LMFDB - Embedded newform 2100.2.f.c.1049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1049,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1049.3
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2100.1049
Dual form			2100.2.f.c.1049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 1.50000i) q^{3} +(1.73205 - 2.00000i) q^{7} +(-1.50000 - 2.59808i) q^{9} -5.19615i q^{11} +1.73205 q^{13} -3.00000i q^{17} +3.46410i q^{19} +(-1.50000 - 4.33013i) q^{21} -5.19615 q^{27} +5.19615i q^{29} -10.3923i q^{31} +(-7.79423 - 4.50000i) q^{33} +8.00000i q^{37} +(1.50000 - 2.59808i) q^{39} +6.00000 q^{41} +10.0000i q^{43} -3.00000i q^{47} +(-1.00000 - 6.92820i) q^{49} +(-4.50000 - 2.59808i) q^{51} -10.3923 q^{53} +(5.19615 + 3.00000i) q^{57} -6.00000 q^{59} +6.92820i q^{61} +(-7.79423 - 1.50000i) q^{63} -2.00000i q^{67} -10.3923i q^{71} -6.92820 q^{73} +(-10.3923 - 9.00000i) q^{77} +13.0000 q^{79} +(-4.50000 + 7.79423i) q^{81} -12.0000i q^{83} +(7.79423 + 4.50000i) q^{87} +(3.00000 - 3.46410i) q^{91} +(-15.5885 - 9.00000i) q^{93} -1.73205 q^{97} +(-13.5000 + 7.79423i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} - 6 q^{21} + 6 q^{39} + 24 q^{41} - 4 q^{49} - 18 q^{51} - 24 q^{59} + 52 q^{79} - 18 q^{81} + 12 q^{91} - 54 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 6 * q^21 + 6 * q^39 + 24 * q^41 - 4 * q^49 - 18 * q^51 - 24 * q^59 + 52 * q^79 - 18 * q^81 + 12 * q^91 - 54 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.866025	−	1.50000i	0.500000	−	0.866025i
$3$	0.866025	−	1.50000i	0.500000	−	0.866025i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$		0			0
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$		0			0
$11$		−	5.19615i		−	1.56670i	−0.621582	−	0.783349i	$-0.713510\pi$
$11$		−	5.19615i		−	1.56670i	0.621582	−	0.783349i	$-0.286490\pi$
$12$		0			0
$13$	1.73205			0.480384			0.240192	−	0.970725i	$-0.422790\pi$
$13$	1.73205			0.480384			0.240192	+	0.970725i	$0.422790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.00000i		−	0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$		−	3.00000i		−	0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	−1.50000	−	4.33013i	−0.327327	−	0.944911i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−5.19615			−1.00000
$28$		0			0
$29$			5.19615i			0.964901i	0.875923	+	0.482451i	$0.160253\pi$
$29$			5.19615i			0.964901i	−0.875923	+	0.482451i	$0.839747\pi$
$30$		0			0
$31$		−	10.3923i		−	1.86651i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$		−	10.3923i		−	1.86651i	0.359211	−	0.933257i	$-0.383046\pi$
$32$		0			0
$33$	−7.79423	−	4.50000i	−1.35680	−	0.783349i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$	1.50000	−	2.59808i	0.240192	−	0.416025i
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$			10.0000i			1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$			10.0000i			1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.00000i		−	0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$		−	3.00000i		−	0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$	−4.50000	−	2.59808i	−0.630126	−	0.363803i
$52$		0			0
$53$	−10.3923			−1.42749			−0.713746	−	0.700404i	$-0.753003\pi$
$53$	−10.3923			−1.42749			−0.713746	+	0.700404i	$0.753003\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5.19615	+	3.00000i	0.688247	+	0.397360i
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	−7.79423	−	1.50000i	−0.981981	−	0.188982i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$	−6.92820			−0.810885			−0.405442	−	0.914121i	$-0.632883\pi$
$73$	−6.92820			−0.810885			−0.405442	+	0.914121i	$0.632883\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.3923	−	9.00000i	−1.18431	−	1.02565i
$78$		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
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	7.79423	+	4.50000i	0.835629	+	0.482451i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	3.00000	−	3.46410i	0.314485	−	0.363137i
$92$		0			0
$93$	−15.5885	−	9.00000i	−1.61645	−	0.933257i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.73205			−0.175863			−0.0879316	−	0.996127i	$-0.528026\pi$
$97$	−1.73205			−0.175863			−0.0879316	+	0.996127i	$0.528026\pi$
$98$		0			0
$99$	−13.5000	+	7.79423i	−1.35680	+	0.783349i
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	−1.73205			−0.170664			−0.0853320	−	0.996353i	$-0.527195\pi$
$103$	−1.73205			−0.170664			−0.0853320	+	0.996353i	$0.527195\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.3923			1.00466			0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923			1.00466			0.502331	+	0.864675i	$0.332476\pi$
$108$		0			0
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$		0			0
$111$	12.0000	+	6.92820i	1.13899	+	0.657596i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.59808	−	4.50000i	−0.240192	−	0.416025i
$118$		0			0
$119$	−6.00000	−	5.19615i	−0.550019	−	0.476331i
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$	5.19615	−	9.00000i	0.468521	−	0.811503i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$	15.0000	+	8.66025i	1.32068	+	0.762493i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	6.92820	+	6.00000i	0.600751	+	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−20.7846			−1.77575			−0.887875	−	0.460086i	$-0.847819\pi$
$137$	−20.7846			−1.77575			−0.887875	+	0.460086i	$0.847819\pi$
$138$		0			0
$139$		−	6.92820i		−	0.587643i	−0.955860	−	0.293821i	$-0.905073\pi$
$139$		−	6.92820i		−	0.587643i	0.955860	−	0.293821i	$-0.0949270\pi$
$140$		0			0
$141$	−4.50000	−	2.59808i	−0.378968	−	0.218797i
$142$		0			0
$143$		−	9.00000i		−	0.752618i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−11.2583	−	4.50000i	−0.928571	−	0.371154i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−5.00000			−0.406894			−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000			−0.406894			−0.203447	+	0.979086i	$0.565214\pi$
$152$		0			0
$153$	−7.79423	+	4.50000i	−0.630126	+	0.363803i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$	−9.00000	+	15.5885i	−0.713746	+	1.23625i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			8.00000i			0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$			8.00000i			0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.0000i			1.16073i	0.814355	+	0.580367i	$0.197091\pi$
$167$			15.0000i			1.16073i	−0.814355	+	0.580367i	$0.802909\pi$
$168$		0			0
$169$	−10.0000			−0.769231
$170$		0			0
$171$	9.00000	−	5.19615i	0.688247	−	0.397360i
$172$		0			0
$173$		−	9.00000i		−	0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$		−	9.00000i		−	0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−5.19615	+	9.00000i	−0.390567	+	0.676481i
$178$		0			0
$179$		−	10.3923i		−	0.776757i	−0.921500	−	0.388379i	$-0.873035\pi$
$179$		−	10.3923i		−	0.776757i	0.921500	−	0.388379i	$-0.126965\pi$
$180$		0			0
$181$		−	17.3205i		−	1.28742i	−0.765268	−	0.643712i	$-0.777394\pi$
$181$		−	17.3205i		−	1.28742i	0.765268	−	0.643712i	$-0.222606\pi$
$182$		0			0
$183$	10.3923	+	6.00000i	0.768221	+	0.443533i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−15.5885			−1.13994
$188$		0			0
$189$	−9.00000	+	10.3923i	−0.654654	+	0.755929i
$190$		0			0
$191$			15.5885i			1.12794i	0.825795	+	0.563971i	$0.190727\pi$
$191$			15.5885i			1.12794i	−0.825795	+	0.563971i	$0.809273\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.3923			0.740421			0.370211	−	0.928948i	$-0.379286\pi$
$197$	10.3923			0.740421			0.370211	+	0.928948i	$0.379286\pi$
$198$		0			0
$199$		−	24.2487i		−	1.71895i	−0.511182	−	0.859473i	$-0.670792\pi$
$199$		−	24.2487i		−	1.71895i	0.511182	−	0.859473i	$-0.329208\pi$
$200$		0			0
$201$	−3.00000	−	1.73205i	−0.211604	−	0.122169i
$202$		0			0
$203$	10.3923	+	9.00000i	0.729397	+	0.631676i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	18.0000			1.24509
$210$		0			0
$211$	25.0000			1.72107			0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000			1.72107			0.860535	+	0.509390i	$0.170129\pi$
$212$		0			0
$213$	−15.5885	−	9.00000i	−1.06810	−	0.616670i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−20.7846	−	18.0000i	−1.41095	−	1.22192i
$218$		0			0
$219$	−6.00000	+	10.3923i	−0.405442	+	0.702247i
$220$		0			0
$221$		−	5.19615i		−	0.349531i
$222$		0			0
$223$	5.19615			0.347960			0.173980	−	0.984749i	$-0.444337\pi$
$223$	5.19615			0.347960			0.173980	+	0.984749i	$0.444337\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			27.0000i			1.79205i	0.444001	+	0.896026i	$0.353559\pi$
$227$			27.0000i			1.79205i	−0.444001	+	0.896026i	$0.646441\pi$
$228$		0			0
$229$		−	10.3923i		−	0.686743i	−0.939200	−	0.343371i	$-0.888431\pi$
$229$		−	10.3923i		−	0.686743i	0.939200	−	0.343371i	$-0.111569\pi$
$230$		0			0
$231$	−22.5000	+	7.79423i	−1.48039	+	0.512823i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	11.2583	−	19.5000i	0.731307	−	1.26666i
$238$		0			0
$239$			25.9808i			1.68056i	0.542156	+	0.840278i	$0.317608\pi$
$239$			25.9808i			1.68056i	−0.542156	+	0.840278i	$0.682392\pi$
$240$		0			0
$241$			6.92820i			0.446285i	0.974786	+	0.223142i	$0.0716315\pi$
$241$			6.92820i			0.446285i	−0.974786	+	0.223142i	$0.928369\pi$
$242$		0			0
$243$	7.79423	+	13.5000i	0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			6.00000i			0.381771i
$248$		0			0
$249$	−18.0000	−	10.3923i	−1.14070	−	0.658586i
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$	16.0000	+	13.8564i	0.994192	+	0.860995i
$260$		0			0
$261$	13.5000	−	7.79423i	0.835629	−	0.482451i
$262$		0			0
$263$	−31.1769			−1.92245			−0.961225	−	0.275764i	$-0.911069\pi$
$263$	−31.1769			−1.92245			−0.961225	+	0.275764i	$0.911069\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$		−	6.92820i		−	0.420858i	−0.977609	−	0.210429i	$-0.932514\pi$
$271$		−	6.92820i		−	0.420858i	0.977609	−	0.210429i	$-0.0674861\pi$
$272$		0			0
$273$	−2.59808	−	7.50000i	−0.157243	−	0.453921i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$		0			0
$279$	−27.0000	+	15.5885i	−1.61645	+	0.933257i
$280$		0			0
$281$			5.19615i			0.309976i	0.987916	+	0.154988i	$0.0495340\pi$
$281$			5.19615i			0.309976i	−0.987916	+	0.154988i	$0.950466\pi$
$282$		0			0
$283$	25.9808			1.54440			0.772198	−	0.635382i	$-0.219157\pi$
$283$	25.9808			1.54440			0.772198	+	0.635382i	$0.219157\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.3923	−	12.0000i	0.613438	−	0.708338i
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$	−1.50000	+	2.59808i	−0.0879316	+	0.152302i
$292$		0			0
$293$		−	3.00000i		−	0.175262i	−0.996153	−	0.0876309i	$-0.972070\pi$
$293$		−	3.00000i		−	0.175262i	0.996153	−	0.0876309i	$-0.0279296\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			27.0000i			1.56670i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	20.0000	+	17.3205i	1.15278	+	0.998337i
$302$		0			0
$303$	15.5885	−	27.0000i	0.895533	−	1.55111i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	25.9808			1.48280			0.741400	−	0.671063i	$-0.234162\pi$
$307$	25.9808			1.48280			0.741400	+	0.671063i	$0.234162\pi$
$308$		0			0
$309$	−1.50000	+	2.59808i	−0.0853320	+	0.147799i
$310$		0			0
$311$	30.0000			1.70114			0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000			1.70114			0.850572	+	0.525859i	$0.176256\pi$
$312$		0			0
$313$	22.5167			1.27272			0.636358	−	0.771393i	$-0.280440\pi$
$313$	22.5167			1.27272			0.636358	+	0.771393i	$0.280440\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.3923			0.583690			0.291845	−	0.956466i	$-0.405731\pi$
$317$	10.3923			0.583690			0.291845	+	0.956466i	$0.405731\pi$
$318$		0			0
$319$	27.0000			1.51171
$320$		0			0
$321$	9.00000	−	15.5885i	0.502331	−	0.870063i
$322$		0			0
$323$	10.3923			0.578243
$324$		0			0
$325$		0			0
$326$		0			0
$327$	9.52628	−	16.5000i	0.526804	−	0.912452i
$328$		0			0
$329$	−6.00000	−	5.19615i	−0.330791	−	0.286473i
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$	20.7846	−	12.0000i	1.13899	−	0.657596i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−54.0000			−2.92426
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		−	17.3205i		−	0.927146i	−0.886059	−	0.463573i	$-0.846567\pi$
$349$		−	17.3205i		−	0.927146i	0.886059	−	0.463573i	$-0.153433\pi$
$350$		0			0
$351$	−9.00000			−0.480384
$352$		0			0
$353$			21.0000i			1.11772i	0.829263	+	0.558859i	$0.188761\pi$
$353$			21.0000i			1.11772i	−0.829263	+	0.558859i	$0.811239\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−12.9904	+	4.50000i	−0.687524	+	0.238165i
$358$		0			0
$359$		−	10.3923i		−	0.548485i	−0.961661	−	0.274242i	$-0.911573\pi$
$359$		−	10.3923i		−	0.548485i	0.961661	−	0.274242i	$-0.0884271\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	−13.8564	+	24.0000i	−0.727273	+	1.25967i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	29.4449			1.53701			0.768505	−	0.639844i	$-0.221001\pi$
$367$	29.4449			1.53701			0.768505	+	0.639844i	$0.221001\pi$
$368$		0			0
$369$	−9.00000	−	15.5885i	−0.468521	−	0.811503i
$370$		0			0
$371$	−18.0000	+	20.7846i	−0.934513	+	1.07908i
$372$		0			0
$373$			20.0000i			1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$			20.0000i			1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			9.00000i			0.463524i
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$	−3.00000	−	1.73205i	−0.153695	−	0.0887357i
$382$		0			0
$383$			24.0000i			1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$			24.0000i			1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	25.9808	−	15.0000i	1.32068	−	0.762493i
$388$		0			0
$389$		−	15.5885i		−	0.790366i	−0.918602	−	0.395183i	$-0.870681\pi$
$389$		−	15.5885i		−	0.790366i	0.918602	−	0.395183i	$-0.129319\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5.19615			−0.260787			−0.130394	−	0.991462i	$-0.541624\pi$
$397$	−5.19615			−0.260787			−0.130394	+	0.991462i	$0.541624\pi$
$398$		0			0
$399$	15.0000	−	5.19615i	0.750939	−	0.260133i
$400$		0			0
$401$		−	5.19615i		−	0.259483i	−0.991548	−	0.129742i	$-0.958585\pi$
$401$		−	5.19615i		−	0.259483i	0.991548	−	0.129742i	$-0.0414148\pi$
$402$		0			0
$403$		−	18.0000i		−	0.896644i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	41.5692			2.06051
$408$		0			0
$409$			3.46410i			0.171289i	0.996326	+	0.0856444i	$0.0272949\pi$
$409$			3.46410i			0.171289i	−0.996326	+	0.0856444i	$0.972705\pi$
$410$		0			0
$411$	−18.0000	+	31.1769i	−0.887875	+	1.53784i
$412$		0			0
$413$	−10.3923	+	12.0000i	−0.511372	+	0.590481i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−10.3923	−	6.00000i	−0.508913	−	0.293821i
$418$		0			0
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$	−13.0000			−0.633581			−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000			−0.633581			−0.316791	+	0.948495i	$0.602605\pi$
$422$		0			0
$423$	−7.79423	+	4.50000i	−0.378968	+	0.218797i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	13.8564	+	12.0000i	0.670559	+	0.580721i
$428$		0			0
$429$	−13.5000	−	7.79423i	−0.651786	−	0.376309i
$430$		0			0
$431$			5.19615i			0.250290i	0.992138	+	0.125145i	$0.0399396\pi$
$431$			5.19615i			0.250290i	−0.992138	+	0.125145i	$0.960060\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	20.7846i		−	0.991995i	−0.868324	−	0.495998i	$-0.834802\pi$
$439$		−	20.7846i		−	0.991995i	0.868324	−	0.495998i	$-0.165198\pi$
$440$		0			0
$441$	−16.5000	+	12.9904i	−0.785714	+	0.618590i
$442$		0			0
$443$	31.1769			1.48126			0.740630	−	0.671913i	$-0.234527\pi$
$443$	31.1769			1.48126			0.740630	+	0.671913i	$0.234527\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	25.9808i		−	1.22611i	−0.790041	−	0.613054i	$-0.789941\pi$
$449$		−	25.9808i		−	1.22611i	0.790041	−	0.613054i	$-0.210059\pi$
$450$		0			0
$451$		−	31.1769i		−	1.46806i
$452$		0			0
$453$	−4.33013	+	7.50000i	−0.203447	+	0.352381i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	10.0000i		−	0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$		−	10.0000i		−	0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$		0			0
$459$			15.5885i			0.727607i
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			9.00000i			0.416470i	0.978079	+	0.208235i	$0.0667719\pi$
$467$			9.00000i			0.416470i	−0.978079	+	0.208235i	$0.933228\pi$
$468$		0			0
$469$	−4.00000	−	3.46410i	−0.184703	−	0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	51.9615			2.38919
$474$		0			0
$475$		0			0
$476$		0			0
$477$	15.5885	+	27.0000i	0.713746	+	1.23625i
$478$		0			0
$479$	12.0000			0.548294			0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000			0.548294			0.274147	+	0.961688i	$0.411605\pi$
$480$		0			0
$481$			13.8564i			0.631798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			8.00000i			0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$			8.00000i			0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$		0			0
$489$	12.0000	+	6.92820i	0.542659	+	0.313304i
$490$		0			0
$491$			5.19615i			0.234499i	0.993102	+	0.117250i	$0.0374077\pi$
$491$			5.19615i			0.234499i	−0.993102	+	0.117250i	$0.962592\pi$
$492$		0			0
$493$	15.5885			0.702069
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−20.7846	−	18.0000i	−0.932317	−	0.807410i
$498$		0			0
$499$	−13.0000			−0.581960			−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000			−0.581960			−0.290980	+	0.956729i	$0.593981\pi$
$500$		0			0
$501$	22.5000	+	12.9904i	1.00523	+	0.580367i
$502$		0			0
$503$		−	3.00000i		−	0.133763i	−0.997761	−	0.0668817i	$-0.978695\pi$
$503$		−	3.00000i		−	0.133763i	0.997761	−	0.0668817i	$-0.0213050\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−8.66025	+	15.0000i	−0.384615	+	0.666173i
$508$		0			0
$509$	−24.0000			−1.06378			−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000			−1.06378			−0.531891	+	0.846813i	$0.678518\pi$
$510$		0			0
$511$	−12.0000	+	13.8564i	−0.530849	+	0.612971i
$512$		0			0
$513$		−	18.0000i		−	0.794719i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−15.5885			−0.685580
$518$		0			0
$519$	−13.5000	−	7.79423i	−0.592584	−	0.342129i
$520$		0			0
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		0			0
$523$	17.3205			0.757373			0.378686	−	0.925525i	$-0.376376\pi$
$523$	17.3205			0.757373			0.378686	+	0.925525i	$0.376376\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−31.1769			−1.35809
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	9.00000	+	15.5885i	0.390567	+	0.676481i
$532$		0			0
$533$	10.3923			0.450141
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−15.5885	−	9.00000i	−0.672692	−	0.388379i
$538$		0			0
$539$	−36.0000	+	5.19615i	−1.55063	+	0.223814i
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	−25.9808	−	15.0000i	−1.11494	−	0.643712i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	32.0000i		−	1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$		−	32.0000i		−	1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$		0			0
$549$	18.0000	−	10.3923i	0.768221	−	0.443533i
$550$		0			0
$551$	−18.0000			−0.766826
$552$		0			0
$553$	22.5167	−	26.0000i	0.957506	−	1.10563i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.7846			0.880672			0.440336	−	0.897833i	$-0.354859\pi$
$557$	20.7846			0.880672			0.440336	+	0.897833i	$0.354859\pi$
$558$		0			0
$559$			17.3205i			0.732579i
$560$		0			0
$561$	−13.5000	+	23.3827i	−0.569970	+	0.987218i
$562$		0			0
$563$			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
$566$		0			0
$567$	7.79423	+	22.5000i	0.327327	+	0.944911i
$568$		0			0
$569$			20.7846i			0.871336i	0.900107	+	0.435668i	$0.143488\pi$
$569$			20.7846i			0.871336i	−0.900107	+	0.435668i	$0.856512\pi$
$570$		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
$573$	23.3827	+	13.5000i	0.976826	+	0.563971i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.73205			0.0721062			0.0360531	−	0.999350i	$-0.488521\pi$
$577$	1.73205			0.0721062			0.0360531	+	0.999350i	$0.488521\pi$
$578$		0			0
$579$	24.0000	+	13.8564i	0.997406	+	0.575853i
$580$		0			0
$581$	−24.0000	−	20.7846i	−0.995688	−	0.862291i
$582$		0			0
$583$			54.0000i			2.23645i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			36.0000i			1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$			36.0000i			1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$		0			0
$589$	36.0000			1.48335
$590$		0			0
$591$	9.00000	−	15.5885i	0.370211	−	0.641223i
$592$		0			0
$593$		−	21.0000i		−	0.862367i	−0.902264	−	0.431183i	$-0.858096\pi$
$593$		−	21.0000i		−	0.862367i	0.902264	−	0.431183i	$-0.141904\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−36.3731	−	21.0000i	−1.48865	−	0.859473i
$598$		0			0
$599$			25.9808i			1.06155i	0.847514	+	0.530773i	$0.178098\pi$
$599$			25.9808i			1.06155i	−0.847514	+	0.530773i	$0.821902\pi$
$600$		0			0
$601$			31.1769i			1.27173i	0.771799	+	0.635866i	$0.219357\pi$
$601$			31.1769i			1.27173i	−0.771799	+	0.635866i	$0.780643\pi$
$602$		0			0
$603$	−5.19615	+	3.00000i	−0.211604	+	0.122169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	8.66025			0.351509			0.175754	−	0.984434i	$-0.443764\pi$
$607$	8.66025			0.351509			0.175754	+	0.984434i	$0.443764\pi$
$608$		0			0
$609$	22.5000	−	7.79423i	0.911746	−	0.315838i
$610$		0			0
$611$		−	5.19615i		−	0.210214i
$612$		0			0
$613$			26.0000i			1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$			26.0000i			1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−41.5692			−1.67351			−0.836757	−	0.547575i	$-0.815551\pi$
$617$	−41.5692			−1.67351			−0.836757	+	0.547575i	$0.815551\pi$
$618$		0			0
$619$			6.92820i			0.278468i	0.990260	+	0.139234i	$0.0444640\pi$
$619$			6.92820i			0.278468i	−0.990260	+	0.139234i	$0.955536\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	15.5885	−	27.0000i	0.622543	−	1.07828i
$628$		0			0
$629$	24.0000			0.956943
$630$		0			0
$631$	17.0000			0.676759			0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000			0.676759			0.338380	+	0.941010i	$0.390121\pi$
$632$		0			0
$633$	21.6506	−	37.5000i	0.860535	−	1.49049i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.73205	−	12.0000i	−0.0686264	−	0.475457i
$638$		0			0
$639$	−27.0000	+	15.5885i	−1.06810	+	0.616670i
$640$		0			0
$641$			20.7846i			0.820943i	0.911873	+	0.410471i	$0.134636\pi$
$641$			20.7846i			0.820943i	−0.911873	+	0.410471i	$0.865364\pi$
$642$		0			0
$643$	15.5885			0.614749			0.307374	−	0.951589i	$-0.400550\pi$
$643$	15.5885			0.614749			0.307374	+	0.951589i	$0.400550\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			36.0000i			1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$			36.0000i			1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$		0			0
$649$			31.1769i			1.22380i
$650$		0			0
$651$	−45.0000	+	15.5885i	−1.76369	+	0.610960i
$652$		0			0
$653$	−10.3923			−0.406682			−0.203341	−	0.979108i	$-0.565180\pi$
$653$	−10.3923			−0.406682			−0.203341	+	0.979108i	$0.565180\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	10.3923	+	18.0000i	0.405442	+	0.702247i
$658$		0			0
$659$			15.5885i			0.607240i	0.952793	+	0.303620i	$0.0981953\pi$
$659$			15.5885i			0.607240i	−0.952793	+	0.303620i	$0.901805\pi$
$660$		0			0
$661$		−	48.4974i		−	1.88633i	−0.332323	−	0.943166i	$-0.607832\pi$
$661$		−	48.4974i		−	1.88633i	0.332323	−	0.943166i	$-0.392168\pi$
$662$		0			0
$663$	−7.79423	−	4.50000i	−0.302703	−	0.174766i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	4.50000	−	7.79423i	0.173980	−	0.301342i
$670$		0			0
$671$	36.0000			1.38976
$672$		0			0
$673$		−	32.0000i		−	1.23351i	−0.787155	−	0.616755i	$-0.788447\pi$
$673$		−	32.0000i		−	1.23351i	0.787155	−	0.616755i	$-0.211553\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	27.0000i		−	1.03769i	−0.854867	−	0.518847i	$-0.826361\pi$
$677$		−	27.0000i		−	1.03769i	0.854867	−	0.518847i	$-0.173639\pi$
$678$		0			0
$679$	−3.00000	+	3.46410i	−0.115129	+	0.132940i
$680$		0			0
$681$	40.5000	+	23.3827i	1.55196	+	0.896026i
$682$		0			0
$683$	20.7846			0.795301			0.397650	−	0.917537i	$-0.369826\pi$
$683$	20.7846			0.795301			0.397650	+	0.917537i	$0.369826\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−15.5885	−	9.00000i	−0.594737	−	0.343371i
$688$		0			0
$689$	−18.0000			−0.685745
$690$		0			0
$691$			41.5692i			1.58137i	0.612225	+	0.790684i	$0.290275\pi$
$691$			41.5692i			1.58137i	−0.612225	+	0.790684i	$0.709725\pi$
$692$		0			0
$693$	−7.79423	+	40.5000i	−0.296078	+	1.53847i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	18.0000i		−	0.681799i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			36.3731i			1.37379i	0.726756	+	0.686896i	$0.241027\pi$
$701$			36.3731i			1.37379i	−0.726756	+	0.686896i	$0.758973\pi$
$702$		0			0
$703$	−27.7128			−1.04521
$704$		0			0
$705$		0			0
$706$		0			0
$707$	31.1769	−	36.0000i	1.17253	−	1.35392i
$708$		0			0
$709$	−13.0000			−0.488225			−0.244113	−	0.969747i	$-0.578497\pi$
$709$	−13.0000			−0.488225			−0.244113	+	0.969747i	$0.578497\pi$
$710$		0			0
$711$	−19.5000	−	33.7750i	−0.731307	−	1.26666i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	38.9711	+	22.5000i	1.45540	+	0.840278i
$718$		0			0
$719$	12.0000			0.447524			0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000			0.447524			0.223762	+	0.974644i	$0.428166\pi$
$720$		0			0
$721$	−3.00000	+	3.46410i	−0.111726	+	0.129010i
$722$		0			0
$723$	10.3923	+	6.00000i	0.386494	+	0.223142i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−51.9615			−1.92715			−0.963573	−	0.267445i	$-0.913821\pi$
$727$	−51.9615			−1.92715			−0.963573	+	0.267445i	$0.913821\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	30.0000			1.10959
$732$		0			0
$733$	−36.3731			−1.34347			−0.671735	−	0.740792i	$-0.734451\pi$
$733$	−36.3731			−1.34347			−0.671735	+	0.740792i	$0.734451\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.3923			−0.382805
$738$		0			0
$739$	−11.0000			−0.404642			−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000			−0.404642			−0.202321	+	0.979319i	$0.564848\pi$
$740$		0			0
$741$	9.00000	+	5.19615i	0.330623	+	0.190885i
$742$		0			0
$743$	−10.3923			−0.381257			−0.190628	−	0.981662i	$-0.561053\pi$
$743$	−10.3923			−0.381257			−0.190628	+	0.981662i	$0.561053\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−31.1769	+	18.0000i	−1.14070	+	0.658586i
$748$		0			0
$749$	18.0000	−	20.7846i	0.657706	−	0.759453i
$750$		0			0
$751$	−11.0000			−0.401396			−0.200698	−	0.979653i	$-0.564321\pi$
$751$	−11.0000			−0.401396			−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$	−20.7846	+	36.0000i	−0.757433	+	1.31191i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			8.00000i			0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$			8.00000i			0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.0000			−0.435000			−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000			−0.435000			−0.217500	+	0.976060i	$0.569790\pi$
$762$		0			0
$763$	19.0526	−	22.0000i	0.689749	−	0.796453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.3923			−0.375244
$768$		0			0
$769$		−	17.3205i		−	0.624593i	−0.949985	−	0.312297i	$-0.898902\pi$
$769$		−	17.3205i		−	0.624593i	0.949985	−	0.312297i	$-0.101098\pi$
$770$		0			0
$771$	−27.0000	−	15.5885i	−0.972381	−	0.561405i
$772$		0			0
$773$		−	15.0000i		−	0.539513i	−0.962929	−	0.269756i	$-0.913057\pi$
$773$		−	15.0000i		−	0.539513i	0.962929	−	0.269756i	$-0.0869431\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	34.6410	−	12.0000i	1.24274	−	0.430498i
$778$		0			0
$779$			20.7846i			0.744686i
$780$		0			0
$781$	−54.0000			−1.93227
$782$		0			0
$783$		−	27.0000i		−	0.964901i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	50.2295			1.79049			0.895244	−	0.445577i	$-0.147001\pi$
$787$	50.2295			1.79049			0.895244	+	0.445577i	$0.147001\pi$
$788$		0			0
$789$	−27.0000	+	46.7654i	−0.961225	+	1.66489i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			12.0000i			0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			9.00000i			0.318796i	0.987214	+	0.159398i	$0.0509554\pi$
$797$			9.00000i			0.318796i	−0.987214	+	0.159398i	$0.949045\pi$
$798$		0			0
$799$	−9.00000			−0.318397
$800$		0			0
$801$		0			0
$802$		0			0
$803$			36.0000i			1.27041i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	10.3923	−	18.0000i	0.365826	−	0.633630i
$808$		0			0
$809$			36.3731i			1.27881i	0.768871	+	0.639404i	$0.220819\pi$
$809$			36.3731i			1.27881i	−0.768871	+	0.639404i	$0.779181\pi$
$810$		0			0
$811$		−	20.7846i		−	0.729846i	−0.931038	−	0.364923i	$-0.881095\pi$
$811$		−	20.7846i		−	0.729846i	0.931038	−	0.364923i	$-0.118905\pi$
$812$		0			0
$813$	−10.3923	−	6.00000i	−0.364474	−	0.210429i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−34.6410			−1.21194
$818$		0			0
$819$	−13.5000	−	2.59808i	−0.471728	−	0.0907841i
$820$		0			0
$821$		−	36.3731i		−	1.26943i	−0.772747	−	0.634714i	$-0.781118\pi$
$821$		−	36.3731i		−	1.26943i	0.772747	−	0.634714i	$-0.218882\pi$
$822$		0			0
$823$			2.00000i			0.0697156i	0.999392	+	0.0348578i	$0.0110978\pi$
$823$			2.00000i			0.0697156i	−0.999392	+	0.0348578i	$0.988902\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−41.5692			−1.44550			−0.722752	−	0.691108i	$-0.757123\pi$
$827$	−41.5692			−1.44550			−0.722752	+	0.691108i	$0.757123\pi$
$828$		0			0
$829$		−	45.0333i		−	1.56407i	−0.623233	−	0.782036i	$-0.714181\pi$
$829$		−	45.0333i		−	1.56407i	0.623233	−	0.782036i	$-0.285819\pi$
$830$		0			0
$831$	−12.0000	−	6.92820i	−0.416275	−	0.240337i
$832$		0			0
$833$	−20.7846	+	3.00000i	−0.720144	+	0.103944i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			54.0000i			1.86651i
$838$		0			0
$839$	−12.0000			−0.414286			−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000			−0.414286			−0.207143	+	0.978311i	$0.566417\pi$
$840$		0			0
$841$	2.00000			0.0689655
$842$		0			0
$843$	7.79423	+	4.50000i	0.268447	+	0.154988i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−27.7128	+	32.0000i	−0.952224	+	1.09953i
$848$		0			0
$849$	22.5000	−	38.9711i	0.772198	−	1.33749i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−13.8564			−0.474434			−0.237217	−	0.971457i	$-0.576235\pi$
$853$	−13.8564			−0.474434			−0.237217	+	0.971457i	$0.576235\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			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
$859$			6.92820i			0.236387i	0.992991	+	0.118194i	$0.0377103\pi$
$859$			6.92820i			0.236387i	−0.992991	+	0.118194i	$0.962290\pi$
$860$		0			0
$861$	−9.00000	−	25.9808i	−0.306719	−	0.885422i
$862$		0			0
$863$	−31.1769			−1.06127			−0.530637	−	0.847599i	$-0.678047\pi$
$863$	−31.1769			−1.06127			−0.530637	+	0.847599i	$0.678047\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	6.92820	−	12.0000i	0.235294	−	0.407541i
$868$		0			0
$869$		−	67.5500i		−	2.29148i
$870$		0			0
$871$		−	3.46410i		−	0.117377i
$872$		0			0
$873$	2.59808	+	4.50000i	0.0879316	+	0.152302i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$		0			0
$879$	−4.50000	−	2.59808i	−0.151781	−	0.0876309i
$880$		0			0
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$		−	44.0000i		−	1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$		−	44.0000i		−	1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	48.0000i		−	1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$		−	48.0000i		−	1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$		0			0
$889$	−4.00000	−	3.46410i	−0.134156	−	0.116182i
$890$		0			0
$891$	40.5000	+	23.3827i	1.35680	+	0.783349i
$892$		0			0
$893$	10.3923			0.347765
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	54.0000			1.80100
$900$		0			0
$901$			31.1769i			1.03865i
$902$		0			0
$903$	43.3013	−	15.0000i	1.44098	−	0.499169i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			32.0000i			1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$			32.0000i			1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$		0			0
$909$	−27.0000	−	46.7654i	−0.895533	−	1.55111i
$910$		0			0
$911$		−	31.1769i		−	1.03294i	−0.856306	−	0.516469i	$-0.827246\pi$
$911$		−	31.1769i		−	1.03294i	0.856306	−	0.516469i	$-0.172754\pi$
$912$		0			0
$913$	−62.3538			−2.06361
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	11.0000			0.362857			0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000			0.362857			0.181428	+	0.983404i	$0.441928\pi$
$920$		0			0
$921$	22.5000	−	38.9711i	0.741400	−	1.28414i
$922$		0			0
$923$		−	18.0000i		−	0.592477i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	2.59808	+	4.50000i	0.0853320	+	0.147799i
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	24.0000	−	3.46410i	0.786568	−	0.113531i
$932$		0			0
$933$	25.9808	−	45.0000i	0.850572	−	1.47323i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	22.5167			0.735587			0.367794	−	0.929907i	$-0.380113\pi$
$937$	22.5167			0.735587			0.367794	+	0.929907i	$0.380113\pi$
$938$		0			0
$939$	19.5000	−	33.7750i	0.636358	−	1.10221i
$940$		0			0
$941$	24.0000			0.782378			0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000			0.782378			0.391189	+	0.920310i	$0.372064\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	10.3923			0.337705			0.168852	−	0.985641i	$-0.445994\pi$
$947$	10.3923			0.337705			0.168852	+	0.985641i	$0.445994\pi$
$948$		0			0
$949$	−12.0000			−0.389536
$950$		0			0
$951$	9.00000	−	15.5885i	0.291845	−	0.505490i
$952$		0			0
$953$	−51.9615			−1.68320			−0.841599	−	0.540102i	$-0.818386\pi$
$953$	−51.9615			−1.68320			−0.841599	+	0.540102i	$0.818386\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	23.3827	−	40.5000i	0.755855	−	1.30918i
$958$		0			0
$959$	−36.0000	+	41.5692i	−1.16250	+	1.34234i
$960$		0			0
$961$	−77.0000			−2.48387
$962$		0			0
$963$	−15.5885	−	27.0000i	−0.502331	−	0.870063i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	2.00000i		−	0.0643157i	−0.999483	−	0.0321578i	$-0.989762\pi$
$967$		−	2.00000i		−	0.0643157i	0.999483	−	0.0321578i	$-0.0102379\pi$
$968$		0			0
$969$	9.00000	−	15.5885i	0.289122	−	0.500773i
$970$		0			0
$971$	6.00000			0.192549			0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000			0.192549			0.0962746	+	0.995355i	$0.469307\pi$
$972$		0			0
$973$	−13.8564	−	12.0000i	−0.444216	−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	51.9615			1.66240			0.831198	−	0.555976i	$-0.187655\pi$
$977$	51.9615			1.66240			0.831198	+	0.555976i	$0.187655\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−16.5000	−	28.5788i	−0.526804	−	0.912452i
$982$		0			0
$983$		−	57.0000i		−	1.81802i	−0.416777	−	0.909009i	$-0.636840\pi$
$983$		−	57.0000i		−	1.81802i	0.416777	−	0.909009i	$-0.363160\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−12.9904	+	4.50000i	−0.413488	+	0.143237i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	28.0000			0.889449			0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000			0.889449			0.444725	+	0.895667i	$0.353302\pi$
$992$		0			0
$993$	3.46410	−	6.00000i	0.109930	−	0.190404i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−15.5885			−0.493691			−0.246846	−	0.969055i	$-0.579394\pi$
$997$	−15.5885			−0.493691			−0.246846	+	0.969055i	$0.579394\pi$
$998$		0			0
$999$		−	41.5692i		−	1.31519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.f.c.1049.3		4
3.2	odd	2		2100.2.f.d.1049.4		4
5.2	odd	4		420.2.d.a.41.1	✓	2
5.3	odd	4		2100.2.d.e.1301.2		2
5.4	even	2	inner	2100.2.f.c.1049.2		4
7.6	odd	2		2100.2.f.d.1049.2		4
15.2	even	4		420.2.d.b.41.1	yes	2
15.8	even	4		2100.2.d.a.1301.2		2
15.14	odd	2		2100.2.f.d.1049.1		4
20.7	even	4		1680.2.f.d.881.2		2
21.20	even	2	inner	2100.2.f.c.1049.1		4
35.13	even	4		2100.2.d.a.1301.1		2
35.27	even	4		420.2.d.b.41.2	yes	2
35.34	odd	2		2100.2.f.d.1049.3		4
60.47	odd	4		1680.2.f.a.881.2		2
105.62	odd	4		420.2.d.a.41.2	yes	2
105.83	odd	4		2100.2.d.e.1301.1		2
105.104	even	2	inner	2100.2.f.c.1049.4		4
140.27	odd	4		1680.2.f.a.881.1		2
420.167	even	4		1680.2.f.d.881.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.d.a.41.1	✓	2	5.2	odd	4
420.2.d.a.41.2	yes	2	105.62	odd	4
420.2.d.b.41.1	yes	2	15.2	even	4
420.2.d.b.41.2	yes	2	35.27	even	4
1680.2.f.a.881.1		2	140.27	odd	4
1680.2.f.a.881.2		2	60.47	odd	4
1680.2.f.d.881.1		2	420.167	even	4
1680.2.f.d.881.2		2	20.7	even	4
2100.2.d.a.1301.1		2	35.13	even	4
2100.2.d.a.1301.2		2	15.8	even	4
2100.2.d.e.1301.1		2	105.83	odd	4
2100.2.d.e.1301.2		2	5.3	odd	4
2100.2.f.c.1049.1		4	21.20	even	2	inner
2100.2.f.c.1049.2		4	5.4	even	2	inner
2100.2.f.c.1049.3		4	1.1	even	1	trivial
2100.2.f.c.1049.4		4	105.104	even	2	inner
2100.2.f.d.1049.1		4	15.14	odd	2
2100.2.f.d.1049.2		4	7.6	odd	2
2100.2.f.d.1049.3		4	35.34	odd	2
2100.2.f.d.1049.4		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 1.50000i) q^{3} +(1.73205 - 2.00000i) q^{7} +(-1.50000 - 2.59808i) q^{9} -5.19615i q^{11} +1.73205 q^{13} -3.00000i q^{17} +3.46410i q^{19} +(-1.50000 - 4.33013i) q^{21} -5.19615 q^{27} +5.19615i q^{29} -10.3923i q^{31} +(-7.79423 - 4.50000i) q^{33} +8.00000i q^{37} +(1.50000 - 2.59808i) q^{39} +6.00000 q^{41} +10.0000i q^{43} -3.00000i q^{47} +(-1.00000 - 6.92820i) q^{49} +(-4.50000 - 2.59808i) q^{51} -10.3923 q^{53} +(5.19615 + 3.00000i) q^{57} -6.00000 q^{59} +6.92820i q^{61} +(-7.79423 - 1.50000i) q^{63} -2.00000i q^{67} -10.3923i q^{71} -6.92820 q^{73} +(-10.3923 - 9.00000i) q^{77} +13.0000 q^{79} +(-4.50000 + 7.79423i) q^{81} -12.0000i q^{83} +(7.79423 + 4.50000i) q^{87} +(3.00000 - 3.46410i) q^{91} +(-15.5885 - 9.00000i) q^{93} -1.73205 q^{97} +(-13.5000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} - 6 q^{21} + 6 q^{39} + 24 q^{41} - 4 q^{49} - 18 q^{51} - 24 q^{59} + 52 q^{79} - 18 q^{81} + 12 q^{91} - 54 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 6 * q^21 + 6 * q^39 + 24 * q^41 - 4 * q^49 - 18 * q^51 - 24 * q^59 + 52 * q^79 - 18 * q^81 + 12 * q^91 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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