LMFDB - Embedded newform 28.3.b.a.13.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,3,Mod(13,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("28.13");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	28.b (of order $2$, degree $1$, 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:	$0.762944740209$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			13.2
Root			$2.44949i$ of defining polynomial
Character	$\chi$	$=$	28.13
Dual form			28.3.b.a.13.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.89898i q^{3} -4.89898i q^{5} +(5.00000 - 4.89898i) q^{7} -15.0000 q^{9} +O(q^{10})$	$q+4.89898i q^{3} -4.89898i q^{5} +(5.00000 - 4.89898i) q^{7} -15.0000 q^{9} -6.00000 q^{11} +4.89898i q^{13} +24.0000 q^{15} -19.5959i q^{17} +24.4949i q^{19} +(24.0000 + 24.4949i) q^{21} -30.0000 q^{23} +1.00000 q^{25} -29.3939i q^{27} -6.00000 q^{29} -29.3939i q^{33} +(-24.0000 - 24.4949i) q^{35} +10.0000 q^{37} -24.0000 q^{39} +48.9898i q^{41} +10.0000 q^{43} +73.4847i q^{45} -19.5959i q^{47} +(1.00000 - 48.9898i) q^{49} +96.0000 q^{51} +90.0000 q^{53} +29.3939i q^{55} -120.000 q^{57} -24.4949i q^{59} +24.4949i q^{61} +(-75.0000 + 73.4847i) q^{63} +24.0000 q^{65} -70.0000 q^{67} -146.969i q^{69} +42.0000 q^{71} -107.778i q^{73} +4.89898i q^{75} +(-30.0000 + 29.3939i) q^{77} +74.0000 q^{79} +9.00000 q^{81} +63.6867i q^{83} -96.0000 q^{85} -29.3939i q^{87} +146.969i q^{89} +(24.0000 + 24.4949i) q^{91} +120.000 q^{95} +78.3837i q^{97} +90.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{7} - 30 q^{9}+O(q^{10}) $ 2 * q + 10 * q^7 - 30 * q^9	$ 2 q + 10 q^{7} - 30 q^{9} - 12 q^{11} + 48 q^{15} + 48 q^{21} - 60 q^{23} + 2 q^{25} - 12 q^{29} - 48 q^{35} + 20 q^{37} - 48 q^{39} + 20 q^{43} + 2 q^{49} + 192 q^{51} + 180 q^{53} - 240 q^{57} - 150 q^{63} + 48 q^{65} - 140 q^{67} + 84 q^{71} - 60 q^{77} + 148 q^{79} + 18 q^{81} - 192 q^{85} + 48 q^{91} + 240 q^{95} + 180 q^{99}+O(q^{100}) $ 2 * q + 10 * q^7 - 30 * q^9 - 12 * q^11 + 48 * q^15 + 48 * q^21 - 60 * q^23 + 2 * q^25 - 12 * q^29 - 48 * q^35 + 20 * q^37 - 48 * q^39 + 20 * q^43 + 2 * q^49 + 192 * q^51 + 180 * q^53 - 240 * q^57 - 150 * q^63 + 48 * q^65 - 140 * q^67 + 84 * q^71 - 60 * q^77 + 148 * q^79 + 18 * q^81 - 192 * q^85 + 48 * q^91 + 240 * q^95 + 180 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/28\mathbb{Z}\right)^\times$.

$n$	$15$	$17$
$\chi(n)$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			4.89898i			1.63299i	0.577350	+	0.816497i	$0.304087\pi$
$3$			4.89898i			1.63299i	−0.577350	+	0.816497i	$0.695913\pi$
$4$		0			0
$5$		−	4.89898i		−	0.979796i	−0.871780	−	0.489898i	$-0.837034\pi$
$5$		−	4.89898i		−	0.979796i	0.871780	−	0.489898i	$-0.162966\pi$
$6$		0			0
$7$	5.00000	−	4.89898i	0.714286	−	0.699854i
$7$	5.00000	−	4.89898i	0.714286	−	0.699854i
$8$		0			0
$9$	−15.0000			−1.66667
$10$		0			0
$11$	−6.00000			−0.545455			−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000			−0.545455			−0.272727	+	0.962091i	$0.587926\pi$
$12$		0			0
$13$			4.89898i			0.376845i	0.982088	+	0.188422i	$0.0603374\pi$
$13$			4.89898i			0.376845i	−0.982088	+	0.188422i	$0.939663\pi$
$14$		0			0
$15$	24.0000			1.60000
$16$		0			0
$17$		−	19.5959i		−	1.15270i	−0.817203	−	0.576351i	$-0.804476\pi$
$17$		−	19.5959i		−	1.15270i	0.817203	−	0.576351i	$-0.195524\pi$
$18$		0			0
$19$			24.4949i			1.28921i	0.764518	+	0.644603i	$0.222977\pi$
$19$			24.4949i			1.28921i	−0.764518	+	0.644603i	$0.777023\pi$
$20$		0			0
$21$	24.0000	+	24.4949i	1.14286	+	1.16642i
$22$		0			0
$23$	−30.0000			−1.30435			−0.652174	−	0.758069i	$-0.726143\pi$
$23$	−30.0000			−1.30435			−0.652174	+	0.758069i	$0.726143\pi$
$24$		0			0
$25$	1.00000			0.0400000
$26$		0			0
$27$		−	29.3939i		−	1.08866i
$28$		0			0
$29$	−6.00000			−0.206897			−0.103448	−	0.994635i	$-0.532988\pi$
$29$	−6.00000			−0.206897			−0.103448	+	0.994635i	$0.532988\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		−	29.3939i		−	0.890724i
$34$		0			0
$35$	−24.0000	−	24.4949i	−0.685714	−	0.699854i
$36$		0			0
$37$	10.0000			0.270270			0.135135	−	0.990827i	$-0.456853\pi$
$37$	10.0000			0.270270			0.135135	+	0.990827i	$0.456853\pi$
$38$		0			0
$39$	−24.0000			−0.615385
$40$		0			0
$41$			48.9898i			1.19487i	0.801916	+	0.597437i	$0.203814\pi$
$41$			48.9898i			1.19487i	−0.801916	+	0.597437i	$0.796186\pi$
$42$		0			0
$43$	10.0000			0.232558			0.116279	−	0.993217i	$-0.462903\pi$
$43$	10.0000			0.232558			0.116279	+	0.993217i	$0.462903\pi$
$44$		0			0
$45$			73.4847i			1.63299i
$46$		0			0
$47$		−	19.5959i		−	0.416934i	−0.978029	−	0.208467i	$-0.933153\pi$
$47$		−	19.5959i		−	0.416934i	0.978029	−	0.208467i	$-0.0668475\pi$
$48$		0			0
$49$	1.00000	−	48.9898i	0.0204082	−	0.999792i
$50$		0			0
$51$	96.0000			1.88235
$52$		0			0
$53$	90.0000			1.69811			0.849057	−	0.528302i	$-0.177171\pi$
$53$	90.0000			1.69811			0.849057	+	0.528302i	$0.177171\pi$
$54$		0			0
$55$			29.3939i			0.534434i
$56$		0			0
$57$	−120.000			−2.10526
$58$		0			0
$59$		−	24.4949i		−	0.415168i	−0.978217	−	0.207584i	$-0.933440\pi$
$59$		−	24.4949i		−	0.415168i	0.978217	−	0.207584i	$-0.0665600\pi$
$60$		0			0
$61$			24.4949i			0.401556i	0.979637	+	0.200778i	$0.0643469\pi$
$61$			24.4949i			0.401556i	−0.979637	+	0.200778i	$0.935653\pi$
$62$		0			0
$63$	−75.0000	+	73.4847i	−1.19048	+	1.16642i
$64$		0			0
$65$	24.0000			0.369231
$66$		0			0
$67$	−70.0000			−1.04478			−0.522388	−	0.852708i	$-0.674959\pi$
$67$	−70.0000			−1.04478			−0.522388	+	0.852708i	$0.674959\pi$
$68$		0			0
$69$		−	146.969i		−	2.12999i
$70$		0			0
$71$	42.0000			0.591549			0.295775	−	0.955258i	$-0.404422\pi$
$71$	42.0000			0.591549			0.295775	+	0.955258i	$0.404422\pi$
$72$		0			0
$73$		−	107.778i		−	1.47640i	−0.674579	−	0.738202i	$-0.735675\pi$
$73$		−	107.778i		−	1.47640i	0.674579	−	0.738202i	$-0.264325\pi$
$74$		0			0
$75$			4.89898i			0.0653197i
$76$		0			0
$77$	−30.0000	+	29.3939i	−0.389610	+	0.381739i
$78$		0			0
$79$	74.0000			0.936709			0.468354	−	0.883541i	$-0.344847\pi$
$79$	74.0000			0.936709			0.468354	+	0.883541i	$0.344847\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			63.6867i			0.767310i	0.923476	+	0.383655i	$0.125335\pi$
$83$			63.6867i			0.767310i	−0.923476	+	0.383655i	$0.874665\pi$
$84$		0			0
$85$	−96.0000			−1.12941
$86$		0			0
$87$		−	29.3939i		−	0.337861i
$88$		0			0
$89$			146.969i			1.65134i	0.564152	+	0.825671i	$0.309203\pi$
$89$			146.969i			1.65134i	−0.564152	+	0.825671i	$0.690797\pi$
$90$		0			0
$91$	24.0000	+	24.4949i	0.263736	+	0.269175i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	120.000			1.26316
$96$		0			0
$97$			78.3837i			0.808079i	0.914742	+	0.404040i	$0.132394\pi$
$97$			78.3837i			0.808079i	−0.914742	+	0.404040i	$0.867606\pi$
$98$		0			0
$99$	90.0000			0.909091
$100$		0			0
$101$			73.4847i			0.727571i	0.931483	+	0.363786i	$0.118516\pi$
$101$			73.4847i			0.727571i	−0.931483	+	0.363786i	$0.881484\pi$
$102$		0			0
$103$		−	166.565i		−	1.61714i	−0.588401	−	0.808569i	$-0.700242\pi$
$103$		−	166.565i		−	1.61714i	0.588401	−	0.808569i	$-0.299758\pi$
$104$		0			0
$105$	120.000	−	117.576i	1.14286	−	1.11977i
$106$		0			0
$107$	−150.000			−1.40187			−0.700935	−	0.713226i	$-0.747234\pi$
$107$	−150.000			−1.40187			−0.700935	+	0.713226i	$0.747234\pi$
$108$		0			0
$109$	−86.0000			−0.788991			−0.394495	−	0.918898i	$-0.629081\pi$
$109$	−86.0000			−0.788991			−0.394495	+	0.918898i	$0.629081\pi$
$110$		0			0
$111$			48.9898i			0.441350i
$112$		0			0
$113$	90.0000			0.796460			0.398230	−	0.917286i	$-0.369625\pi$
$113$	90.0000			0.796460			0.398230	+	0.917286i	$0.369625\pi$
$114$		0			0
$115$			146.969i			1.27799i
$116$		0			0
$117$		−	73.4847i		−	0.628074i
$118$		0			0
$119$	−96.0000	−	97.9796i	−0.806723	−	0.823358i
$120$		0			0
$121$	−85.0000			−0.702479
$122$		0			0
$123$	−240.000			−1.95122
$124$		0			0
$125$		−	127.373i		−	1.01899i
$126$		0			0
$127$	50.0000			0.393701			0.196850	−	0.980434i	$-0.436929\pi$
$127$	50.0000			0.393701			0.196850	+	0.980434i	$0.436929\pi$
$128$		0			0
$129$			48.9898i			0.379766i
$130$		0			0
$131$		−	73.4847i		−	0.560952i	−0.959861	−	0.280476i	$-0.909508\pi$
$131$		−	73.4847i		−	0.560952i	0.959861	−	0.280476i	$-0.0904923\pi$
$132$		0			0
$133$	120.000	+	122.474i	0.902256	+	0.920861i
$134$		0			0
$135$	−144.000			−1.06667
$136$		0			0
$137$	210.000			1.53285			0.766423	−	0.642336i	$-0.222035\pi$
$137$	210.000			1.53285			0.766423	+	0.642336i	$0.222035\pi$
$138$		0			0
$139$		−	122.474i		−	0.881111i	−0.897725	−	0.440556i	$-0.854781\pi$
$139$		−	122.474i		−	0.881111i	0.897725	−	0.440556i	$-0.145219\pi$
$140$		0			0
$141$	96.0000			0.680851
$142$		0			0
$143$		−	29.3939i		−	0.205552i
$144$		0			0
$145$			29.3939i			0.202716i
$146$		0			0
$147$	240.000	+	4.89898i	1.63265	+	0.0333264i
$148$		0			0
$149$	−198.000			−1.32886			−0.664430	−	0.747351i	$-0.731325\pi$
$149$	−198.000			−1.32886			−0.664430	+	0.747351i	$0.731325\pi$
$150$		0			0
$151$	34.0000			0.225166			0.112583	−	0.993642i	$-0.464088\pi$
$151$	34.0000			0.225166			0.112583	+	0.993642i	$0.464088\pi$
$152$		0			0
$153$			293.939i			1.92117i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	132.272i		−	0.842500i	−0.906945	−	0.421250i	$-0.861592\pi$
$157$		−	132.272i		−	0.842500i	0.906945	−	0.421250i	$-0.138408\pi$
$158$		0			0
$159$			440.908i			2.77301i
$160$		0			0
$161$	−150.000	+	146.969i	−0.931677	+	0.912853i
$162$		0			0
$163$	170.000			1.04294			0.521472	−	0.853268i	$-0.325383\pi$
$163$	170.000			1.04294			0.521472	+	0.853268i	$0.325383\pi$
$164$		0			0
$165$	−144.000			−0.872727
$166$		0			0
$167$		−	166.565i		−	0.997397i	−0.866776	−	0.498699i	$-0.833811\pi$
$167$		−	166.565i		−	0.997397i	0.866776	−	0.498699i	$-0.166189\pi$
$168$		0			0
$169$	145.000			0.857988
$170$		0			0
$171$		−	367.423i		−	2.14868i
$172$		0			0
$173$		−	191.060i		−	1.10439i	−0.833714	−	0.552197i	$-0.813790\pi$
$173$		−	191.060i		−	1.10439i	0.833714	−	0.552197i	$-0.186210\pi$
$174$		0			0
$175$	5.00000	−	4.89898i	0.0285714	−	0.0279942i
$176$		0			0
$177$	120.000			0.677966
$178$		0			0
$179$	42.0000			0.234637			0.117318	−	0.993094i	$-0.462570\pi$
$179$	42.0000			0.234637			0.117318	+	0.993094i	$0.462570\pi$
$180$		0			0
$181$		−	24.4949i		−	0.135331i	−0.997708	−	0.0676655i	$-0.978445\pi$
$181$		−	24.4949i		−	0.135331i	0.997708	−	0.0676655i	$-0.0215551\pi$
$182$		0			0
$183$	−120.000			−0.655738
$184$		0			0
$185$		−	48.9898i		−	0.264810i
$186$		0			0
$187$			117.576i			0.628746i
$188$		0			0
$189$	−144.000	−	146.969i	−0.761905	−	0.777616i
$190$		0			0
$191$	−246.000			−1.28796			−0.643979	−	0.765043i	$-0.722718\pi$
$191$	−246.000			−1.28796			−0.643979	+	0.765043i	$0.722718\pi$
$192$		0			0
$193$	−230.000			−1.19171			−0.595855	−	0.803092i	$-0.703187\pi$
$193$	−230.000			−1.19171			−0.595855	+	0.803092i	$0.703187\pi$
$194$		0			0
$195$			117.576i			0.602951i
$196$		0			0
$197$	−150.000			−0.761421			−0.380711	−	0.924694i	$-0.624321\pi$
$197$	−150.000			−0.761421			−0.380711	+	0.924694i	$0.624321\pi$
$198$		0			0
$199$			342.929i			1.72326i	0.507538	+	0.861630i	$0.330556\pi$
$199$			342.929i			1.72326i	−0.507538	+	0.861630i	$0.669444\pi$
$200$		0			0
$201$		−	342.929i		−	1.70611i
$202$		0			0
$203$	−30.0000	+	29.3939i	−0.147783	+	0.144797i
$204$		0			0
$205$	240.000			1.17073
$206$		0			0
$207$	450.000			2.17391
$208$		0			0
$209$		−	146.969i		−	0.703203i
$210$		0			0
$211$	−326.000			−1.54502			−0.772512	−	0.635000i	$-0.781000\pi$
$211$	−326.000			−1.54502			−0.772512	+	0.635000i	$0.781000\pi$
$212$		0			0
$213$			205.757i			0.965996i
$214$		0			0
$215$		−	48.9898i		−	0.227860i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	528.000			2.41096
$220$		0			0
$221$	96.0000			0.434389
$222$		0			0
$223$		−	117.576i		−	0.527244i	−0.964626	−	0.263622i	$-0.915083\pi$
$223$		−	117.576i		−	0.527244i	0.964626	−	0.263622i	$-0.0849172\pi$
$224$		0			0
$225$	−15.0000			−0.0666667
$226$		0			0
$227$			396.817i			1.74809i	0.485841	+	0.874047i	$0.338513\pi$
$227$			396.817i			1.74809i	−0.485841	+	0.874047i	$0.661487\pi$
$228$		0			0
$229$			73.4847i			0.320894i	0.987044	+	0.160447i	$0.0512935\pi$
$229$			73.4847i			0.320894i	−0.987044	+	0.160447i	$0.948706\pi$
$230$		0			0
$231$	−144.000	−	146.969i	−0.623377	−	0.636231i
$232$		0			0
$233$	210.000			0.901288			0.450644	−	0.892704i	$-0.351194\pi$
$233$	210.000			0.901288			0.450644	+	0.892704i	$0.351194\pi$
$234$		0			0
$235$	−96.0000			−0.408511
$236$		0			0
$237$			362.524i			1.52964i
$238$		0			0
$239$	114.000			0.476987			0.238494	−	0.971144i	$-0.423346\pi$
$239$	114.000			0.476987			0.238494	+	0.971144i	$0.423346\pi$
$240$		0			0
$241$			97.9796i			0.406554i	0.979121	+	0.203277i	$0.0651592\pi$
$241$			97.9796i			0.406554i	−0.979121	+	0.203277i	$0.934841\pi$
$242$		0			0
$243$		−	220.454i		−	0.907218i
$244$		0			0
$245$	−240.000	−	4.89898i	−0.979592	−	0.0199958i
$246$		0			0
$247$	−120.000			−0.485830
$248$		0			0
$249$	−312.000			−1.25301
$250$		0			0
$251$			171.464i			0.683125i	0.939859	+	0.341562i	$0.110956\pi$
$251$			171.464i			0.683125i	−0.939859	+	0.341562i	$0.889044\pi$
$252$		0			0
$253$	180.000			0.711462
$254$		0			0
$255$		−	470.302i		−	1.84432i
$256$		0			0
$257$			235.151i			0.914984i	0.889214	+	0.457492i	$0.151252\pi$
$257$			235.151i			0.914984i	−0.889214	+	0.457492i	$0.848748\pi$
$258$		0			0
$259$	50.0000	−	48.9898i	0.193050	−	0.189150i
$260$		0			0
$261$	90.0000			0.344828
$262$		0			0
$263$	330.000			1.25475			0.627376	−	0.778716i	$-0.284129\pi$
$263$	330.000			1.25475			0.627376	+	0.778716i	$0.284129\pi$
$264$		0			0
$265$		−	440.908i		−	1.66380i
$266$		0			0
$267$	−720.000			−2.69663
$268$		0			0
$269$		−	465.403i		−	1.73012i	−0.501666	−	0.865061i	$-0.667279\pi$
$269$		−	465.403i		−	1.73012i	0.501666	−	0.865061i	$-0.332721\pi$
$270$		0			0
$271$		−	97.9796i		−	0.361548i	−0.983525	−	0.180774i	$-0.942140\pi$
$271$		−	97.9796i		−	0.361548i	0.983525	−	0.180774i	$-0.0578603\pi$
$272$		0			0
$273$	−120.000	+	117.576i	−0.439560	+	0.430680i
$274$		0			0
$275$	−6.00000			−0.0218182
$276$		0			0
$277$	170.000			0.613718			0.306859	−	0.951755i	$-0.400722\pi$
$277$	170.000			0.613718			0.306859	+	0.951755i	$0.400722\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	114.000			0.405694			0.202847	−	0.979210i	$-0.434981\pi$
$281$	114.000			0.405694			0.202847	+	0.979210i	$0.434981\pi$
$282$		0			0
$283$		−	338.030i		−	1.19445i	−0.802073	−	0.597225i	$-0.796270\pi$
$283$		−	338.030i		−	1.19445i	0.802073	−	0.597225i	$-0.203730\pi$
$284$		0			0
$285$			587.878i			2.06273i
$286$		0			0
$287$	240.000	+	244.949i	0.836237	+	0.853481i
$288$		0			0
$289$	−95.0000			−0.328720
$290$		0			0
$291$	−384.000			−1.31959
$292$		0			0
$293$			151.868i			0.518322i	0.965834	+	0.259161i	$0.0834460\pi$
$293$			151.868i			0.518322i	−0.965834	+	0.259161i	$0.916554\pi$
$294$		0			0
$295$	−120.000			−0.406780
$296$		0			0
$297$			176.363i			0.593816i
$298$		0			0
$299$		−	146.969i		−	0.491536i
$300$		0			0
$301$	50.0000	−	48.9898i	0.166113	−	0.162757i
$302$		0			0
$303$	−360.000			−1.18812
$304$		0			0
$305$	120.000			0.393443
$306$		0			0
$307$			102.879i			0.335109i	0.985863	+	0.167555i	$0.0535871\pi$
$307$			102.879i			0.335109i	−0.985863	+	0.167555i	$0.946413\pi$
$308$		0			0
$309$	816.000			2.64078
$310$		0			0
$311$			244.949i			0.787617i	0.919192	+	0.393809i	$0.128843\pi$
$311$			244.949i			0.787617i	−0.919192	+	0.393809i	$0.871157\pi$
$312$		0			0
$313$			225.353i			0.719978i	0.932957	+	0.359989i	$0.117220\pi$
$313$			225.353i			0.719978i	−0.932957	+	0.359989i	$0.882780\pi$
$314$		0			0
$315$	360.000	+	367.423i	1.14286	+	1.16642i
$316$		0			0
$317$	90.0000			0.283912			0.141956	−	0.989873i	$-0.454661\pi$
$317$	90.0000			0.283912			0.141956	+	0.989873i	$0.454661\pi$
$318$		0			0
$319$	36.0000			0.112853
$320$		0			0
$321$		−	734.847i		−	2.28924i
$322$		0			0
$323$	480.000			1.48607
$324$		0			0
$325$			4.89898i			0.0150738i
$326$		0			0
$327$		−	421.312i		−	1.28842i
$328$		0			0
$329$	−96.0000	−	97.9796i	−0.291793	−	0.297810i
$330$		0			0
$331$	−38.0000			−0.114804			−0.0574018	−	0.998351i	$-0.518282\pi$
$331$	−38.0000			−0.114804			−0.0574018	+	0.998351i	$0.518282\pi$
$332$		0			0
$333$	−150.000			−0.450450
$334$		0			0
$335$			342.929i			1.02367i
$336$		0			0
$337$	−230.000			−0.682493			−0.341246	−	0.939974i	$-0.610849\pi$
$337$	−230.000			−0.682493			−0.341246	+	0.939974i	$0.610849\pi$
$338$		0			0
$339$			440.908i			1.30061i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−235.000	−	249.848i	−0.685131	−	0.728420i
$344$		0			0
$345$	−720.000			−2.08696
$346$		0			0
$347$	−150.000			−0.432277			−0.216138	−	0.976363i	$-0.569346\pi$
$347$	−150.000			−0.432277			−0.216138	+	0.976363i	$0.569346\pi$
$348$		0			0
$349$			612.372i			1.75465i	0.479898	+	0.877324i	$0.340674\pi$
$349$			612.372i			1.75465i	−0.479898	+	0.877324i	$0.659326\pi$
$350$		0			0
$351$	144.000			0.410256
$352$		0			0
$353$			274.343i			0.777175i	0.921412	+	0.388588i	$0.127037\pi$
$353$			274.343i			0.777175i	−0.921412	+	0.388588i	$0.872963\pi$
$354$		0			0
$355$		−	205.757i		−	0.579598i
$356$		0			0
$357$	480.000	−	470.302i	1.34454	−	1.31737i
$358$		0			0
$359$	−318.000			−0.885794			−0.442897	−	0.896573i	$-0.646049\pi$
$359$	−318.000			−0.885794			−0.442897	+	0.896573i	$0.646049\pi$
$360$		0			0
$361$	−239.000			−0.662050
$362$		0			0
$363$		−	416.413i		−	1.14714i
$364$		0			0
$365$	−528.000			−1.44658
$366$		0			0
$367$		−	215.555i		−	0.587344i	−0.955906	−	0.293672i	$-0.905123\pi$
$367$		−	215.555i		−	0.587344i	0.955906	−	0.293672i	$-0.0948773\pi$
$368$		0			0
$369$		−	734.847i		−	1.99146i
$370$		0			0
$371$	450.000	−	440.908i	1.21294	−	1.18843i
$372$		0			0
$373$	−310.000			−0.831099			−0.415550	−	0.909571i	$-0.636411\pi$
$373$	−310.000			−0.831099			−0.415550	+	0.909571i	$0.636411\pi$
$374$		0			0
$375$	624.000			1.66400
$376$		0			0
$377$		−	29.3939i		−	0.0779678i
$378$		0			0
$379$	442.000			1.16623			0.583113	−	0.812391i	$-0.301834\pi$
$379$	442.000			1.16623			0.583113	+	0.812391i	$0.301834\pi$
$380$		0			0
$381$			244.949i			0.642911i
$382$		0			0
$383$			431.110i			1.12561i	0.826588	+	0.562807i	$0.190279\pi$
$383$			431.110i			1.12561i	−0.826588	+	0.562807i	$0.809721\pi$
$384$		0			0
$385$	144.000	+	146.969i	0.374026	+	0.381739i
$386$		0			0
$387$	−150.000			−0.387597
$388$		0			0
$389$	−726.000			−1.86632			−0.933162	−	0.359456i	$-0.882962\pi$
$389$	−726.000			−1.86632			−0.933162	+	0.359456i	$0.882962\pi$
$390$		0			0
$391$			587.878i			1.50352i
$392$		0			0
$393$	360.000			0.916031
$394$		0			0
$395$		−	362.524i		−	0.917783i
$396$		0			0
$397$		−	191.060i		−	0.481260i	−0.970617	−	0.240630i	$-0.922646\pi$
$397$		−	191.060i		−	0.481260i	0.970617	−	0.240630i	$-0.0773540\pi$
$398$		0			0
$399$	−600.000	+	587.878i	−1.50376	+	1.47338i
$400$		0			0
$401$	474.000			1.18204			0.591022	−	0.806655i	$-0.298724\pi$
$401$	474.000			1.18204			0.591022	+	0.806655i	$0.298724\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	44.0908i		−	0.108866i
$406$		0			0
$407$	−60.0000			−0.147420
$408$		0			0
$409$		−	48.9898i		−	0.119779i	−0.998205	−	0.0598897i	$-0.980925\pi$
$409$		−	48.9898i		−	0.119779i	0.998205	−	0.0598897i	$-0.0190749\pi$
$410$		0			0
$411$			1028.79i			2.50313i
$412$		0			0
$413$	−120.000	−	122.474i	−0.290557	−	0.296548i
$414$		0			0
$415$	312.000			0.751807
$416$		0			0
$417$	600.000			1.43885
$418$		0			0
$419$		−	269.444i		−	0.643064i	−0.946899	−	0.321532i	$-0.895802\pi$
$419$		−	269.444i		−	0.643064i	0.946899	−	0.321532i	$-0.104198\pi$
$420$		0			0
$421$	314.000			0.745843			0.372922	−	0.927863i	$-0.378356\pi$
$421$	314.000			0.745843			0.372922	+	0.927863i	$0.378356\pi$
$422$		0			0
$423$			293.939i			0.694891i
$424$		0			0
$425$		−	19.5959i		−	0.0461080i
$426$		0			0
$427$	120.000	+	122.474i	0.281030	+	0.286825i
$428$		0			0
$429$	144.000			0.335664
$430$		0			0
$431$	−366.000			−0.849188			−0.424594	−	0.905384i	$-0.639583\pi$
$431$	−366.000			−0.849188			−0.424594	+	0.905384i	$0.639583\pi$
$432$		0			0
$433$		−	58.7878i		−	0.135768i	−0.997693	−	0.0678842i	$-0.978375\pi$
$433$		−	58.7878i		−	0.135768i	0.997693	−	0.0678842i	$-0.0216249\pi$
$434$		0			0
$435$	−144.000			−0.331034
$436$		0			0
$437$		−	734.847i		−	1.68157i
$438$		0			0
$439$		−	146.969i		−	0.334782i	−0.985891	−	0.167391i	$-0.946466\pi$
$439$		−	146.969i		−	0.334782i	0.985891	−	0.167391i	$-0.0535343\pi$
$440$		0			0
$441$	−15.0000	+	734.847i	−0.0340136	+	1.66632i
$442$		0			0
$443$	90.0000			0.203160			0.101580	−	0.994827i	$-0.467610\pi$
$443$	90.0000			0.203160			0.101580	+	0.994827i	$0.467610\pi$
$444$		0			0
$445$	720.000			1.61798
$446$		0			0
$447$		−	969.998i		−	2.17002i
$448$		0			0
$449$	−606.000			−1.34967			−0.674833	−	0.737971i	$-0.735784\pi$
$449$	−606.000			−1.34967			−0.674833	+	0.737971i	$0.735784\pi$
$450$		0			0
$451$		−	293.939i		−	0.651749i
$452$		0			0
$453$			166.565i			0.367694i
$454$		0			0
$455$	120.000	−	117.576i	0.263736	−	0.258408i
$456$		0			0
$457$	−230.000			−0.503282			−0.251641	−	0.967821i	$-0.580970\pi$
$457$	−230.000			−0.503282			−0.251641	+	0.967821i	$0.580970\pi$
$458$		0			0
$459$	−576.000			−1.25490
$460$		0			0
$461$			122.474i			0.265671i	0.991138	+	0.132836i	$0.0424083\pi$
$461$			122.474i			0.265671i	−0.991138	+	0.132836i	$0.957592\pi$
$462$		0			0
$463$	−310.000			−0.669546			−0.334773	−	0.942299i	$-0.608660\pi$
$463$	−310.000			−0.669546			−0.334773	+	0.942299i	$0.608660\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			102.879i			0.220297i	0.993915	+	0.110148i	$0.0351326\pi$
$467$			102.879i			0.220297i	−0.993915	+	0.110148i	$0.964867\pi$
$468$		0			0
$469$	−350.000	+	342.929i	−0.746269	+	0.731191i
$470$		0			0
$471$	648.000			1.37580
$472$		0			0
$473$	−60.0000			−0.126850
$474$		0			0
$475$			24.4949i			0.0515682i
$476$		0			0
$477$	−1350.00			−2.83019
$478$		0			0
$479$		−	391.918i		−	0.818201i	−0.912489	−	0.409101i	$-0.865843\pi$
$479$		−	391.918i		−	0.818201i	0.912489	−	0.409101i	$-0.134157\pi$
$480$		0			0
$481$			48.9898i			0.101850i
$482$		0			0
$483$	−720.000	−	734.847i	−1.49068	−	1.52142i
$484$		0			0
$485$	384.000			0.791753
$486$		0			0
$487$	130.000			0.266940			0.133470	−	0.991053i	$-0.457388\pi$
$487$	130.000			0.266940			0.133470	+	0.991053i	$0.457388\pi$
$488$		0			0
$489$			832.827i			1.70312i
$490$		0			0
$491$	234.000			0.476578			0.238289	−	0.971194i	$-0.423413\pi$
$491$	234.000			0.476578			0.238289	+	0.971194i	$0.423413\pi$
$492$		0			0
$493$			117.576i			0.238490i
$494$		0			0
$495$		−	440.908i		−	0.890724i
$496$		0			0
$497$	210.000	−	205.757i	0.422535	−	0.413998i
$498$		0			0
$499$	394.000			0.789579			0.394790	−	0.918772i	$-0.370817\pi$
$499$	394.000			0.789579			0.394790	+	0.918772i	$0.370817\pi$
$500$		0			0
$501$	816.000			1.62874
$502$		0			0
$503$		−	68.5857i		−	0.136353i	−0.997673	−	0.0681767i	$-0.978282\pi$
$503$		−	68.5857i		−	0.136353i	0.997673	−	0.0681767i	$-0.0217181\pi$
$504$		0			0
$505$	360.000			0.712871
$506$		0			0
$507$			710.352i			1.40109i
$508$		0			0
$509$		−	759.342i		−	1.49183i	−0.666041	−	0.745915i	$-0.732012\pi$
$509$		−	759.342i		−	1.49183i	0.666041	−	0.745915i	$-0.267988\pi$
$510$		0			0
$511$	−528.000	−	538.888i	−1.03327	−	1.05457i
$512$		0			0
$513$	720.000			1.40351
$514$		0			0
$515$	−816.000			−1.58447
$516$		0			0
$517$			117.576i			0.227419i
$518$		0			0
$519$	936.000			1.80347
$520$		0			0
$521$			636.867i			1.22239i	0.791478	+	0.611197i	$0.209312\pi$
$521$			636.867i			1.22239i	−0.791478	+	0.611197i	$0.790688\pi$
$522$		0			0
$523$			504.595i			0.964809i	0.875949	+	0.482404i	$0.160236\pi$
$523$			504.595i			0.964809i	−0.875949	+	0.482404i	$0.839764\pi$
$524$		0			0
$525$	24.0000	+	24.4949i	0.0457143	+	0.0466569i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	371.000			0.701323
$530$		0			0
$531$			367.423i			0.691946i
$532$		0			0
$533$	−240.000			−0.450281
$534$		0			0
$535$			734.847i			1.37355i
$536$		0			0
$537$			205.757i			0.383160i
$538$		0			0
$539$	−6.00000	+	293.939i	−0.0111317	+	0.545341i
$540$		0			0
$541$	−406.000			−0.750462			−0.375231	−	0.926931i	$-0.622437\pi$
$541$	−406.000			−0.750462			−0.375231	+	0.926931i	$0.622437\pi$
$542$		0			0
$543$	120.000			0.220994
$544$		0			0
$545$			421.312i			0.773050i
$546$		0			0
$547$	−790.000			−1.44424			−0.722121	−	0.691767i	$-0.756832\pi$
$547$	−790.000			−1.44424			−0.722121	+	0.691767i	$0.756832\pi$
$548$		0			0
$549$		−	367.423i		−	0.669259i
$550$		0			0
$551$		−	146.969i		−	0.266732i
$552$		0			0
$553$	370.000	−	362.524i	0.669078	−	0.655560i
$554$		0			0
$555$	240.000			0.432432
$556$		0			0
$557$	−150.000			−0.269300			−0.134650	−	0.990893i	$-0.542991\pi$
$557$	−150.000			−0.269300			−0.134650	+	0.990893i	$0.542991\pi$
$558$		0			0
$559$			48.9898i			0.0876383i
$560$		0			0
$561$	−576.000			−1.02674
$562$		0			0
$563$			455.605i			0.809245i	0.914484	+	0.404623i	$0.132597\pi$
$563$			455.605i			0.809245i	−0.914484	+	0.404623i	$0.867403\pi$
$564$		0			0
$565$		−	440.908i		−	0.780368i
$566$		0			0
$567$	45.0000	−	44.0908i	0.0793651	−	0.0777616i
$568$		0			0
$569$	762.000			1.33919			0.669596	−	0.742726i	$-0.266467\pi$
$569$	762.000			1.33919			0.669596	+	0.742726i	$0.266467\pi$
$570$		0			0
$571$	842.000			1.47461			0.737303	−	0.675562i	$-0.236099\pi$
$571$	842.000			1.47461			0.737303	+	0.675562i	$0.236099\pi$
$572$		0			0
$573$		−	1205.15i		−	2.10323i
$574$		0			0
$575$	−30.0000			−0.0521739
$576$		0			0
$577$		−	313.535i		−	0.543388i	−0.962384	−	0.271694i	$-0.912416\pi$
$577$		−	313.535i		−	0.543388i	0.962384	−	0.271694i	$-0.0875838\pi$
$578$		0			0
$579$		−	1126.77i		−	1.94605i
$580$		0			0
$581$	312.000	+	318.434i	0.537005	+	0.548079i
$582$		0			0
$583$	−540.000			−0.926244
$584$		0			0
$585$	−360.000			−0.615385
$586$		0			0
$587$		−	279.242i		−	0.475710i	−0.971301	−	0.237855i	$-0.923556\pi$
$587$		−	279.242i		−	0.475710i	0.971301	−	0.237855i	$-0.0764443\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		−	734.847i		−	1.24340i
$592$		0			0
$593$			960.200i			1.61922i	0.586965	+	0.809612i	$0.300323\pi$
$593$			960.200i			1.61922i	−0.586965	+	0.809612i	$0.699677\pi$
$594$		0			0
$595$	−480.000	+	470.302i	−0.806723	+	0.790424i
$596$		0			0
$597$	−1680.00			−2.81407
$598$		0			0
$599$	−726.000			−1.21202			−0.606010	−	0.795457i	$-0.707231\pi$
$599$	−726.000			−1.21202			−0.606010	+	0.795457i	$0.707231\pi$
$600$		0			0
$601$		−	1028.79i		−	1.71179i	−0.517150	−	0.855895i	$-0.673007\pi$
$601$		−	1028.79i		−	1.71179i	0.517150	−	0.855895i	$-0.326993\pi$
$602$		0			0
$603$	1050.00			1.74129
$604$		0			0
$605$			416.413i			0.688286i
$606$		0			0
$607$			666.261i			1.09763i	0.835944	+	0.548815i	$0.184921\pi$
$607$			666.261i			1.09763i	−0.835944	+	0.548815i	$0.815079\pi$
$608$		0			0
$609$	−144.000	−	146.969i	−0.236453	−	0.241329i
$610$		0			0
$611$	96.0000			0.157119
$612$		0			0
$613$	−1030.00			−1.68026			−0.840131	−	0.542384i	$-0.817522\pi$
$613$	−1030.00			−1.68026			−0.840131	+	0.542384i	$0.817522\pi$
$614$		0			0
$615$			1175.76i			1.91180i
$616$		0			0
$617$	90.0000			0.145867			0.0729335	−	0.997337i	$-0.476764\pi$
$617$	90.0000			0.145867			0.0729335	+	0.997337i	$0.476764\pi$
$618$		0			0
$619$		−	906.311i		−	1.46415i	−0.681222	−	0.732077i	$-0.738551\pi$
$619$		−	906.311i		−	1.46415i	0.681222	−	0.732077i	$-0.261449\pi$
$620$		0			0
$621$			881.816i			1.41999i
$622$		0			0
$623$	720.000	+	734.847i	1.15570	+	1.17953i
$624$		0			0
$625$	−599.000			−0.958400
$626$		0			0
$627$	720.000			1.14833
$628$		0			0
$629$		−	195.959i		−	0.311541i
$630$		0			0
$631$	1034.00			1.63867			0.819334	−	0.573316i	$-0.194343\pi$
$631$	1034.00			1.63867			0.819334	+	0.573316i	$0.194343\pi$
$632$		0			0
$633$		−	1597.07i		−	2.52301i
$634$		0			0
$635$		−	244.949i		−	0.385746i
$636$		0			0
$637$	240.000	+	4.89898i	0.376766	+	0.00769071i
$638$		0			0
$639$	−630.000			−0.985915
$640$		0			0
$641$	282.000			0.439938			0.219969	−	0.975507i	$-0.429404\pi$
$641$	282.000			0.439938			0.219969	+	0.975507i	$0.429404\pi$
$642$		0			0
$643$		−	191.060i		−	0.297139i	−0.988902	−	0.148569i	$-0.952533\pi$
$643$		−	191.060i		−	0.297139i	0.988902	−	0.148569i	$-0.0474668\pi$
$644$		0			0
$645$	240.000			0.372093
$646$		0			0
$647$		−	205.757i		−	0.318017i	−0.987277	−	0.159009i	$-0.949170\pi$
$647$		−	205.757i		−	0.318017i	0.987277	−	0.159009i	$-0.0508298\pi$
$648$		0			0
$649$			146.969i			0.226455i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1290.00			1.97550			0.987749	−	0.156052i	$-0.0498767\pi$
$653$	1290.00			1.97550			0.987749	+	0.156052i	$0.0498767\pi$
$654$		0			0
$655$	−360.000			−0.549618
$656$		0			0
$657$			1616.66i			2.46067i
$658$		0			0
$659$	474.000			0.719272			0.359636	−	0.933093i	$-0.382901\pi$
$659$	474.000			0.719272			0.359636	+	0.933093i	$0.382901\pi$
$660$		0			0
$661$			1151.26i			1.74169i	0.491554	+	0.870847i	$0.336429\pi$
$661$			1151.26i			1.74169i	−0.491554	+	0.870847i	$0.663571\pi$
$662$		0			0
$663$			470.302i			0.709354i
$664$		0			0
$665$	600.000	−	587.878i	0.902256	−	0.884026i
$666$		0			0
$667$	180.000			0.269865
$668$		0			0
$669$	576.000			0.860987
$670$		0			0
$671$		−	146.969i		−	0.219030i
$672$		0			0
$673$	−350.000			−0.520059			−0.260030	−	0.965601i	$-0.583732\pi$
$673$	−350.000			−0.520059			−0.260030	+	0.965601i	$0.583732\pi$
$674$		0			0
$675$		−	29.3939i		−	0.0435465i
$676$		0			0
$677$		−	44.0908i		−	0.0651268i	−0.999470	−	0.0325634i	$-0.989633\pi$
$677$		−	44.0908i		−	0.0651268i	0.999470	−	0.0325634i	$-0.0103671\pi$
$678$		0			0
$679$	384.000	+	391.918i	0.565538	+	0.577199i
$680$		0			0
$681$	−1944.00			−2.85463
$682$		0			0
$683$	−630.000			−0.922401			−0.461201	−	0.887296i	$-0.652581\pi$
$683$	−630.000			−0.922401			−0.461201	+	0.887296i	$0.652581\pi$
$684$		0			0
$685$		−	1028.79i		−	1.50188i
$686$		0			0
$687$	−360.000			−0.524017
$688$		0			0
$689$			440.908i			0.639925i
$690$		0			0
$691$		−	367.423i		−	0.531727i	−0.964011	−	0.265864i	$-0.914343\pi$
$691$		−	367.423i		−	0.531727i	0.964011	−	0.265864i	$-0.0856571\pi$
$692$		0			0
$693$	450.000	−	440.908i	0.649351	−	0.636231i
$694$		0			0
$695$	−600.000			−0.863309
$696$		0			0
$697$	960.000			1.37733
$698$		0			0
$699$			1028.79i			1.47180i
$700$		0			0
$701$	234.000			0.333809			0.166904	−	0.985973i	$-0.446623\pi$
$701$	234.000			0.333809			0.166904	+	0.985973i	$0.446623\pi$
$702$		0			0
$703$			244.949i			0.348434i
$704$		0			0
$705$		−	470.302i		−	0.667095i
$706$		0			0
$707$	360.000	+	367.423i	0.509194	+	0.519694i
$708$		0			0
$709$	−326.000			−0.459803			−0.229901	−	0.973214i	$-0.573840\pi$
$709$	−326.000			−0.459803			−0.229901	+	0.973214i	$0.573840\pi$
$710$		0			0
$711$	−1110.00			−1.56118
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−144.000			−0.201399
$716$		0			0
$717$			558.484i			0.778917i
$718$		0			0
$719$		−	685.857i		−	0.953904i	−0.878929	−	0.476952i	$-0.841742\pi$
$719$		−	685.857i		−	0.953904i	0.878929	−	0.476952i	$-0.158258\pi$
$720$		0			0
$721$	−816.000	−	832.827i	−1.13176	−	1.15510i
$722$		0			0
$723$	−480.000			−0.663900
$724$		0			0
$725$	−6.00000			−0.00827586
$726$		0			0
$727$		−	1087.57i		−	1.49597i	−0.663713	−	0.747987i	$-0.731020\pi$
$727$		−	1087.57i		−	1.49597i	0.663713	−	0.747987i	$-0.268980\pi$
$728$		0			0
$729$	1161.00			1.59259
$730$		0			0
$731$		−	195.959i		−	0.268070i
$732$		0			0
$733$		−	484.999i		−	0.661663i	−0.943690	−	0.330831i	$-0.892671\pi$
$733$		−	484.999i		−	0.661663i	0.943690	−	0.330831i	$-0.107329\pi$
$734$		0			0
$735$	24.0000	−	1175.76i	0.0326531	−	1.59967i
$736$		0			0
$737$	420.000			0.569878
$738$		0			0
$739$	74.0000			0.100135			0.0500677	−	0.998746i	$-0.484056\pi$
$739$	74.0000			0.100135			0.0500677	+	0.998746i	$0.484056\pi$
$740$		0			0
$741$		−	587.878i		−	0.793357i
$742$		0			0
$743$	−510.000			−0.686406			−0.343203	−	0.939261i	$-0.611512\pi$
$743$	−510.000			−0.686406			−0.343203	+	0.939261i	$0.611512\pi$
$744$		0			0
$745$			969.998i			1.30201i
$746$		0			0
$747$		−	955.301i		−	1.27885i
$748$		0			0
$749$	−750.000	+	734.847i	−1.00134	+	0.981104i
$750$		0			0
$751$	754.000			1.00399			0.501997	−	0.864869i	$-0.332599\pi$
$751$	754.000			1.00399			0.501997	+	0.864869i	$0.332599\pi$
$752$		0			0
$753$	−840.000			−1.11554
$754$		0			0
$755$		−	166.565i		−	0.220616i
$756$		0			0
$757$	730.000			0.964333			0.482166	−	0.876080i	$-0.339850\pi$
$757$	730.000			0.964333			0.482166	+	0.876080i	$0.339850\pi$
$758$		0			0
$759$			881.816i			1.16181i
$760$		0			0
$761$		−	244.949i		−	0.321878i	−0.986964	−	0.160939i	$-0.948548\pi$
$761$		−	244.949i		−	0.321878i	0.986964	−	0.160939i	$-0.0514522\pi$
$762$		0			0
$763$	−430.000	+	421.312i	−0.563565	+	0.552179i
$764$		0			0
$765$	1440.00			1.88235
$766$		0			0
$767$	120.000			0.156454
$768$		0			0
$769$			979.796i			1.27412i	0.770815	+	0.637058i	$0.219849\pi$
$769$			979.796i			1.27412i	−0.770815	+	0.637058i	$0.780151\pi$
$770$		0			0
$771$	−1152.00			−1.49416
$772$		0			0
$773$			935.705i			1.21049i	0.796041	+	0.605243i	$0.206924\pi$
$773$			935.705i			1.21049i	−0.796041	+	0.605243i	$0.793076\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	240.000	+	244.949i	0.308880	+	0.315250i
$778$		0			0
$779$	−1200.00			−1.54044
$780$		0			0
$781$	−252.000			−0.322663
$782$		0			0
$783$			176.363i			0.225240i
$784$		0			0
$785$	−648.000			−0.825478
$786$		0			0
$787$		−	289.040i		−	0.367268i	−0.982995	−	0.183634i	$-0.941214\pi$
$787$		−	289.040i		−	0.367268i	0.982995	−	0.183634i	$-0.0587861\pi$
$788$		0			0
$789$			1616.66i			2.04900i
$790$		0			0
$791$	450.000	−	440.908i	0.568900	−	0.557406i
$792$		0			0
$793$	−120.000			−0.151324
$794$		0			0
$795$	2160.00			2.71698
$796$		0			0
$797$			102.879i			0.129082i	0.997915	+	0.0645411i	$0.0205584\pi$
$797$			102.879i			0.129082i	−0.997915	+	0.0645411i	$0.979442\pi$
$798$		0			0
$799$	−384.000			−0.480601
$800$		0			0
$801$		−	2204.54i		−	2.75224i
$802$		0			0
$803$			646.665i			0.805312i
$804$		0			0
$805$	720.000	+	734.847i	0.894410	+	0.912853i
$806$		0			0
$807$	2280.00			2.82528
$808$		0			0
$809$	−486.000			−0.600742			−0.300371	−	0.953823i	$-0.597110\pi$
$809$	−486.000			−0.600742			−0.300371	+	0.953823i	$0.597110\pi$
$810$		0			0
$811$			661.362i			0.815490i	0.913096	+	0.407745i	$0.133685\pi$
$811$			661.362i			0.815490i	−0.913096	+	0.407745i	$0.866315\pi$
$812$		0			0
$813$	480.000			0.590406
$814$		0			0
$815$		−	832.827i		−	1.02187i
$816$		0			0
$817$			244.949i			0.299815i
$818$		0			0
$819$	−360.000	−	367.423i	−0.439560	−	0.448624i
$820$		0			0
$821$	234.000			0.285018			0.142509	−	0.989793i	$-0.454483\pi$
$821$	234.000			0.285018			0.142509	+	0.989793i	$0.454483\pi$
$822$		0			0
$823$	1130.00			1.37303			0.686513	−	0.727118i	$-0.259141\pi$
$823$	1130.00			1.37303			0.686513	+	0.727118i	$0.259141\pi$
$824$		0			0
$825$		−	29.3939i		−	0.0356289i
$826$		0			0
$827$	90.0000			0.108827			0.0544135	−	0.998518i	$-0.482671\pi$
$827$	90.0000			0.108827			0.0544135	+	0.998518i	$0.482671\pi$
$828$		0			0
$829$		−	759.342i		−	0.915973i	−0.888959	−	0.457987i	$-0.848571\pi$
$829$		−	759.342i		−	0.915973i	0.888959	−	0.457987i	$-0.151429\pi$
$830$		0			0
$831$			832.827i			1.00220i
$832$		0			0
$833$	−960.000	−	19.5959i	−1.15246	−	0.0235245i
$834$		0			0
$835$	−816.000			−0.977246
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1224.74i		−	1.45977i	−0.683571	−	0.729884i	$-0.739574\pi$
$839$		−	1224.74i		−	1.45977i	0.683571	−	0.729884i	$-0.260426\pi$
$840$		0			0
$841$	−805.000			−0.957194
$842$		0			0
$843$			558.484i			0.662495i
$844$		0			0
$845$		−	710.352i		−	0.840653i
$846$		0			0
$847$	−425.000	+	416.413i	−0.501771	+	0.491633i
$848$		0			0
$849$	1656.00			1.95053
$850$		0			0
$851$	−300.000			−0.352526
$852$		0			0
$853$			1386.41i			1.62534i	0.582727	+	0.812668i	$0.301986\pi$
$853$			1386.41i			1.62534i	−0.582727	+	0.812668i	$0.698014\pi$
$854$		0			0
$855$	−1800.00			−2.10526
$856$		0			0
$857$		−	989.594i		−	1.15472i	−0.816490	−	0.577359i	$-0.804083\pi$
$857$		−	989.594i		−	1.15472i	0.816490	−	0.577359i	$-0.195917\pi$
$858$		0			0
$859$			1151.26i			1.34023i	0.742256	+	0.670117i	$0.233756\pi$
$859$			1151.26i			1.34023i	−0.742256	+	0.670117i	$0.766244\pi$
$860$		0			0
$861$	−1200.00	+	1175.76i	−1.39373	+	1.36557i
$862$		0			0
$863$	330.000			0.382387			0.191194	−	0.981552i	$-0.438764\pi$
$863$	330.000			0.382387			0.191194	+	0.981552i	$0.438764\pi$
$864$		0			0
$865$	−936.000			−1.08208
$866$		0			0
$867$		−	465.403i		−	0.536797i
$868$		0			0
$869$	−444.000			−0.510932
$870$		0			0
$871$		−	342.929i		−	0.393718i
$872$		0			0
$873$		−	1175.76i		−	1.34680i
$874$		0			0
$875$	−624.000	−	636.867i	−0.713143	−	0.727848i
$876$		0			0
$877$	410.000			0.467503			0.233751	−	0.972296i	$-0.424900\pi$
$877$	410.000			0.467503			0.233751	+	0.972296i	$0.424900\pi$
$878$		0			0
$879$	−744.000			−0.846416
$880$		0			0
$881$		−	881.816i		−	1.00093i	−0.865758	−	0.500463i	$-0.833163\pi$
$881$		−	881.816i		−	1.00093i	0.865758	−	0.500463i	$-0.166837\pi$
$882$		0			0
$883$	730.000			0.826727			0.413364	−	0.910566i	$-0.364354\pi$
$883$	730.000			0.826727			0.413364	+	0.910566i	$0.364354\pi$
$884$		0			0
$885$		−	587.878i		−	0.664268i
$886$		0			0
$887$			1067.98i			1.20403i	0.798484	+	0.602017i	$0.205636\pi$
$887$			1067.98i			1.20403i	−0.798484	+	0.602017i	$0.794364\pi$
$888$		0			0
$889$	250.000	−	244.949i	0.281215	−	0.275533i
$890$		0			0
$891$	−54.0000			−0.0606061
$892$		0			0
$893$	480.000			0.537514
$894$		0			0
$895$		−	205.757i		−	0.229896i
$896$		0			0
$897$	720.000			0.802676
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	1763.63i		−	1.95742i
$902$		0			0
$903$	240.000	+	244.949i	0.265781	+	0.271261i
$904$		0			0
$905$	−120.000			−0.132597
$906$		0			0
$907$	250.000			0.275634			0.137817	−	0.990458i	$-0.455991\pi$
$907$	250.000			0.275634			0.137817	+	0.990458i	$0.455991\pi$
$908$		0			0
$909$		−	1102.27i		−	1.21262i
$910$		0			0
$911$	−78.0000			−0.0856202			−0.0428101	−	0.999083i	$-0.513631\pi$
$911$	−78.0000			−0.0856202			−0.0428101	+	0.999083i	$0.513631\pi$
$912$		0			0
$913$		−	382.120i		−	0.418533i
$914$		0			0
$915$			587.878i			0.642489i
$916$		0			0
$917$	−360.000	−	367.423i	−0.392585	−	0.400680i
$918$		0			0
$919$	1354.00			1.47334			0.736670	−	0.676252i	$-0.236397\pi$
$919$	1354.00			1.47334			0.736670	+	0.676252i	$0.236397\pi$
$920$		0			0
$921$	−504.000			−0.547231
$922$		0			0
$923$			205.757i			0.222922i
$924$		0			0
$925$	10.0000			0.0108108
$926$		0			0
$927$			2498.48i			2.69523i
$928$		0			0
$929$			391.918i			0.421871i	0.977500	+	0.210936i	$0.0676510\pi$
$929$			391.918i			0.421871i	−0.977500	+	0.210936i	$0.932349\pi$
$930$		0			0
$931$	1200.00	+	24.4949i	1.28894	+	0.0263103i
$932$		0			0
$933$	−1200.00			−1.28617
$934$		0			0
$935$	576.000			0.616043
$936$		0			0
$937$			715.251i			0.763342i	0.924298	+	0.381671i	$0.124651\pi$
$937$			715.251i			0.763342i	−0.924298	+	0.381671i	$0.875349\pi$
$938$		0			0
$939$	−1104.00			−1.17572
$940$		0			0
$941$			1298.23i			1.37963i	0.723987	+	0.689814i	$0.242308\pi$
$941$			1298.23i			1.37963i	−0.723987	+	0.689814i	$0.757692\pi$
$942$		0			0
$943$		−	1469.69i		−	1.55853i
$944$		0			0
$945$	−720.000	+	705.453i	−0.761905	+	0.746511i
$946$		0			0
$947$	−390.000			−0.411827			−0.205913	−	0.978570i	$-0.566017\pi$
$947$	−390.000			−0.411827			−0.205913	+	0.978570i	$0.566017\pi$
$948$		0			0
$949$	528.000			0.556375
$950$		0			0
$951$			440.908i			0.463626i
$952$		0			0
$953$	−270.000			−0.283316			−0.141658	−	0.989916i	$-0.545243\pi$
$953$	−270.000			−0.283316			−0.141658	+	0.989916i	$0.545243\pi$
$954$		0			0
$955$			1205.15i			1.26194i
$956$		0			0
$957$			176.363i			0.184288i
$958$		0			0
$959$	1050.00	−	1028.79i	1.09489	−	1.07277i
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$	2250.00			2.33645
$964$		0			0
$965$			1126.77i			1.16763i
$966$		0			0
$967$	−1150.00			−1.18925			−0.594623	−	0.804005i	$-0.702698\pi$
$967$	−1150.00			−1.18925			−0.594623	+	0.804005i	$0.702698\pi$
$968$		0			0
$969$			2351.51i			2.42674i
$970$		0			0
$971$		−	906.311i		−	0.933379i	−0.884421	−	0.466690i	$-0.845447\pi$
$971$		−	906.311i		−	0.933379i	0.884421	−	0.466690i	$-0.154553\pi$
$972$		0			0
$973$	−600.000	−	612.372i	−0.616650	−	0.629365i
$974$		0			0
$975$	−24.0000			−0.0246154
$976$		0			0
$977$	−30.0000			−0.0307062			−0.0153531	−	0.999882i	$-0.504887\pi$
$977$	−30.0000			−0.0307062			−0.0153531	+	0.999882i	$0.504887\pi$
$978$		0			0
$979$		−	881.816i		−	0.900732i
$980$		0			0
$981$	1290.00			1.31498
$982$		0			0
$983$		−	1244.34i		−	1.26586i	−0.774209	−	0.632930i	$-0.781852\pi$
$983$		−	1244.34i		−	1.26586i	0.774209	−	0.632930i	$-0.218148\pi$
$984$		0			0
$985$			734.847i			0.746037i
$986$		0			0
$987$	480.000	−	470.302i	0.486322	−	0.476496i
$988$		0			0
$989$	−300.000			−0.303337
$990$		0			0
$991$	−1718.00			−1.73360			−0.866801	−	0.498654i	$-0.833828\pi$
$991$	−1718.00			−1.73360			−0.866801	+	0.498654i	$0.833828\pi$
$992$		0			0
$993$		−	186.161i		−	0.187474i
$994$		0			0
$995$	1680.00			1.68844
$996$		0			0
$997$		−	1219.85i		−	1.22352i	−0.791045	−	0.611758i	$-0.790463\pi$
$997$		−	1219.85i		−	1.22352i	0.791045	−	0.611758i	$-0.209537\pi$
$998$		0			0
$999$		−	293.939i		−	0.294233i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.3.b.a.13.2	yes	2
3.2	odd	2		252.3.d.c.181.2		2
4.3	odd	2		112.3.c.b.97.1		2
5.2	odd	4		700.3.h.a.349.4		4
5.3	odd	4		700.3.h.a.349.1		4
5.4	even	2		700.3.d.a.601.1		2
7.2	even	3		196.3.h.b.129.1		4
7.3	odd	6		196.3.h.b.117.1		4
7.4	even	3		196.3.h.b.117.2		4
7.5	odd	6		196.3.h.b.129.2		4
7.6	odd	2	inner	28.3.b.a.13.1	✓	2
8.3	odd	2		448.3.c.c.321.2		2
8.5	even	2		448.3.c.d.321.1		2
12.11	even	2		1008.3.f.c.433.2		2
21.2	odd	6		1764.3.z.i.325.2		4
21.5	even	6		1764.3.z.i.325.1		4
21.11	odd	6		1764.3.z.i.901.1		4
21.17	even	6		1764.3.z.i.901.2		4
21.20	even	2		252.3.d.c.181.1		2
28.3	even	6		784.3.s.d.705.2		4
28.11	odd	6		784.3.s.d.705.1		4
28.19	even	6		784.3.s.d.129.1		4
28.23	odd	6		784.3.s.d.129.2		4
28.27	even	2		112.3.c.b.97.2		2
35.13	even	4		700.3.h.a.349.3		4
35.27	even	4		700.3.h.a.349.2		4
35.34	odd	2		700.3.d.a.601.2		2
56.13	odd	2		448.3.c.d.321.2		2
56.27	even	2		448.3.c.c.321.1		2
84.83	odd	2		1008.3.f.c.433.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.b.a.13.1	✓	2	7.6	odd	2	inner
28.3.b.a.13.2	yes	2	1.1	even	1	trivial
112.3.c.b.97.1		2	4.3	odd	2
112.3.c.b.97.2		2	28.27	even	2
196.3.h.b.117.1		4	7.3	odd	6
196.3.h.b.117.2		4	7.4	even	3
196.3.h.b.129.1		4	7.2	even	3
196.3.h.b.129.2		4	7.5	odd	6
252.3.d.c.181.1		2	21.20	even	2
252.3.d.c.181.2		2	3.2	odd	2
448.3.c.c.321.1		2	56.27	even	2
448.3.c.c.321.2		2	8.3	odd	2
448.3.c.d.321.1		2	8.5	even	2
448.3.c.d.321.2		2	56.13	odd	2
700.3.d.a.601.1		2	5.4	even	2
700.3.d.a.601.2		2	35.34	odd	2
700.3.h.a.349.1		4	5.3	odd	4
700.3.h.a.349.2		4	35.27	even	4
700.3.h.a.349.3		4	35.13	even	4
700.3.h.a.349.4		4	5.2	odd	4
784.3.s.d.129.1		4	28.19	even	6
784.3.s.d.129.2		4	28.23	odd	6
784.3.s.d.705.1		4	28.11	odd	6
784.3.s.d.705.2		4	28.3	even	6
1008.3.f.c.433.1		2	84.83	odd	2
1008.3.f.c.433.2		2	12.11	even	2
1764.3.z.i.325.1		4	21.5	even	6
1764.3.z.i.325.2		4	21.2	odd	6
1764.3.z.i.901.1		4	21.11	odd	6
1764.3.z.i.901.2		4	21.17	even	6

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:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	28.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.89898i q^{3} -4.89898i q^{5} +(5.00000 - 4.89898i) q^{7} -15.0000 q^{9} +O(q^{10})\)	\(q+4.89898i q^{3} -4.89898i q^{5} +(5.00000 - 4.89898i) q^{7} -15.0000 q^{9} -6.00000 q^{11} +4.89898i q^{13} +24.0000 q^{15} -19.5959i q^{17} +24.4949i q^{19} +(24.0000 + 24.4949i) q^{21} -30.0000 q^{23} +1.00000 q^{25} -29.3939i q^{27} -6.00000 q^{29} -29.3939i q^{33} +(-24.0000 - 24.4949i) q^{35} +10.0000 q^{37} -24.0000 q^{39} +48.9898i q^{41} +10.0000 q^{43} +73.4847i q^{45} -19.5959i q^{47} +(1.00000 - 48.9898i) q^{49} +96.0000 q^{51} +90.0000 q^{53} +29.3939i q^{55} -120.000 q^{57} -24.4949i q^{59} +24.4949i q^{61} +(-75.0000 + 73.4847i) q^{63} +24.0000 q^{65} -70.0000 q^{67} -146.969i q^{69} +42.0000 q^{71} -107.778i q^{73} +4.89898i q^{75} +(-30.0000 + 29.3939i) q^{77} +74.0000 q^{79} +9.00000 q^{81} +63.6867i q^{83} -96.0000 q^{85} -29.3939i q^{87} +146.969i q^{89} +(24.0000 + 24.4949i) q^{91} +120.000 q^{95} +78.3837i q^{97} +90.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{7} - 30 q^{9}+O(q^{10}) \) 2 * q + 10 * q^7 - 30 * q^9	\( 2 q + 10 q^{7} - 30 q^{9} - 12 q^{11} + 48 q^{15} + 48 q^{21} - 60 q^{23} + 2 q^{25} - 12 q^{29} - 48 q^{35} + 20 q^{37} - 48 q^{39} + 20 q^{43} + 2 q^{49} + 192 q^{51} + 180 q^{53} - 240 q^{57} - 150 q^{63} + 48 q^{65} - 140 q^{67} + 84 q^{71} - 60 q^{77} + 148 q^{79} + 18 q^{81} - 192 q^{85} + 48 q^{91} + 240 q^{95} + 180 q^{99}+O(q^{100}) \) 2 * q + 10 * q^7 - 30 * q^9 - 12 * q^11 + 48 * q^15 + 48 * q^21 - 60 * q^23 + 2 * q^25 - 12 * q^29 - 48 * q^35 + 20 * q^37 - 48 * q^39 + 20 * q^43 + 2 * q^49 + 192 * q^51 + 180 * q^53 - 240 * q^57 - 150 * q^63 + 48 * q^65 - 140 * q^67 + 84 * q^71 - 60 * q^77 + 148 * q^79 + 18 * q^81 - 192 * q^85 + 48 * q^91 + 240 * q^95 + 180 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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