LMFDB - Embedded newform 2850.2.d.f.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.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:	$22.7573645761$
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 570)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.f.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +4.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} -10.0000 q^{29} +1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} +2.00000 q^{41} +4.00000i q^{42} -4.00000i q^{43} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} +1.00000i q^{57} -10.0000i q^{58} -8.00000 q^{59} +6.00000 q^{61} +4.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -2.00000i q^{68} +1.00000i q^{72} -14.0000i q^{73} +2.00000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} -12.0000i q^{83} -4.00000 q^{84} +4.00000 q^{86} -10.0000i q^{87} -10.0000 q^{89} +8.00000 q^{91} -1.00000 q^{96} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{14} + 2 q^{16} + 2 q^{19} + 8 q^{21} + 2 q^{24} - 4 q^{26} - 20 q^{29} - 4 q^{34} + 2 q^{36} - 4 q^{39} + 4 q^{41} - 18 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 16 q^{59} + 12 q^{61} - 2 q^{64} + 4 q^{74} - 2 q^{76} + 2 q^{81} - 8 q^{84} + 8 q^{86} - 20 q^{89} + 16 q^{91} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^14 + 2 * q^16 + 2 * q^19 + 8 * q^21 + 2 * q^24 - 4 * q^26 - 20 * q^29 - 4 * q^34 + 2 * q^36 - 4 * q^39 + 4 * q^41 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 16 * q^59 + 12 * q^61 - 2 * q^64 + 4 * q^74 - 2 * q^76 + 2 * q^81 - 8 * q^84 + 8 * q^86 - 20 * q^89 + 16 * q^91 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$.

$n$	$1027$	$1351$	$1901$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	− 1.00000i	− 0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	4.00000i	0.755929i
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	1.00000i	0.162221i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	4.00000i	0.617213i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 2.00000i	− 0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	1.00000i	0.132453i
$58$	− 10.0000i	− 1.31306i
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	4.00000	0.431331
$87$	− 10.0000i	− 1.07211i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 20.0000i	− 1.97066i	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	− 20.0000i	− 1.97066i	0.170664	−	0.985329i	$-0.445409\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	1.00000i	0.0962250i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 4.00000i	− 0.377964i
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	10.0000	0.928477
$117$	− 2.00000i	− 0.184900i
$118$	− 8.00000i	− 0.736460i
$119$	8.00000	0.733359
$120$	0	0
$121$	−11.0000	−1.00000
$122$	6.00000i	0.543214i
$123$	2.00000i	0.180334i
$124$	0	0
$125$	0	0
$126$	−4.00000	−0.356348
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	12.0000	1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	− 9.00000i	− 0.742307i
$148$	2.00000i	0.164399i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	− 4.00000i	− 0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	4.00000i	0.304997i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	0	0
$177$	− 8.00000i	− 0.601317i
$178$	− 10.0000i	− 0.749532i
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	8.00000i	0.592999i
$183$	6.00000i	0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	9.00000	0.642857
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	− 14.0000i	− 0.985037i
$203$	40.0000i	2.80745i
$204$	2.00000	0.140028
$205$	0	0
$206$	20.0000	1.39347
$207$	0	0
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	2.00000i	0.135457i
$219$	14.0000	0.946032
$220$	0	0
$221$	−4.00000	−0.269069
$222$	2.00000i	0.134231i
$223$	− 12.0000i	− 0.803579i	−0.915732	−	0.401790i	$-0.868388\pi$
$223$	− 12.0000i	− 0.803579i	0.915732	−	0.401790i	$-0.131612\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	2.00000	0.133038
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	10.0000i	0.656532i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	8.00000	0.520756
$237$	0	0
$238$	8.00000i	0.518563i
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000i	0.0641500i
$244$	−6.00000	−0.384111
$245$	0	0
$246$	−2.00000	−0.127515
$247$	2.00000i	0.127257i
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	− 4.00000i	− 0.251976i
$253$	0	0
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	4.00000i	0.249029i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	32.0000i	1.97320i	0.163144	+	0.986602i	$0.447836\pi$
$263$	32.0000i	1.97320i	−0.163144	+	0.986602i	$0.552164\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	− 10.0000i	− 0.611990i
$268$	12.0000i	0.733017i
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	2.00000i	0.121268i
$273$	8.00000i	0.484182i
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 12.0000i	− 0.719712i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.00000i	− 0.472225i
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	2.00000	0.117242
$292$	14.0000i	0.819288i
$293$	26.0000i	1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$	26.0000i	1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	−16.0000	−0.922225
$302$	− 16.0000i	− 0.920697i
$303$	− 14.0000i	− 0.804279i
$304$	1.00000	0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	20.0000	1.13776
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	2.00000i	0.113228i
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	6.00000i	0.336463i
$319$	0	0
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	2.00000i	0.111283i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	2.00000i	0.110600i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	12.0000i	0.658586i
$333$	2.00000i	0.109599i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	4.00000	0.218218
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	9.00000i	0.489535i
$339$	2.00000	0.108625
$340$	0	0
$341$	0	0
$342$	− 1.00000i	− 0.0540738i
$343$	8.00000i	0.431959i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	10.0000i	0.536056i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	10.0000	0.529999
$357$	8.00000i	0.423405i
$358$	16.0000i	0.845626i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 10.0000i	− 0.525588i
$363$	− 11.0000i	− 0.577350i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	−6.00000	−0.313625
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 20.0000i	− 1.03005i
$378$	− 4.00000i	− 0.205738i
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	8.00000i	0.409316i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	22.0000	1.11977
$387$	4.00000i	0.203331i
$388$	2.00000i	0.101535i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	9.00000i	0.454569i
$393$	0	0
$394$	10.0000	0.503793
$395$	0	0
$396$	0	0
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	8.00000i	0.401004i
$399$	4.00000	0.200250
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	12.0000i	0.598506i
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−40.0000	−1.98517
$407$	0	0
$408$	2.00000i	0.0990148i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	20.0000i	0.985329i
$413$	32.0000i	1.57462i
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 12.0000i	− 0.587643i
$418$	0	0
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 24.0000i	− 1.16144i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 38.0000i	− 1.82616i	−0.407777	−	0.913082i	$-0.633696\pi$
$433$	− 38.0000i	− 1.82616i	0.407777	−	0.913082i	$-0.366304\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	14.0000i	0.668946i
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	− 4.00000i	− 0.190261i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	12.0000	0.568216
$447$	6.00000i	0.283790i
$448$	4.00000i	0.188982i
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	0	0
$452$	2.00000i	0.0940721i
$453$	− 16.0000i	− 0.751746i
$454$	20.0000	0.938647
$455$	0	0
$456$	1.00000	0.0468293
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	− 22.0000i	− 1.02799i
$459$	2.00000	0.0933520
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	36.0000i	1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$	36.0000i	1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	−6.00000	−0.277945
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	2.00000i	0.0924500i
$469$	−48.0000	−2.21643
$470$	0	0
$471$	−22.0000	−1.01371
$472$	8.00000i	0.368230i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	6.00000i	0.274721i
$478$	− 8.00000i	− 0.365911i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	− 6.00000i	− 0.271607i
$489$	−12.0000	−0.542659
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	− 20.0000i	− 0.900755i
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	0	0
$497$	0	0
$498$	12.0000i	0.537733i
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	24.0000i	1.07117i
$503$	8.00000i	0.356702i	0.983967	+	0.178351i	$0.0570763\pi$
$503$	8.00000i	0.356702i	−0.983967	+	0.178351i	$0.942924\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	4.00000i	0.177471i
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	1.00000i	0.0441942i
$513$	− 1.00000i	− 0.0441511i
$514$	14.0000	0.617514
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	− 8.00000i	− 0.351500i
$519$	14.0000	0.614532
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	10.0000i	0.437688i
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	0	0
$525$	0	0
$526$	−32.0000	−1.39527
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	8.00000	0.347170
$532$	4.00000i	0.173422i
$533$	4.00000i	0.173259i
$534$	10.0000	0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	16.0000i	0.690451i
$538$	− 10.0000i	− 0.431131i
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	− 10.0000i	− 0.429141i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	− 18.0000i	− 0.768922i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−10.0000	−0.426014
$552$	0	0
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	12.0000	0.508913
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	− 6.00000i	− 0.253095i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	− 4.00000i	− 0.167984i
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	8.00000	0.333914
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	13.0000i	0.540729i
$579$	22.0000	0.914289
$580$	0	0
$581$	−48.0000	−1.99138
$582$	2.00000i	0.0829027i
$583$	0	0
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−26.0000	−1.07405
$587$	20.0000i	0.825488i	0.910847	+	0.412744i	$0.135430\pi$
$587$	20.0000i	0.825488i	−0.910847	+	0.412744i	$0.864570\pi$
$588$	9.00000i	0.371154i
$589$	0	0
$590$	0	0
$591$	10.0000	0.411345
$592$	− 2.00000i	− 0.0821995i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	8.00000i	0.327418i
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 16.0000i	− 0.652111i
$603$	12.0000i	0.488678i
$604$	16.0000	0.651031
$605$	0	0
$606$	14.0000	0.568711
$607$	− 12.0000i	− 0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$	− 12.0000i	− 0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$	1.00000i	0.0405554i
$609$	−40.0000	−1.62088
$610$	0	0
$611$	0	0
$612$	2.00000i	0.0808452i
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	20.0000i	0.804518i
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	0	0
$621$	0	0
$622$	− 32.0000i	− 1.28308i
$623$	40.0000i	1.60257i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	14.0000	0.559553
$627$	0	0
$628$	− 22.0000i	− 0.877896i
$629$	4.00000	0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	20.0000i	0.794929i
$634$	18.0000	0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	4.00000i	0.157867i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 12.0000i	− 0.469956i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	2.00000	0.0780869
$657$	14.0000i	0.546192i
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	− 12.0000i	− 0.466393i
$663$	− 4.00000i	− 0.155347i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	− 16.0000i	− 0.619059i
$669$	12.0000	0.463947
$670$	0	0
$671$	0	0
$672$	4.00000i	0.154303i
$673$	− 38.0000i	− 1.46479i	−0.680879	−	0.732396i	$-0.738402\pi$
$673$	− 38.0000i	− 1.46479i	0.680879	−	0.732396i	$-0.261598\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	2.00000i	0.0768095i
$679$	−8.00000	−0.307012
$680$	0	0
$681$	20.0000	0.766402
$682$	0	0
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−8.00000	−0.305441
$687$	− 22.0000i	− 0.839352i
$688$	− 4.00000i	− 0.152499i
$689$	12.0000	0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	14.0000i	0.532200i
$693$	0	0
$694$	−28.0000	−1.06287
$695$	0	0
$696$	−10.0000	−0.379049
$697$	4.00000i	0.151511i
$698$	10.0000i	0.378506i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	2.00000i	0.0754851i
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	56.0000i	2.10610i
$708$	8.00000i	0.300658i
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	0	0
$712$	10.0000i	0.374766i
$713$	0	0
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−16.0000	−0.597948
$717$	− 8.00000i	− 0.298765i
$718$	8.00000i	0.298557i
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	−80.0000	−2.97936
$722$	1.00000i	0.0372161i
$723$	2.00000i	0.0743808i
$724$	10.0000	0.371647
$725$	0	0
$726$	11.0000	0.408248
$727$	4.00000i	0.148352i	0.997245	+	0.0741759i	$0.0236326\pi$
$727$	4.00000i	0.148352i	−0.997245	+	0.0741759i	$0.976367\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	− 6.00000i	− 0.221766i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	0	0
$737$	0	0
$738$	− 2.00000i	− 0.0736210i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	− 24.0000i	− 0.881068i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	12.0000i	0.439057i
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	24.0000i	0.874609i
$754$	20.0000	0.728357
$755$	0	0
$756$	4.00000	0.145479
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	− 28.0000i	− 1.01701i
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	4.00000i	0.144905i
$763$	− 8.00000i	− 0.289619i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	0	0
$767$	− 16.0000i	− 0.577727i
$768$	1.00000i	0.0360844i
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	22.0000i	0.791797i
$773$	10.0000i	0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$	10.0000i	0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	− 8.00000i	− 0.286998i
$778$	6.00000i	0.215110i
$779$	2.00000	0.0716574
$780$	0	0
$781$	0	0
$782$	0	0
$783$	10.0000i	0.357371i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	10.0000i	0.356235i
$789$	−32.0000	−1.13923
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	12.0000i	0.426132i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	−8.00000	−0.283552
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	4.00000i	0.141598i
$799$	0	0
$800$	0	0
$801$	10.0000	0.353333
$802$	18.0000i	0.635602i
$803$	0	0
$804$	−12.0000	−0.423207
$805$	0	0
$806$	0	0
$807$	− 10.0000i	− 0.352017i
$808$	14.0000i	0.492518i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	− 40.0000i	− 1.40372i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 4.00000i	− 0.139942i
$818$	− 26.0000i	− 0.909069i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	− 18.0000i	− 0.627822i
$823$	− 36.0000i	− 1.25488i	−0.778664	−	0.627441i	$-0.784103\pi$
$823$	− 36.0000i	− 1.25488i	0.778664	−	0.627441i	$-0.215897\pi$
$824$	−20.0000	−0.696733
$825$	0	0
$826$	−32.0000	−1.11342
$827$	4.00000i	0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$	4.00000i	0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	− 2.00000i	− 0.0693375i
$833$	− 18.0000i	− 0.623663i
$834$	12.0000	0.415526
$835$	0	0
$836$	0	0
$837$	0	0
$838$	16.0000i	0.552711i
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	− 34.0000i	− 1.17172i
$843$	− 6.00000i	− 0.206651i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	44.0000i	1.51186i
$848$	− 6.00000i	− 0.206041i
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	24.0000	0.821263
$855$	0	0
$856$	−4.00000	−0.136717
$857$	− 14.0000i	− 0.478231i	−0.970991	−	0.239115i	$-0.923143\pi$
$857$	− 14.0000i	− 0.478231i	0.970991	−	0.239115i	$-0.0768574\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	32.0000i	1.08992i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	38.0000	1.29129
$867$	13.0000i	0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	− 2.00000i	− 0.0677285i
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	−14.0000	−0.473016
$877$	− 50.0000i	− 1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$	− 50.0000i	− 1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$	0	0
$879$	−26.0000	−0.876958
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	9.00000i	0.303046i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−4.00000	−0.134383
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	12.0000i	0.401790i
$893$	0	0
$894$	−6.00000	−0.200670
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	− 34.0000i	− 1.13459i
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	− 16.0000i	− 0.532447i
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	16.0000	0.531564
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	20.0000i	0.663723i
$909$	14.0000	0.464351
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	1.00000i	0.0331133i
$913$	0	0
$914$	−6.00000	−0.198462
$915$	0	0
$916$	22.0000	0.726900
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−36.0000	−1.18303
$927$	20.0000i	0.656886i
$928$	− 10.0000i	− 0.328266i
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	− 6.00000i	− 0.196537i
$933$	− 32.0000i	− 1.04763i
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	− 48.0000i	− 1.56726i
$939$	14.0000	0.456873
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	− 22.0000i	− 0.716799i
$943$	0	0
$944$	−8.00000	−0.260378
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	18.0000	0.583690
$952$	− 8.00000i	− 0.259281i
$953$	− 26.0000i	− 0.842223i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	− 26.0000i	− 0.842223i	0.907009	−	0.421111i	$-0.138360\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	24.0000i	0.775405i
$959$	72.0000	2.32500
$960$	0	0
$961$	−31.0000	−1.00000
$962$	4.00000i	0.128965i
$963$	4.00000i	0.128898i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	28.0000i	0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$	28.0000i	0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$	11.0000i	0.353553i
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	16.0000	0.513464	0.256732	−	0.966483i	$-0.417354\pi$
$971$	16.0000	0.513464	0.256732	+	0.966483i	$0.417354\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	48.0000i	1.53881i
$974$	−12.0000	−0.384505
$975$	0	0
$976$	6.00000	0.192055
$977$	− 22.0000i	− 0.703842i	−0.936030	−	0.351921i	$-0.885529\pi$
$977$	− 22.0000i	− 0.703842i	0.936030	−	0.351921i	$-0.114471\pi$
$978$	− 12.0000i	− 0.383718i
$979$	0	0
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	24.0000i	0.765871i
$983$	− 8.00000i	− 0.255160i	−0.991828	−	0.127580i	$-0.959279\pi$
$983$	− 8.00000i	− 0.255160i	0.991828	−	0.127580i	$-0.0407210\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	20.0000	0.636930
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	0	0
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	− 34.0000i	− 1.07679i	−0.842692	−	0.538395i	$-0.819031\pi$
$997$	− 34.0000i	− 1.07679i	0.842692	−	0.538395i	$-0.180969\pi$
$998$	− 12.0000i	− 0.379853i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.f.799.2		2
5.2	odd	4		570.2.a.d.1.1	✓	1
5.3	odd	4		2850.2.a.p.1.1		1
5.4	even	2	inner	2850.2.d.f.799.1		2
15.2	even	4		1710.2.a.t.1.1		1
15.8	even	4		8550.2.a.b.1.1		1
20.7	even	4		4560.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.d.1.1	✓	1	5.2	odd	4
1710.2.a.t.1.1		1	15.2	even	4
2850.2.a.p.1.1		1	5.3	odd	4
2850.2.d.f.799.1		2	5.4	even	2	inner
2850.2.d.f.799.2		2	1.1	even	1	trivial
4560.2.a.a.1.1		1	20.7	even	4
8550.2.a.b.1.1		1	15.8	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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +4.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} -10.0000 q^{29} +1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} +2.00000 q^{41} +4.00000i q^{42} -4.00000i q^{43} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} +1.00000i q^{57} -10.0000i q^{58} -8.00000 q^{59} +6.00000 q^{61} +4.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -2.00000i q^{68} +1.00000i q^{72} -14.0000i q^{73} +2.00000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} -12.0000i q^{83} -4.00000 q^{84} +4.00000 q^{86} -10.0000i q^{87} -10.0000 q^{89} +8.00000 q^{91} -1.00000 q^{96} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{14} + 2 q^{16} + 2 q^{19} + 8 q^{21} + 2 q^{24} - 4 q^{26} - 20 q^{29} - 4 q^{34} + 2 q^{36} - 4 q^{39} + 4 q^{41} - 18 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 16 q^{59} + 12 q^{61} - 2 q^{64} + 4 q^{74} - 2 q^{76} + 2 q^{81} - 8 q^{84} + 8 q^{86} - 20 q^{89} + 16 q^{91} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^14 + 2 * q^16 + 2 * q^19 + 8 * q^21 + 2 * q^24 - 4 * q^26 - 20 * q^29 - 4 * q^34 + 2 * q^36 - 4 * q^39 + 4 * q^41 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 16 * q^59 + 12 * q^61 - 2 * q^64 + 4 * q^74 - 2 * q^76 + 2 * q^81 - 8 * q^84 + 8 * q^86 - 20 * q^89 + 16 * q^91 - 2 * q^96

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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