0.388018i
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	− 8.00000i	− 0.343629i
$543$	0	0
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	− 13.0000i	− 0.555840i	−0.960604	−	0.277920i	$-0.910355\pi$
$547$	− 13.0000i	− 0.555840i	0.960604	−	0.277920i	$-0.0896450\pi$
$548$	− 6.00000i	− 0.256307i
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	26.0000i	1.10563i
$554$	−10.0000	−0.424859
$555$	0	0
$556$	2.00000	0.0848189
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	0	0
$561$	0	0
$562$	− 30.0000i	− 1.26547i
$563$	6.00000i	0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$	6.00000i	0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	0	0
$565$	0	0
$566$	−32.0000	−1.34506
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	− 15.0000i	− 0.627182i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	32.0000i	1.33218i	0.745873	+	0.666089i	$0.232033\pi$
$577$	32.0000i	1.33218i	−0.745873	+	0.666089i	$0.767967\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	36.0000i	1.49097i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	18.0000	0.743573
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	0	0
$592$	− 10.0000i	− 0.410997i
$593$	− 27.0000i	− 1.10876i	−0.832265	−	0.554379i	$-0.812956\pi$
$593$	− 27.0000i	− 1.10876i	0.832265	−	0.554379i	$-0.187044\pi$
$594$	0	0
$595$	0	0
$596$	−21.0000	−0.860194
$597$	0	0
$598$	45.0000i	1.84019i
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	2.00000i	0.0815139i
$603$	0	0
$604$	−11.0000	−0.447584
$605$	0	0
$606$	0	0
$607$	− 34.0000i	− 1.38002i	−0.723801	−	0.690009i	$-0.757607\pi$
$607$	− 34.0000i	− 1.38002i	0.723801	−	0.690009i	$-0.242393\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	−45.0000	−1.82051
$612$	0	0
$613$	19.0000i	0.767403i	0.923457	+	0.383701i	$0.125351\pi$
$613$	19.0000i	0.767403i	−0.923457	+	0.383701i	$0.874649\pi$
$614$	11.0000	0.443924
$615$	0	0
$616$	6.00000	0.241747
$617$	− 3.00000i	− 0.120775i	−0.998175	−	0.0603877i	$-0.980766\pi$
$617$	− 3.00000i	− 0.120775i	0.998175	−	0.0603877i	$-0.0192337\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	− 24.0000i	− 0.961540i
$624$	0	0
$625$	0	0
$626$	−8.00000	−0.319744
$627$	0	0
$628$	13.0000i	0.518756i
$629$	30.0000	1.19618
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	13.0000i	0.517112i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 15.0000i	− 0.594322i
$638$	− 9.00000i	− 0.356313i
$639$	0	0
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	− 23.0000i	− 0.907031i	−0.891248	−	0.453516i	$-0.850170\pi$
$643$	− 23.0000i	− 0.907031i	0.891248	−	0.453516i	$-0.149830\pi$
$644$	−18.0000	−0.709299
$645$	0	0
$646$	12.0000	0.472134
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	11.0000i	0.430793i
$653$	36.0000i	1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$	36.0000i	1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	− 18.0000i	− 0.701713i
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	− 14.0000i	− 0.544125i
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	0	0
$667$	27.0000i	1.04544i
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	22.0000i	0.848038i	0.905653	+	0.424019i	$0.139381\pi$
$673$	22.0000i	0.848038i	−0.905653	+	0.424019i	$0.860619\pi$
$674$	32.0000	1.23259
$675$	0	0
$676$	12.0000	0.461538
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	15.0000i	0.574380i
$683$	− 48.0000i	− 1.83667i	−0.395805	−	0.918334i	$-0.629534\pi$
$683$	− 48.0000i	− 1.83667i	0.395805	−	0.918334i	$-0.370466\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	1.00000i	0.0381246i
$689$	−60.0000	−2.28582
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	14.0000i	0.529908i
$699$	0	0
$700$	0	0
$701$	−21.0000	−0.793159	−0.396580	−	0.918000i	$-0.629803\pi$
$701$	−21.0000	−0.793159	−0.396580	+	0.918000i	$0.629803\pi$
$702$	0	0
$703$	− 40.0000i	− 1.50863i
$704$	3.00000	0.113067
$705$	0	0
$706$	3.00000	0.112906
$707$	− 30.0000i	− 1.12827i
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	0	0
$712$	− 12.0000i	− 0.449719i
$713$	− 45.0000i	− 1.68526i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	− 12.0000i	− 0.447836i
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	3.00000i	0.111648i
$723$	0	0
$724$	−20.0000	−0.743294
$725$	0	0
$726$	0	0
$727$	44.0000i	1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$	44.0000i	1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$	10.0000i	0.370625i
$729$	0	0
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	−9.00000	−0.331744
$737$	12.0000i	0.442026i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.881068i
$743$	− 27.0000i	− 0.990534i	−0.868741	−	0.495267i	$-0.835070\pi$
$743$	− 27.0000i	− 0.990534i	0.868741	−	0.495267i	$-0.164930\pi$
$744$	0	0
$745$	0	0
$746$	1.00000	0.0366126
$747$	0	0
$748$	9.00000i	0.329073i
$749$	12.0000	0.438470
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	− 9.00000i	− 0.328196i
$753$	0	0
$754$	15.0000	0.546268
$755$	0	0
$756$	0	0
$757$	− 13.0000i	− 0.472493i	−0.971693	−	0.236247i	$-0.924083\pi$
$757$	− 13.0000i	− 0.472493i	0.971693	−	0.236247i	$-0.0759173\pi$
$758$	− 16.0000i	− 0.581146i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 40.0000i	− 1.44810i
$764$	6.00000	0.217072
$765$	0	0
$766$	−9.00000	−0.325183
$767$	− 60.0000i	− 2.16647i
$768$	0	0
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	0	0
$772$	− 10.0000i	− 0.359908i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	− 9.00000i	− 0.322666i
$779$	0	0
$780$	0	0
$781$	36.0000	1.28818
$782$	− 27.0000i	− 0.965518i
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	− 1.00000i	− 0.0356462i	−0.999841	−	0.0178231i	$-0.994326\pi$
$787$	− 1.00000i	− 0.0356462i	0.999841	−	0.0178231i	$-0.00567356\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	30.0000	1.06668
$792$	0	0
$793$	− 10.0000i	− 0.355110i
$794$	−7.00000	−0.248421
$795$	0	0
$796$	−7.00000	−0.248108
$797$	36.0000i	1.27519i	0.770374	+	0.637593i	$0.220070\pi$
$797$	36.0000i	1.27519i	−0.770374	+	0.637593i	$0.779930\pi$
$798$	0	0
$799$	27.0000	0.955191
$800$	0	0
$801$	0	0
$802$	36.0000i	1.27120i
$803$	− 30.0000i	− 1.05868i
$804$	0	0
$805$	0	0
$806$	−25.0000	−0.880587
$807$	0	0
$808$	− 15.0000i	− 0.527698i
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	6.00000i	0.210559i
$813$	0	0
$814$	30.0000	1.05150
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	23.0000i	0.804176i
$819$	0	0
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	2.00000	0.0696733
$825$	0	0
$826$	24.0000	0.835067
$827$	42.0000i	1.46048i	0.683189	+	0.730242i	$0.260592\pi$
$827$	42.0000i	1.46048i	−0.683189	+	0.730242i	$0.739408\pi$
$828$	0	0
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	0	0
$831$	0	0
$832$	5.00000i	0.173344i
$833$	9.00000i	0.311832i
$834$	0	0
$835$	0	0
$836$	12.0000	0.415029
$837$	0	0
$838$	3.00000i	0.103633i
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	4.00000i	0.137849i
$843$	0	0
$844$	22.0000	0.757271
$845$	0	0
$846$	0	0
$847$	− 4.00000i	− 0.137442i
$848$	− 12.0000i	− 0.412082i
$849$	0	0
$850$	0	0
$851$	−90.0000	−3.08516
$852$	0	0
$853$	19.0000i	0.650548i	0.945620	+	0.325274i	$0.105456\pi$
$853$	19.0000i	0.650548i	−0.945620	+	0.325274i	$0.894544\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	6.00000	0.205076
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	0	0
$859$	−2.00000	−0.0682391	−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000	−0.0682391	−0.0341196	+	0.999418i	$0.510863\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.0000i	0.714848i	0.933942	+	0.357424i	$0.116345\pi$
$863$	21.0000i	0.714848i	−0.933942	+	0.357424i	$0.883655\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	− 10.0000i	− 0.339422i
$869$	−39.0000	−1.32298
$870$	0	0
$871$	−20.0000	−0.677674
$872$	− 20.0000i	− 0.677285i
$873$	0	0
$874$	−36.0000	−1.21772
$875$	0	0
$876$	0	0
$877$	− 1.00000i	− 0.0337676i	−0.999857	−	0.0168838i	$-0.994625\pi$
$877$	− 1.00000i	− 0.0337676i	0.999857	−	0.0168838i	$-0.00537454\pi$
$878$	8.00000i	0.269987i
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	−15.0000	−0.504505
$885$	0	0
$886$	−12.0000	−0.403148
$887$	3.00000i	0.100730i	0.998731	+	0.0503651i	$0.0160385\pi$
$887$	3.00000i	0.100730i	−0.998731	+	0.0503651i	$0.983962\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	8.00000i	0.267860i
$893$	− 36.0000i	− 1.20469i
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	6.00000i	0.200223i
$899$	−15.0000	−0.500278
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	15.0000	0.498893
$905$	0	0
$906$	0	0
$907$	59.0000i	1.95906i	0.201291	+	0.979531i	$0.435486\pi$
$907$	59.0000i	1.95906i	−0.201291	+	0.979531i	$0.564514\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	− 18.0000i	− 0.595713i
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−4.00000	−0.132164
$917$	6.00000i	0.198137i
$918$	0	0
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	0	0
$921$	0	0
$922$	18.0000i	0.592798i
$923$	60.0000i	1.97492i
$924$	0	0
$925$	0	0
$926$	−38.0000	−1.24876
$927$	0	0
$928$	3.00000i	0.0984798i
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	− 8.00000i	− 0.261209i
$939$	0	0
$940$	0	0
$941$	−33.0000	−1.07577	−0.537885	−	0.843018i	$-0.680776\pi$
$941$	−33.0000	−1.07577	−0.537885	+	0.843018i	$0.680776\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	42.0000i	1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$	42.0000i	1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	0	0
$949$	50.0000	1.62307
$950$	0	0
$951$	0	0
$952$	− 6.00000i	− 0.194461i
$953$	15.0000i	0.485898i	0.970039	+	0.242949i	$0.0781147\pi$
$953$	15.0000i	0.485898i	−0.970039	+	0.242949i	$0.921885\pi$
$954$	0	0
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	− 36.0000i	− 1.16311i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	50.0000i	1.61206i
$963$	0	0
$964$	19.0000	0.611949
$965$	0	0
$966$	0	0
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	57.0000	1.82922	0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000	1.82922	0.914609	+	0.404341i	$0.132499\pi$
$972$	0	0
$973$	− 4.00000i	− 0.128234i
$974$	20.0000	0.640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 33.0000i	− 1.05576i	−0.849318	−	0.527882i	$-0.822986\pi$
$977$	− 33.0000i	− 1.05576i	0.849318	−	0.527882i	$-0.177014\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	0	0
$982$	− 12.0000i	− 0.382935i
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	0	0
$985$	0	0
$986$	−9.00000	−0.286618
$987$	0	0
$988$	20.0000i	0.636285i
$989$	9.00000	0.286183
$990$	0	0
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	− 5.00000i	− 0.158750i
$993$	0	0
$994$	−24.0000	−0.761234
$995$	0	0
$996$	0	0
$997$	− 37.0000i	− 1.17180i	−0.810383	−	0.585901i	$-0.800741\pi$
$997$	− 37.0000i	− 1.17180i	0.810383	−	0.585901i	$-0.199259\pi$
$998$	− 4.00000i	− 0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.c.c.649.1		2
3.2	odd	2		1350.2.c.j.649.2		2
5.2	odd	4		1350.2.a.o.1.1		1
5.3	odd	4		270.2.a.b.1.1	✓	1
5.4	even	2	inner	1350.2.c.c.649.2		2
15.2	even	4		1350.2.a.d.1.1		1
15.8	even	4		270.2.a.c.1.1	yes	1
15.14	odd	2		1350.2.c.j.649.1		2
20.3	even	4		2160.2.a.q.1.1		1
40.3	even	4		8640.2.a.e.1.1		1
40.13	odd	4		8640.2.a.y.1.1		1
45.13	odd	12		810.2.e.i.541.1		2
45.23	even	12		810.2.e.d.541.1		2
45.38	even	12		810.2.e.d.271.1		2
45.43	odd	12		810.2.e.i.271.1		2
60.23	odd	4		2160.2.a.b.1.1		1
120.53	even	4		8640.2.a.bx.1.1		1
120.83	odd	4		8640.2.a.bn.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.a.b.1.1	✓	1	5.3	odd	4
270.2.a.c.1.1	yes	1	15.8	even	4
810.2.e.d.271.1		2	45.38	even	12
810.2.e.d.541.1		2	45.23	even	12
810.2.e.i.271.1		2	45.43	odd	12
810.2.e.i.541.1		2	45.13	odd	12
1350.2.a.d.1.1		1	15.2	even	4
1350.2.a.o.1.1		1	5.2	odd	4
1350.2.c.c.649.1		2	1.1	even	1	trivial
1350.2.c.c.649.2		2	5.4	even	2	inner
1350.2.c.j.649.1		2	15.14	odd	2
1350.2.c.j.649.2		2	3.2	odd	2
2160.2.a.b.1.1		1	60.23	odd	4
2160.2.a.q.1.1		1	20.3	even	4
8640.2.a.e.1.1		1	40.3	even	4
8640.2.a.y.1.1		1	40.13	odd	4
8640.2.a.bn.1.1		1	120.83	odd	4
8640.2.a.bx.1.1		1	120.53	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} -3.00000 q^{11} -5.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +4.00000 q^{19} +3.00000i q^{22} -9.00000i q^{23} -5.00000 q^{26} -2.00000i q^{28} -3.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} -10.0000i q^{37} -4.00000i q^{38} +1.00000i q^{43} +3.00000 q^{44} -9.00000 q^{46} -9.00000i q^{47} +3.00000 q^{49} +5.00000i q^{52} -12.0000i q^{53} -2.00000 q^{56} +3.00000i q^{58} +12.0000 q^{59} +2.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} -3.00000i q^{68} -12.0000 q^{71} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -6.00000i q^{77} +13.0000 q^{79} +6.00000i q^{83} +1.00000 q^{86} -3.00000i q^{88} -12.0000 q^{89} +10.0000 q^{91} +9.00000i q^{92} -9.00000 q^{94} +2.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} - 10 q^{26} - 6 q^{29} + 10 q^{31} + 6 q^{34} + 6 q^{44} - 18 q^{46} + 6 q^{49} - 4 q^{56} + 24 q^{59} + 4 q^{61} - 2 q^{64} - 24 q^{71} - 20 q^{74} - 8 q^{76} + 26 q^{79} + 2 q^{86} - 24 q^{89} + 20 q^{91} - 18 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 - 10 * q^26 - 6 * q^29 + 10 * q^31 + 6 * q^34 + 6 * q^44 - 18 * q^46 + 6 * q^49 - 4 * q^56 + 24 * q^59 + 4 * q^61 - 2 * q^64 - 24 * q^71 - 20 * q^74 - 8 * q^76 + 26 * q^79 + 2 * q^86 - 24 * q^89 + 20 * q^91 - 18 * q^94

Introduction

L-functions

Modular forms

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