LMFDB - Embedded newform 1764.3.bk.c.557.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(557,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("1764.557");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.bk (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			557.2
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.557
Dual form			1764.3.bk.c.1745.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.89898 - 2.82843i) q^{5} +O(q^{10})$	$q+(4.89898 - 2.82843i) q^{5} +(1.22474 + 0.707107i) q^{11} +16.0000 q^{13} +(14.6969 + 8.48528i) q^{17} +(4.00000 + 6.92820i) q^{19} +(11.0227 - 6.36396i) q^{23} +(3.50000 - 6.06218i) q^{25} +24.0416i q^{29} +(-28.0000 + 48.4974i) q^{31} +(-12.0000 - 20.7846i) q^{37} -50.9117i q^{41} +40.0000 q^{43} +(-9.79796 + 5.65685i) q^{47} +(37.9671 + 21.9203i) q^{53} +8.00000 q^{55} +(9.79796 + 5.65685i) q^{59} +(20.0000 + 34.6410i) q^{61} +(78.3837 - 45.2548i) q^{65} +(13.0000 - 22.5167i) q^{67} +134.350i q^{71} +(44.0000 - 76.2102i) q^{73} +(-41.0000 - 71.0141i) q^{79} +101.823i q^{83} +96.0000 q^{85} +(53.8888 - 31.1127i) q^{89} +(39.1918 + 22.6274i) q^{95} -40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 64 q^{13} + 16 q^{19} + 14 q^{25} - 112 q^{31} - 48 q^{37} + 160 q^{43} + 32 q^{55} + 80 q^{61} + 52 q^{67} + 176 q^{73} - 164 q^{79} + 384 q^{85} - 160 q^{97}+O(q^{100}) $ 4 * q + 64 * q^13 + 16 * q^19 + 14 * q^25 - 112 * q^31 - 48 * q^37 + 160 * q^43 + 32 * q^55 + 80 * q^61 + 52 * q^67 + 176 * q^73 - 164 * q^79 + 384 * q^85 - 160 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	4.89898	−	2.82843i	0.979796	−	0.565685i	0.0775874	−	0.996986i	$-0.475278\pi$
$5$	4.89898	−	2.82843i	0.979796	−	0.565685i	0.902209	+	0.431300i	$0.141945\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.22474	+	0.707107i	0.111340	+	0.0642824i	0.554636	−	0.832093i	$-0.312857\pi$
$11$	1.22474	+	0.707107i	0.111340	+	0.0642824i	−0.443296	+	0.896375i	$0.646191\pi$
$12$		0			0
$13$	16.0000			1.23077			0.615385	−	0.788227i	$-0.289001\pi$
$13$	16.0000			1.23077			0.615385	+	0.788227i	$0.289001\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.6969	+	8.48528i	0.864526	+	0.499134i	0.865525	−	0.500865i	$-0.166985\pi$
$17$	14.6969	+	8.48528i	0.864526	+	0.499134i	−0.000999453		1.00000i	$0.500318\pi$
$18$		0			0
$19$	4.00000	+	6.92820i	0.210526	+	0.364642i	0.951879	−	0.306473i	$-0.0991489\pi$
$19$	4.00000	+	6.92820i	0.210526	+	0.364642i	−0.741353	+	0.671115i	$0.765816\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	11.0227	−	6.36396i	0.479248	−	0.276694i	−0.240855	−	0.970561i	$-0.577428\pi$
$23$	11.0227	−	6.36396i	0.479248	−	0.276694i	0.720103	+	0.693867i	$0.244095\pi$
$24$		0			0
$25$	3.50000	−	6.06218i	0.140000	−	0.242487i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			24.0416i			0.829022i	0.910044	+	0.414511i	$0.136047\pi$
$29$			24.0416i			0.829022i	−0.910044	+	0.414511i	$0.863953\pi$
$30$		0			0
$31$	−28.0000	+	48.4974i	−0.903226	+	1.56443i	−0.0799446	+	0.996799i	$0.525474\pi$
$31$	−28.0000	+	48.4974i	−0.903226	+	1.56443i	−0.823281	+	0.567634i	$0.807859\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−12.0000	−	20.7846i	−0.324324	−	0.561746i	0.657051	−	0.753846i	$-0.271804\pi$
$37$	−12.0000	−	20.7846i	−0.324324	−	0.561746i	−0.981375	+	0.192100i	$0.938470\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	50.9117i		−	1.24175i	−0.783910	−	0.620874i	$-0.786778\pi$
$41$		−	50.9117i		−	1.24175i	0.783910	−	0.620874i	$-0.213222\pi$
$42$		0			0
$43$	40.0000			0.930233			0.465116	−	0.885250i	$-0.346013\pi$
$43$	40.0000			0.930233			0.465116	+	0.885250i	$0.346013\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.79796	+	5.65685i	−0.208467	+	0.120359i	−0.600599	−	0.799551i	$-0.705071\pi$
$47$	−9.79796	+	5.65685i	−0.208467	+	0.120359i	0.392132	+	0.919909i	$0.371738\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	37.9671	+	21.9203i	0.716360	+	0.413591i	0.813412	−	0.581689i	$-0.197608\pi$
$53$	37.9671	+	21.9203i	0.716360	+	0.413591i	−0.0970513	+	0.995279i	$0.530941\pi$
$54$		0			0
$55$	8.00000			0.145455
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.79796	+	5.65685i	0.166067	+	0.0958789i	0.580730	−	0.814096i	$-0.302767\pi$
$59$	9.79796	+	5.65685i	0.166067	+	0.0958789i	−0.414663	+	0.909975i	$0.636101\pi$
$60$		0			0
$61$	20.0000	+	34.6410i	0.327869	+	0.567886i	0.982089	−	0.188419i	$-0.0603363\pi$
$61$	20.0000	+	34.6410i	0.327869	+	0.567886i	−0.654220	+	0.756304i	$0.727003\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	78.3837	−	45.2548i	1.20590	−	0.696228i
$66$		0			0
$67$	13.0000	−	22.5167i	0.194030	−	0.336070i	−0.752552	−	0.658533i	$-0.771177\pi$
$67$	13.0000	−	22.5167i	0.194030	−	0.336070i	0.946582	+	0.322463i	$0.104511\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			134.350i			1.89226i	0.323791	+	0.946129i	$0.395043\pi$
$71$			134.350i			1.89226i	−0.323791	+	0.946129i	$0.604957\pi$
$72$		0			0
$73$	44.0000	−	76.2102i	0.602740	−	1.04398i	−0.389665	−	0.920957i	$-0.627409\pi$
$73$	44.0000	−	76.2102i	0.602740	−	1.04398i	0.992404	−	0.123019i	$-0.0392576\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−41.0000	−	71.0141i	−0.518987	−	0.898912i	−0.999757	−	0.0220653i	$-0.992976\pi$
$79$	−41.0000	−	71.0141i	−0.518987	−	0.898912i	0.480769	−	0.876847i	$-0.340357\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			101.823i			1.22679i	0.789777	+	0.613394i	$0.210196\pi$
$83$			101.823i			1.22679i	−0.789777	+	0.613394i	$0.789804\pi$
$84$		0			0
$85$	96.0000			1.12941
$86$		0			0
$87$		0			0
$88$		0			0
$89$	53.8888	−	31.1127i	0.605492	−	0.349581i	−0.165707	−	0.986175i	$-0.552991\pi$
$89$	53.8888	−	31.1127i	0.605492	−	0.349581i	0.771199	+	0.636594i	$0.219657\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	39.1918	+	22.6274i	0.412546	+	0.238183i
$96$		0			0
$97$	−40.0000			−0.412371			−0.206186	−	0.978513i	$-0.566105\pi$
$97$	−40.0000			−0.412371			−0.206186	+	0.978513i	$0.566105\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−93.0806	−	53.7401i	−0.921590	−	0.532080i	−0.0374482	−	0.999299i	$-0.511923\pi$
$101$	−93.0806	−	53.7401i	−0.921590	−	0.532080i	−0.884142	+	0.467218i	$0.845256\pi$
$102$		0			0
$103$	−92.0000	−	159.349i	−0.893204	−	1.54707i	−0.836012	−	0.548711i	$-0.815119\pi$
$103$	−92.0000	−	159.349i	−0.893204	−	1.54707i	−0.0571920	−	0.998363i	$-0.518215\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	126.149	−	72.8320i	1.17896	−	0.680673i	0.223186	−	0.974776i	$-0.428354\pi$
$107$	126.149	−	72.8320i	1.17896	−	0.680673i	0.955774	+	0.294103i	$0.0950209\pi$
$108$		0			0
$109$	52.0000	−	90.0666i	0.477064	−	0.826299i	−0.522590	−	0.852584i	$-0.675034\pi$
$109$	52.0000	−	90.0666i	0.477064	−	0.826299i	0.999655	+	0.0262845i	$0.00836757\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			171.120i			1.51433i	0.653221	+	0.757167i	$0.273417\pi$
$113$			171.120i			1.51433i	−0.653221	+	0.757167i	$0.726583\pi$
$114$		0			0
$115$	36.0000	−	62.3538i	0.313043	−	0.542207i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−59.5000	−	103.057i	−0.491736	−	0.851711i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			101.823i			0.814587i
$126$		0			0
$127$	206.000			1.62205			0.811024	−	0.585013i	$-0.198911\pi$
$127$	206.000			1.62205			0.811024	+	0.585013i	$0.198911\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−176.363	+	101.823i	−1.34628	+	0.777278i	−0.987721	−	0.156228i	$-0.950067\pi$
$131$	−176.363	+	101.823i	−1.34628	+	0.777278i	−0.358563	+	0.933505i	$0.616733\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	155.543	+	89.8026i	1.13535	+	0.655493i	0.945274	−	0.326277i	$-0.105794\pi$
$137$	155.543	+	89.8026i	1.13535	+	0.655493i	0.190073	+	0.981770i	$0.439127\pi$
$138$		0			0
$139$	−192.000			−1.38129			−0.690647	−	0.723192i	$-0.742674\pi$
$139$	−192.000			−1.38129			−0.690647	+	0.723192i	$0.742674\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	19.5959	+	11.3137i	0.137034	+	0.0791168i
$144$		0			0
$145$	68.0000	+	117.779i	0.468966	+	0.812272i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	204.532	−	118.087i	1.37270	−	0.792529i	0.381433	−	0.924396i	$-0.375431\pi$
$149$	204.532	−	118.087i	1.37270	−	0.792529i	0.991267	+	0.131867i	$0.0420973\pi$
$150$		0			0
$151$	−8.00000	+	13.8564i	−0.0529801	+	0.0917643i	−0.891299	−	0.453416i	$-0.850205\pi$
$151$	−8.00000	+	13.8564i	−0.0529801	+	0.0917643i	0.838319	+	0.545180i	$0.183539\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			316.784i			2.04377i
$156$		0			0
$157$	28.0000	−	48.4974i	0.178344	−	0.308901i	−0.762969	−	0.646434i	$-0.776259\pi$
$157$	28.0000	−	48.4974i	0.178344	−	0.308901i	0.941313	+	0.337534i	$0.109593\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−51.0000	−	88.3346i	−0.312883	−	0.541930i	0.666102	−	0.745861i	$-0.267962\pi$
$163$	−51.0000	−	88.3346i	−0.312883	−	0.541930i	−0.978985	+	0.203931i	$0.934628\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	260.215i		−	1.55818i	−0.626915	−	0.779088i	$-0.715683\pi$
$167$		−	260.215i		−	1.55818i	0.626915	−	0.779088i	$-0.284317\pi$
$168$		0			0
$169$	87.0000			0.514793
$170$		0			0
$171$		0			0
$172$		0			0
$173$	83.2827	−	48.0833i	0.481403	−	0.277938i	−0.239598	−	0.970872i	$-0.577016\pi$
$173$	83.2827	−	48.0833i	0.481403	−	0.277938i	0.721001	+	0.692934i	$0.243682\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−18.3712	−	10.6066i	−0.102632	−	0.0592548i	0.447805	−	0.894131i	$-0.352206\pi$
$179$	−18.3712	−	10.6066i	−0.102632	−	0.0592548i	−0.550438	+	0.834876i	$0.685539\pi$
$180$		0			0
$181$	128.000			0.707182			0.353591	−	0.935400i	$-0.384960\pi$
$181$	128.000			0.707182			0.353591	+	0.935400i	$0.384960\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−117.576	−	67.8823i	−0.635543	−	0.366931i
$186$		0			0
$187$	12.0000	+	20.7846i	0.0641711	+	0.111148i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	99.2043	−	57.2756i	0.519394	−	0.299873i	−0.217292	−	0.976107i	$-0.569723\pi$
$191$	99.2043	−	57.2756i	0.519394	−	0.299873i	0.736687	+	0.676234i	$0.236389\pi$
$192$		0			0
$193$	104.000	−	180.133i	0.538860	−	0.933333i	−0.460106	−	0.887864i	$-0.652188\pi$
$193$	104.000	−	180.133i	0.538860	−	0.933333i	0.998966	−	0.0454689i	$-0.0144782\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			123.037i			0.624551i	0.949992	+	0.312276i	$0.101091\pi$
$197$			123.037i			0.624551i	−0.949992	+	0.312276i	$0.898909\pi$
$198$		0			0
$199$	−88.0000	+	152.420i	−0.442211	+	0.765932i	−0.997853	−	0.0654895i	$-0.979139\pi$
$199$	−88.0000	+	152.420i	−0.442211	+	0.765932i	0.555642	+	0.831422i	$0.312472\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−144.000	−	249.415i	−0.702439	−	1.21666i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			11.3137i			0.0541326i
$210$		0			0
$211$	248.000			1.17536			0.587678	−	0.809095i	$-0.300042\pi$
$211$	248.000			1.17536			0.587678	+	0.809095i	$0.300042\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	195.959	−	113.137i	0.911438	−	0.526219i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	235.151	+	135.765i	1.06403	+	0.614319i
$222$		0			0
$223$	272.000			1.21973			0.609865	−	0.792505i	$-0.291223\pi$
$223$	272.000			1.21973			0.609865	+	0.792505i	$0.291223\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	342.929	+	197.990i	1.51070	+	0.872202i	0.999922	+	0.0124887i	$0.00397537\pi$
$227$	342.929	+	197.990i	1.51070	+	0.872202i	0.510777	+	0.859714i	$0.329358\pi$
$228$		0			0
$229$	40.0000	+	69.2820i	0.174672	+	0.302542i	0.940048	−	0.341042i	$-0.110780\pi$
$229$	40.0000	+	69.2820i	0.174672	+	0.302542i	−0.765375	+	0.643584i	$0.777447\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	175.139	−	101.116i	0.751667	−	0.433975i	−0.0746286	−	0.997211i	$-0.523777\pi$
$233$	175.139	−	101.116i	0.751667	−	0.433975i	0.826296	+	0.563236i	$0.190444\pi$
$234$		0			0
$235$	−32.0000	+	55.4256i	−0.136170	+	0.235854i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			326.683i			1.36688i	0.730009	+	0.683438i	$0.239516\pi$
$239$			326.683i			1.36688i	−0.730009	+	0.683438i	$0.760484\pi$
$240$		0			0
$241$	−124.000	+	214.774i	−0.514523	+	0.891180i	0.485335	+	0.874328i	$0.338698\pi$
$241$	−124.000	+	214.774i	−0.514523	+	0.891180i	−0.999858	+	0.0168515i	$0.994636\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	64.0000	+	110.851i	0.259109	+	0.448790i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			294.156i			1.17194i	0.810333	+	0.585969i	$0.199286\pi$
$251$			294.156i			1.17194i	−0.810333	+	0.585969i	$0.800714\pi$
$252$		0			0
$253$	18.0000			0.0711462
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−151.868	+	87.6812i	−0.590927	+	0.341172i	−0.765464	−	0.643479i	$-0.777491\pi$
$257$	−151.868	+	87.6812i	−0.590927	+	0.341172i	0.174537	+	0.984651i	$0.444157\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−138.396	−	79.9031i	−0.526221	−	0.303814i	0.213255	−	0.976997i	$-0.431593\pi$
$263$	−138.396	−	79.9031i	−0.526221	−	0.303814i	−0.739476	+	0.673183i	$0.764927\pi$
$264$		0			0
$265$	248.000			0.935849
$266$		0			0
$267$		0			0
$268$		0			0
$269$	259.646	+	149.907i	0.965226	+	0.557274i	0.897778	−	0.440449i	$-0.145181\pi$
$269$	259.646	+	149.907i	0.965226	+	0.557274i	0.0674488	+	0.997723i	$0.478514\pi$
$270$		0			0
$271$	−44.0000	−	76.2102i	−0.162362	−	0.281219i	0.773354	−	0.633975i	$-0.218578\pi$
$271$	−44.0000	−	76.2102i	−0.162362	−	0.281219i	−0.935715	+	0.352756i	$0.885244\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.57321	−	4.94975i	0.0311753	−	0.0179991i
$276$		0			0
$277$	19.0000	−	32.9090i	0.0685921	−	0.118805i	−0.829690	−	0.558225i	$-0.811483\pi$
$277$	19.0000	−	32.9090i	0.0685921	−	0.118805i	0.898282	+	0.439420i	$0.144816\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	182.434i		−	0.649230i	−0.945846	−	0.324615i	$-0.894765\pi$
$281$		−	182.434i		−	0.649230i	0.945846	−	0.324615i	$-0.105235\pi$
$282$		0			0
$283$	−76.0000	+	131.636i	−0.268551	+	0.465144i	−0.968488	−	0.249061i	$-0.919878\pi$
$283$	−76.0000	+	131.636i	−0.268551	+	0.465144i	0.699937	+	0.714205i	$0.253212\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.00173010	−	0.00299663i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	458.205i		−	1.56384i	−0.623379	−	0.781920i	$-0.714240\pi$
$293$		−	458.205i		−	1.56384i	0.623379	−	0.781920i	$-0.285760\pi$
$294$		0			0
$295$	64.0000			0.216949
$296$		0			0
$297$		0			0
$298$		0			0
$299$	176.363	−	101.823i	0.589844	−	0.340546i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	195.959	+	113.137i	0.642489	+	0.370941i
$306$		0			0
$307$	−232.000			−0.755700			−0.377850	−	0.925867i	$-0.623337\pi$
$307$	−232.000			−0.755700			−0.377850	+	0.925867i	$0.623337\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−421.312	−	243.245i	−1.35470	−	0.782137i	−0.365798	−	0.930694i	$-0.619204\pi$
$311$	−421.312	−	243.245i	−1.35470	−	0.782137i	−0.988904	+	0.148557i	$0.952537\pi$
$312$		0			0
$313$	−72.0000	−	124.708i	−0.230032	−	0.398427i	0.727785	−	0.685805i	$-0.240550\pi$
$313$	−72.0000	−	124.708i	−0.230032	−	0.398427i	−0.957817	+	0.287378i	$0.907216\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	69.8105	−	40.3051i	0.220222	−	0.127145i	−0.385831	−	0.922570i	$-0.626085\pi$
$317$	69.8105	−	40.3051i	0.220222	−	0.127145i	0.606053	+	0.795424i	$0.292752\pi$
$318$		0			0
$319$	−17.0000	+	29.4449i	−0.0532915	+	0.0923036i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			135.765i			0.420324i
$324$		0			0
$325$	56.0000	−	96.9948i	0.172308	−	0.298446i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−100.000	−	173.205i	−0.302115	−	0.523278i	0.674500	−	0.738275i	$-0.264359\pi$
$331$	−100.000	−	173.205i	−0.302115	−	0.523278i	−0.976615	+	0.214997i	$0.931026\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	147.078i		−	0.439039i
$336$		0			0
$337$	−208.000			−0.617211			−0.308605	−	0.951190i	$-0.599862\pi$
$337$	−208.000			−0.617211			−0.308605	+	0.951190i	$0.599862\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−68.5857	+	39.5980i	−0.201131	+	0.116123i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−400.492	−	231.224i	−1.15415	−	0.666351i	−0.204258	−	0.978917i	$-0.565478\pi$
$347$	−400.492	−	231.224i	−1.15415	−	0.666351i	−0.949896	+	0.312566i	$0.898812\pi$
$348$		0			0
$349$	−456.000			−1.30659			−0.653295	−	0.757103i	$-0.726614\pi$
$349$	−456.000			−1.30659			−0.653295	+	0.757103i	$0.726614\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.6969	+	8.48528i	0.0416344	+	0.0240376i	0.520673	−	0.853756i	$-0.325681\pi$
$353$	14.6969	+	8.48528i	0.0416344	+	0.0240376i	−0.479038	+	0.877794i	$0.659015\pi$
$354$		0			0
$355$	380.000	+	658.179i	1.07042	+	1.85403i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−380.896	+	219.910i	−1.06099	+	0.612563i	−0.925706	−	0.378244i	$-0.876528\pi$
$359$	−380.896	+	219.910i	−1.06099	+	0.612563i	−0.135285	+	0.990807i	$0.543195\pi$
$360$		0			0
$361$	148.500	−	257.210i	0.411357	−	0.712492i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	497.803i		−	1.36384i
$366$		0			0
$367$	248.000	−	429.549i	0.675749	−	1.17043i	−0.300500	−	0.953782i	$-0.597154\pi$
$367$	248.000	−	429.549i	0.675749	−	1.17043i	0.976249	−	0.216650i	$-0.0695131\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	157.000	+	271.932i	0.420912	+	0.729040i	0.996029	−	0.0890308i	$-0.0283769\pi$
$373$	157.000	+	271.932i	0.420912	+	0.729040i	−0.575117	+	0.818071i	$0.695044\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			384.666i			1.02033i
$378$		0			0
$379$	24.0000			0.0633245			0.0316623	−	0.999499i	$-0.489920\pi$
$379$	24.0000			0.0633245			0.0316623	+	0.999499i	$0.489920\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−470.302	+	271.529i	−1.22794	+	0.708953i	−0.966599	−	0.256293i	$-0.917499\pi$
$383$	−470.302	+	271.529i	−1.22794	+	0.708953i	−0.261343	+	0.965246i	$0.584166\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	216.780	+	125.158i	0.557275	+	0.321743i	0.752051	−	0.659105i	$-0.229065\pi$
$389$	216.780	+	125.158i	0.557275	+	0.321743i	−0.194776	+	0.980848i	$0.562398\pi$
$390$		0			0
$391$	216.000			0.552430
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−401.716	−	231.931i	−1.01700	−	0.587167i
$396$		0			0
$397$	−332.000	−	575.041i	−0.836272	−	1.44847i	−0.892991	−	0.450075i	$-0.851397\pi$
$397$	−332.000	−	575.041i	−0.836272	−	1.44847i	0.0567185	−	0.998390i	$-0.481936\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−106.553	+	61.5183i	−0.265718	+	0.153412i	−0.626940	−	0.779068i	$-0.715693\pi$
$401$	−106.553	+	61.5183i	−0.265718	+	0.153412i	0.361222	+	0.932480i	$0.382360\pi$
$402$		0			0
$403$	−448.000	+	775.959i	−1.11166	+	1.92546i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	33.9411i		−	0.0833934i
$408$		0			0
$409$	156.000	−	270.200i	0.381418	−	0.660636i	−0.609847	−	0.792519i	$-0.708769\pi$
$409$	156.000	−	270.200i	0.381418	−	0.660636i	0.991265	+	0.131884i	$0.0421025\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	288.000	+	498.831i	0.693976	+	1.20200i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			226.274i			0.540034i	0.962856	+	0.270017i	$0.0870293\pi$
$419$			226.274i			0.540034i	−0.962856	+	0.270017i	$0.912971\pi$
$420$		0			0
$421$	−762.000			−1.80998			−0.904988	−	0.425437i	$-0.860120\pi$
$421$	−762.000			−1.80998			−0.904988	+	0.425437i	$0.860120\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	102.879	−	59.3970i	0.242067	−	0.139758i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−324.557	−	187.383i	−0.753033	−	0.434764i	0.0737555	−	0.997276i	$-0.476502\pi$
$431$	−324.557	−	187.383i	−0.753033	−	0.434764i	−0.826789	+	0.562512i	$0.809835\pi$
$432$		0			0
$433$	−464.000			−1.07159			−0.535797	−	0.844347i	$-0.679989\pi$
$433$	−464.000			−1.07159			−0.535797	+	0.844347i	$0.679989\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	88.1816	+	50.9117i	0.201789	+	0.116503i
$438$		0			0
$439$	−240.000	−	415.692i	−0.546697	−	0.946907i	−0.998498	−	0.0547883i	$-0.982552\pi$
$439$	−240.000	−	415.692i	−0.546697	−	0.946907i	0.451801	−	0.892119i	$-0.350782\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−393.143	+	226.981i	−0.887456	+	0.512373i	−0.873110	−	0.487524i	$-0.837900\pi$
$443$	−393.143	+	226.981i	−0.887456	+	0.512373i	−0.0143466	+	0.999897i	$0.504567\pi$
$444$		0			0
$445$	176.000	−	304.841i	0.395506	−	0.685036i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	835.800i		−	1.86147i	−0.365694	−	0.930735i	$-0.619168\pi$
$449$		−	835.800i		−	1.86147i	0.365694	−	0.930735i	$-0.380832\pi$
$450$		0			0
$451$	36.0000	−	62.3538i	0.0798226	−	0.138257i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−208.000	−	360.267i	−0.455142	−	0.788329i	0.543554	−	0.839374i	$-0.317078\pi$
$457$	−208.000	−	360.267i	−0.455142	−	0.788329i	−0.998696	+	0.0510446i	$0.983745\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			661.852i			1.43569i	0.696204	+	0.717844i	$0.254871\pi$
$461$			661.852i			1.43569i	−0.696204	+	0.717844i	$0.745129\pi$
$462$		0			0
$463$	146.000			0.315335			0.157667	−	0.987492i	$-0.449603\pi$
$463$	146.000			0.315335			0.157667	+	0.987492i	$0.449603\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	509.494	−	294.156i	1.09099	−	0.629885i	0.157153	−	0.987574i	$-0.449769\pi$
$467$	509.494	−	294.156i	1.09099	−	0.629885i	0.933841	+	0.357689i	$0.116435\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	48.9898	+	28.2843i	0.103573	+	0.0597976i
$474$		0			0
$475$	56.0000			0.117895
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−58.7878	−	33.9411i	−0.122730	−	0.0708583i	0.437378	−	0.899278i	$-0.355907\pi$
$479$	−58.7878	−	33.9411i	−0.122730	−	0.0708583i	−0.560108	+	0.828419i	$0.689240\pi$
$480$		0			0
$481$	−192.000	−	332.554i	−0.399168	−	0.691380i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−195.959	+	113.137i	−0.404040	+	0.233272i
$486$		0			0
$487$	−56.0000	+	96.9948i	−0.114990	+	0.199168i	−0.917776	−	0.397099i	$-0.870017\pi$
$487$	−56.0000	+	96.9948i	−0.114990	+	0.199168i	0.802786	+	0.596267i	$0.203350\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	428.507i		−	0.872722i	−0.899772	−	0.436361i	$-0.856267\pi$
$491$		−	428.507i		−	0.872722i	0.899772	−	0.436361i	$-0.143733\pi$
$492$		0			0
$493$	−204.000	+	353.338i	−0.413793	+	0.716711i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	372.000	+	644.323i	0.745491	+	1.29123i	0.949965	+	0.312356i	$0.101118\pi$
$499$	372.000	+	644.323i	0.745491	+	1.29123i	−0.204474	+	0.978872i	$0.565548\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	192.333i		−	0.382372i	−0.981554	−	0.191186i	$-0.938767\pi$
$503$		−	192.333i		−	0.382372i	0.981554	−	0.191186i	$-0.0612333\pi$
$504$		0			0
$505$	−608.000			−1.20396
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−240.050	+	138.593i	−0.471611	+	0.272285i	−0.716914	−	0.697162i	$-0.754446\pi$
$509$	−240.050	+	138.593i	−0.471611	+	0.272285i	0.245303	+	0.969446i	$0.421113\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−901.412	−	520.431i	−1.75032	−	1.01054i
$516$		0			0
$517$	−16.0000			−0.0309478
$518$		0			0
$519$		0			0
$520$		0			0
$521$	328.232	+	189.505i	0.630003	+	0.363732i	0.780753	−	0.624839i	$-0.214836\pi$
$521$	328.232	+	189.505i	0.630003	+	0.363732i	−0.150750	+	0.988572i	$0.548169\pi$
$522$		0			0
$523$	256.000	+	443.405i	0.489484	+	0.847811i	0.999927	−	0.0121008i	$-0.00385191\pi$
$523$	256.000	+	443.405i	0.489484	+	0.847811i	−0.510443	+	0.859912i	$0.670519\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−823.029	+	475.176i	−1.56172	+	0.901662i
$528$		0			0
$529$	−183.500	+	317.831i	−0.346881	+	0.600815i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	814.587i		−	1.52831i
$534$		0			0
$535$	412.000	−	713.605i	0.770093	−	1.33384i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	235.000	+	407.032i	0.434381	+	0.752370i	0.997245	−	0.0741799i	$-0.0236339\pi$
$541$	235.000	+	407.032i	0.434381	+	0.752370i	−0.562864	+	0.826549i	$0.690301\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	588.313i		−	1.07947i
$546$		0			0
$547$	−614.000			−1.12249			−0.561243	−	0.827651i	$-0.689677\pi$
$547$	−614.000			−1.12249			−0.561243	+	0.827651i	$0.689677\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−166.565	+	96.1665i	−0.302296	+	0.174531i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	275.568	+	159.099i	0.494735	+	0.285636i	0.726537	−	0.687128i	$-0.241129\pi$
$557$	275.568	+	159.099i	0.494735	+	0.285636i	−0.231801	+	0.972763i	$0.574462\pi$
$558$		0			0
$559$	640.000			1.14490
$560$		0			0
$561$		0			0
$562$		0			0
$563$	225.353	+	130.108i	0.400272	+	0.231097i	0.686601	−	0.727034i	$-0.259102\pi$
$563$	225.353	+	130.108i	0.400272	+	0.231097i	−0.286329	+	0.958131i	$0.592435\pi$
$564$		0			0
$565$	484.000	+	838.313i	0.856637	+	1.48374i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−246.174	+	142.128i	−0.432643	+	0.249786i	−0.700472	−	0.713680i	$-0.747027\pi$
$569$	−246.174	+	142.128i	−0.432643	+	0.249786i	0.267829	+	0.963466i	$0.413694\pi$
$570$		0			0
$571$	−75.0000	+	129.904i	−0.131349	+	0.227502i	−0.924197	−	0.381917i	$-0.875264\pi$
$571$	−75.0000	+	129.904i	−0.131349	+	0.227502i	0.792848	+	0.609419i	$0.208597\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	89.0955i		−	0.154949i
$576$		0			0
$577$	−424.000	+	734.390i	−0.734835	+	1.27277i	0.219960	+	0.975509i	$0.429407\pi$
$577$	−424.000	+	734.390i	−0.734835	+	1.27277i	−0.954796	+	0.297263i	$0.903926\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	31.0000	+	53.6936i	0.0531732	+	0.0920988i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	622.254i		−	1.06006i	−0.847980	−	0.530029i	$-0.822181\pi$
$587$		−	622.254i		−	1.06006i	0.847980	−	0.530029i	$-0.177819\pi$
$588$		0			0
$589$	−448.000			−0.760611
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−249.848	+	144.250i	−0.421329	+	0.243254i	−0.695646	−	0.718385i	$-0.744882\pi$
$593$	−249.848	+	144.250i	−0.421329	+	0.243254i	0.274317	+	0.961639i	$0.411548\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−393.143	−	226.981i	−0.656332	−	0.378934i	0.134546	−	0.990907i	$-0.457042\pi$
$599$	−393.143	−	226.981i	−0.656332	−	0.378934i	−0.790878	+	0.611974i	$0.790376\pi$
$600$		0			0
$601$	208.000			0.346090			0.173045	−	0.984914i	$-0.444639\pi$
$601$	208.000			0.346090			0.173045	+	0.984914i	$0.444639\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−582.979	−	336.583i	−0.963601	−	0.556335i
$606$		0			0
$607$	24.0000	+	41.5692i	0.0395387	+	0.0684831i	0.885118	−	0.465367i	$-0.154078\pi$
$607$	24.0000	+	41.5692i	0.0395387	+	0.0684831i	−0.845579	+	0.533851i	$0.820744\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−156.767	+	90.5097i	−0.256575	+	0.148134i
$612$		0			0
$613$	77.0000	−	133.368i	0.125612	−	0.217566i	−0.796360	−	0.604823i	$-0.793244\pi$
$613$	77.0000	−	133.368i	0.125612	−	0.217566i	0.921972	+	0.387257i	$0.126577\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1076.22i		−	1.74427i	−0.489263	−	0.872137i	$-0.662734\pi$
$617$		−	1076.22i		−	1.74427i	0.489263	−	0.872137i	$-0.337266\pi$
$618$		0			0
$619$	−472.000	+	817.528i	−0.762520	+	1.32072i	0.179028	+	0.983844i	$0.442705\pi$
$619$	−472.000	+	817.528i	−0.762520	+	1.32072i	−0.941548	+	0.336880i	$0.890628\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	375.500	+	650.385i	0.600800	+	1.04062i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	407.294i		−	0.647525i
$630$		0			0
$631$	126.000			0.199683			0.0998415	−	0.995003i	$-0.468166\pi$
$631$	126.000			0.199683			0.0998415	+	0.995003i	$0.468166\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1009.19	−	582.656i	1.58928	−	0.917568i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	402.941	+	232.638i	0.628613	+	0.362930i	0.780215	−	0.625512i	$-0.215110\pi$
$641$	402.941	+	232.638i	0.628613	+	0.362930i	−0.151602	+	0.988442i	$0.548443\pi$
$642$		0			0
$643$	−552.000			−0.858476			−0.429238	−	0.903191i	$-0.641218\pi$
$643$	−552.000			−0.858476			−0.429238	+	0.903191i	$0.641218\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	440.908	+	254.558i	0.681465	+	0.393444i	0.800407	−	0.599457i	$-0.204617\pi$
$647$	440.908	+	254.558i	0.681465	+	0.393444i	−0.118942	+	0.992901i	$0.537950\pi$
$648$		0			0
$649$	8.00000	+	13.8564i	0.0123267	+	0.0213504i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−273.118	+	157.685i	−0.418251	+	0.241478i	−0.694329	−	0.719658i	$-0.744299\pi$
$653$	−273.118	+	157.685i	−0.418251	+	0.241478i	0.276078	+	0.961135i	$0.410965\pi$
$654$		0			0
$655$	−576.000	+	997.661i	−0.879389	+	1.52315i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	270.115i		−	0.409886i	−0.978774	−	0.204943i	$-0.934299\pi$
$659$		−	270.115i		−	0.409886i	0.978774	−	0.204943i	$-0.0657009\pi$
$660$		0			0
$661$	−404.000	+	699.749i	−0.611195	+	1.05862i	0.379844	+	0.925050i	$0.375978\pi$
$661$	−404.000	+	699.749i	−0.611195	+	1.05862i	−0.991039	+	0.133571i	$0.957356\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	153.000	+	265.004i	0.229385	+	0.397307i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			56.5685i			0.0843048i
$672$		0			0
$673$	−832.000			−1.23626			−0.618128	−	0.786078i	$-0.712109\pi$
$673$	−832.000			−1.23626			−0.618128	+	0.786078i	$0.712109\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−367.423	+	212.132i	−0.542723	+	0.313341i	−0.746182	−	0.665742i	$-0.768115\pi$
$677$	−367.423	+	212.132i	−0.542723	+	0.313341i	0.203459	+	0.979084i	$0.434782\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1118.19	−	645.588i	−1.63718	−	0.945225i	−0.981799	−	0.189924i	$-0.939176\pi$
$683$	−1118.19	−	645.588i	−1.63718	−	0.945225i	−0.655378	−	0.755301i	$-0.727491\pi$
$684$		0			0
$685$	1016.00			1.48321
$686$		0			0
$687$		0			0
$688$		0			0
$689$	607.473	+	350.725i	0.881674	+	0.509035i
$690$		0			0
$691$	−136.000	−	235.559i	−0.196816	−	0.340896i	0.750678	−	0.660668i	$-0.229727\pi$
$691$	−136.000	−	235.559i	−0.196816	−	0.340896i	−0.947494	+	0.319772i	$0.896394\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−940.604	+	543.058i	−1.35339	+	0.781378i
$696$		0			0
$697$	432.000	−	748.246i	0.619799	−	1.07352i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	982.878i		−	1.40211i	−0.713108	−	0.701055i	$-0.752713\pi$
$701$		−	982.878i		−	1.40211i	0.713108	−	0.701055i	$-0.247287\pi$
$702$		0			0
$703$	96.0000	−	166.277i	0.136558	−	0.236525i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−596.000	−	1032.30i	−0.840621	−	1.45600i	−0.889371	−	0.457187i	$-0.848857\pi$
$709$	−596.000	−	1032.30i	−0.840621	−	1.45600i	0.0487502	−	0.998811i	$-0.484476\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			712.764i			0.999668i
$714$		0			0
$715$	128.000			0.179021
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−646.665	+	373.352i	−0.899395	+	0.519266i	−0.877004	−	0.480483i	$-0.840461\pi$
$719$	−646.665	+	373.352i	−0.899395	+	0.519266i	−0.0223914	+	0.999749i	$0.507128\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	145.745	+	84.1457i	0.201027	+	0.116063i
$726$		0			0
$727$	184.000			0.253095			0.126547	−	0.991961i	$-0.459610\pi$
$727$	184.000			0.253095			0.126547	+	0.991961i	$0.459610\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	587.878	+	339.411i	0.804210	+	0.464311i
$732$		0			0
$733$	−400.000	−	692.820i	−0.545703	−	0.945185i	−0.998562	−	0.0536028i	$-0.982930\pi$
$733$	−400.000	−	692.820i	−0.545703	−	0.945185i	0.452860	−	0.891582i	$-0.350404\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	31.8434	−	18.3848i	0.0432067	−	0.0249454i
$738$		0			0
$739$	−173.000	+	299.645i	−0.234100	+	0.405473i	−0.959011	−	0.283370i	$-0.908548\pi$
$739$	−173.000	+	299.645i	−0.234100	+	0.405473i	0.724911	+	0.688843i	$0.241881\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			895.197i			1.20484i	0.798179	+	0.602421i	$0.205797\pi$
$743$			895.197i			1.20484i	−0.798179	+	0.602421i	$0.794203\pi$
$744$		0			0
$745$	668.000	−	1157.01i	0.896644	−	1.55303i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	624.000	+	1080.80i	0.830892	+	1.43915i	0.897332	+	0.441357i	$0.145503\pi$
$751$	624.000	+	1080.80i	0.830892	+	1.43915i	−0.0664395	+	0.997790i	$0.521164\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			90.5097i			0.119880i
$756$		0			0
$757$	−392.000			−0.517834			−0.258917	−	0.965900i	$-0.583366\pi$
$757$	−392.000			−0.517834			−0.258917	+	0.965900i	$0.583366\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	827.928	−	478.004i	1.08795	−	0.628126i	0.154918	−	0.987927i	$-0.450489\pi$
$761$	827.928	−	478.004i	1.08795	−	0.628126i	0.933029	+	0.359801i	$0.117155\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	156.767	+	90.5097i	0.204390	+	0.118005i
$768$		0			0
$769$	256.000			0.332900			0.166450	−	0.986050i	$-0.446770\pi$
$769$	256.000			0.332900			0.166450	+	0.986050i	$0.446770\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−279.242	−	161.220i	−0.361244	−	0.208564i	0.308382	−	0.951263i	$-0.400212\pi$
$773$	−279.242	−	161.220i	−0.361244	−	0.208564i	−0.669626	+	0.742698i	$0.733546\pi$
$774$		0			0
$775$	196.000	+	339.482i	0.252903	+	0.438041i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	352.727	−	203.647i	0.452794	−	0.261421i
$780$		0			0
$781$	−95.0000	+	164.545i	−0.121639	+	0.210685i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	316.784i		−	0.403546i
$786$		0			0
$787$	−400.000	+	692.820i	−0.508259	+	0.880331i	0.491695	+	0.870767i	$0.336377\pi$
$787$	−400.000	+	692.820i	−0.508259	+	0.880331i	−0.999954	+	0.00956332i	$0.996956\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	320.000	+	554.256i	0.403531	+	0.698936i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			797.616i			1.00077i	0.865802	+	0.500387i	$0.166809\pi$
$797$			797.616i			1.00077i	−0.865802	+	0.500387i	$0.833191\pi$
$798$		0			0
$799$	−192.000			−0.240300
$800$		0			0
$801$		0			0
$802$		0			0
$803$	107.778	−	62.2254i	0.134219	−	0.0774912i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	422.537	+	243.952i	0.522295	+	0.301547i	0.737873	−	0.674939i	$-0.235830\pi$
$809$	422.537	+	243.952i	0.522295	+	0.301547i	−0.215578	+	0.976487i	$0.569164\pi$
$810$		0			0
$811$	−1232.00			−1.51911			−0.759556	−	0.650442i	$-0.774584\pi$
$811$	−1232.00			−1.51911			−0.759556	+	0.650442i	$0.774584\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−499.696	−	288.500i	−0.613124	−	0.353987i
$816$		0			0
$817$	160.000	+	277.128i	0.195838	+	0.339202i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	410.290	−	236.881i	0.499744	−	0.288527i	−0.228864	−	0.973458i	$-0.573501\pi$
$821$	410.290	−	236.881i	0.499744	−	0.288527i	0.728608	+	0.684931i	$0.240168\pi$
$822$		0			0
$823$	127.000	−	219.970i	0.154313	−	0.267279i	−0.778495	−	0.627650i	$-0.784017\pi$
$823$	127.000	−	219.970i	0.154313	−	0.267279i	0.932809	+	0.360372i	$0.117350\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.7279i			0.0153905i	0.999970	+	0.00769524i	$0.00244949\pi$
$827$			12.7279i			0.0153905i	−0.999970	+	0.00769524i	$0.997551\pi$
$828$		0			0
$829$	752.000	−	1302.50i	0.907117	−	1.57117i	0.0890670	−	0.996026i	$-0.471611\pi$
$829$	752.000	−	1302.50i	0.907117	−	1.57117i	0.818050	−	0.575147i	$-0.195055\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−736.000	−	1274.79i	−0.881437	−	1.52669i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	203.647i		−	0.242726i	−0.992608	−	0.121363i	$-0.961274\pi$
$839$		−	203.647i		−	0.242726i	0.992608	−	0.121363i	$-0.0387264\pi$
$840$		0			0
$841$	263.000			0.312723
$842$		0			0
$843$		0			0
$844$		0			0
$845$	426.211	−	246.073i	0.504392	−	0.291211i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−264.545	−	152.735i	−0.310864	−	0.179477i
$852$		0			0
$853$	−536.000			−0.628370			−0.314185	−	0.949362i	$-0.601731\pi$
$853$	−536.000			−0.628370			−0.314185	+	0.949362i	$0.601731\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1337.42	−	772.161i	−1.56059	−	0.901004i	−0.997198	−	0.0748113i	$-0.976165\pi$
$857$	−1337.42	−	772.161i	−1.56059	−	0.901004i	−0.563387	−	0.826193i	$-0.690502\pi$
$858$		0			0
$859$	−396.000	−	685.892i	−0.461001	−	0.798477i	0.538010	−	0.842938i	$-0.319176\pi$
$859$	−396.000	−	685.892i	−0.461001	−	0.798477i	−0.999011	+	0.0444610i	$0.985843\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−782.612	+	451.841i	−0.906850	+	0.523570i	−0.879416	−	0.476053i	$-0.842067\pi$
$863$	−782.612	+	451.841i	−0.906850	+	0.523570i	−0.0274340	+	0.999624i	$0.508734\pi$
$864$		0			0
$865$	272.000	−	471.118i	0.314451	−	0.544645i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	115.966i		−	0.133447i
$870$		0			0
$871$	208.000	−	360.267i	0.238806	−	0.413624i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	171.000	+	296.181i	0.194983	+	0.337720i	0.946895	−	0.321543i	$-0.104202\pi$
$877$	171.000	+	296.181i	0.194983	+	0.337720i	−0.751912	+	0.659263i	$0.770868\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1487.75i		−	1.68871i	−0.535785	−	0.844355i	$-0.679984\pi$
$881$		−	1487.75i		−	1.68871i	0.535785	−	0.844355i	$-0.320016\pi$
$882$		0			0
$883$	1096.00			1.24122			0.620612	−	0.784118i	$-0.286884\pi$
$883$	1096.00			1.24122			0.620612	+	0.784118i	$0.286884\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−969.998	+	560.029i	−1.09357	+	0.631374i	−0.934525	−	0.355897i	$-0.884175\pi$
$887$	−969.998	+	560.029i	−1.09357	+	0.631374i	−0.159046	+	0.987271i	$0.550842\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−78.3837	−	45.2548i	−0.0877757	−	0.0506773i
$894$		0			0
$895$	−120.000			−0.134078
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1165.96	−	673.166i	−1.29695	−	0.748794i
$900$		0			0
$901$	372.000	+	644.323i	0.412875	+	0.715120i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	627.069	−	362.039i	0.692894	−	0.400043i
$906$		0			0
$907$	−764.000	+	1323.29i	−0.842337	+	1.45897i	0.0455762	+	0.998961i	$0.485488\pi$
$907$	−764.000	+	1323.29i	−0.842337	+	1.45897i	−0.887914	+	0.460010i	$0.847846\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			552.958i			0.606979i	0.952835	+	0.303489i	$0.0981517\pi$
$911$			552.958i			0.606979i	−0.952835	+	0.303489i	$0.901848\pi$
$912$		0			0
$913$	−72.0000	+	124.708i	−0.0788609	+	0.136591i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	336.000	+	581.969i	0.365615	+	0.633263i	0.988875	−	0.148751i	$-0.0475254\pi$
$919$	336.000	+	581.969i	0.365615	+	0.633263i	−0.623260	+	0.782015i	$0.714192\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2149.60i			2.32893i
$924$		0			0
$925$	−168.000			−0.181622
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−896.513	+	517.602i	−0.965030	+	0.557161i	−0.897718	−	0.440571i	$-0.854776\pi$
$929$	−896.513	+	517.602i	−0.965030	+	0.557161i	−0.0673128	+	0.997732i	$0.521443\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	117.576	+	67.8823i	0.125749	+	0.0726013i
$936$		0			0
$937$	−960.000			−1.02455			−0.512273	−	0.858823i	$-0.671196\pi$
$937$	−960.000			−1.02455			−0.512273	+	0.858823i	$0.671196\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	837.725	+	483.661i	0.890250	+	0.513986i	0.874024	−	0.485882i	$-0.161502\pi$
$941$	837.725	+	483.661i	0.890250	+	0.513986i	0.0162259	+	0.999868i	$0.494835\pi$
$942$		0			0
$943$	−324.000	−	561.184i	−0.343584	−	0.595105i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1174.53	−	678.115i	1.24026	−	0.716067i	0.271116	−	0.962547i	$-0.412607\pi$
$947$	1174.53	−	678.115i	1.24026	−	0.716067i	0.969148	+	0.246480i	$0.0792738\pi$
$948$		0			0
$949$	704.000	−	1219.36i	0.741834	−	1.28489i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1107.33i			1.16194i	0.813925	+	0.580970i	$0.197327\pi$
$953$			1107.33i			1.16194i	−0.813925	+	0.580970i	$0.802673\pi$
$954$		0			0
$955$	324.000	−	561.184i	0.339267	−	0.587628i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1087.50	−	1883.61i	−1.13163	−	1.96005i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1176.63i		−	1.21930i
$966$		0			0
$967$	994.000			1.02792			0.513961	−	0.857814i	$-0.328178\pi$
$967$	994.000			1.02792			0.513961	+	0.857814i	$0.328178\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1538.28	+	888.126i	−1.58422	+	0.914651i	−0.589989	+	0.807411i	$0.700868\pi$
$971$	−1538.28	+	888.126i	−1.58422	+	0.914651i	−0.994233	+	0.107240i	$0.965799\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	941.829	+	543.765i	0.964001	+	0.556566i	0.897402	−	0.441214i	$-0.145452\pi$
$977$	941.829	+	543.765i	0.964001	+	0.556566i	0.0665988	+	0.997780i	$0.478785\pi$
$978$		0			0
$979$	88.0000			0.0898876
$980$		0			0
$981$		0			0
$982$		0			0
$983$	950.402	+	548.715i	0.966838	+	0.558204i	0.898271	−	0.439442i	$-0.144824\pi$
$983$	950.402	+	548.715i	0.966838	+	0.558204i	0.0685674	+	0.997646i	$0.478157\pi$
$984$		0			0
$985$	348.000	+	602.754i	0.353299	+	0.611933i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	440.908	−	254.558i	0.445812	−	0.257390i
$990$		0			0
$991$	824.000	−	1427.21i	0.831483	−	1.44017i	−0.0653785	−	0.997861i	$-0.520825\pi$
$991$	824.000	−	1427.21i	0.831483	−	1.44017i	0.896862	−	0.442311i	$-0.145841\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			995.606i			1.00061i
$996$		0			0
$997$	124.000	−	214.774i	0.124373	−	0.215421i	−0.797115	−	0.603828i	$-0.793641\pi$
$997$	124.000	−	214.774i	0.124373	−	0.215421i	0.921488	+	0.388407i	$0.126975\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.bk.c.557.2		4
3.2	odd	2	inner	1764.3.bk.c.557.1		4
7.2	even	3	inner	1764.3.bk.c.1745.1		4
7.3	odd	6		1764.3.c.b.197.1	✓	2
7.4	even	3		1764.3.c.c.197.2	yes	2
7.5	odd	6		1764.3.bk.b.1745.2		4
7.6	odd	2		1764.3.bk.b.557.1		4
21.2	odd	6	inner	1764.3.bk.c.1745.2		4
21.5	even	6		1764.3.bk.b.1745.1		4
21.11	odd	6		1764.3.c.c.197.1	yes	2
21.17	even	6		1764.3.c.b.197.2	yes	2
21.20	even	2		1764.3.bk.b.557.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.c.b.197.1	✓	2	7.3	odd	6
1764.3.c.b.197.2	yes	2	21.17	even	6
1764.3.c.c.197.1	yes	2	21.11	odd	6
1764.3.c.c.197.2	yes	2	7.4	even	3
1764.3.bk.b.557.1		4	7.6	odd	2
1764.3.bk.b.557.2		4	21.20	even	2
1764.3.bk.b.1745.1		4	21.5	even	6
1764.3.bk.b.1745.2		4	7.5	odd	6
1764.3.bk.c.557.1		4	3.2	odd	2	inner
1764.3.bk.c.557.2		4	1.1	even	1	trivial
1764.3.bk.c.1745.1		4	7.2	even	3	inner
1764.3.bk.c.1745.2		4	21.2	odd	6	inner

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.bk (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.89898 - 2.82843i) q^{5} +O(q^{10})\)	\(q+(4.89898 - 2.82843i) q^{5} +(1.22474 + 0.707107i) q^{11} +16.0000 q^{13} +(14.6969 + 8.48528i) q^{17} +(4.00000 + 6.92820i) q^{19} +(11.0227 - 6.36396i) q^{23} +(3.50000 - 6.06218i) q^{25} +24.0416i q^{29} +(-28.0000 + 48.4974i) q^{31} +(-12.0000 - 20.7846i) q^{37} -50.9117i q^{41} +40.0000 q^{43} +(-9.79796 + 5.65685i) q^{47} +(37.9671 + 21.9203i) q^{53} +8.00000 q^{55} +(9.79796 + 5.65685i) q^{59} +(20.0000 + 34.6410i) q^{61} +(78.3837 - 45.2548i) q^{65} +(13.0000 - 22.5167i) q^{67} +134.350i q^{71} +(44.0000 - 76.2102i) q^{73} +(-41.0000 - 71.0141i) q^{79} +101.823i q^{83} +96.0000 q^{85} +(53.8888 - 31.1127i) q^{89} +(39.1918 + 22.6274i) q^{95} -40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 64 q^{13} + 16 q^{19} + 14 q^{25} - 112 q^{31} - 48 q^{37} + 160 q^{43} + 32 q^{55} + 80 q^{61} + 52 q^{67} + 176 q^{73} - 164 q^{79} + 384 q^{85} - 160 q^{97}+O(q^{100}) \) 4 * q + 64 * q^13 + 16 * q^19 + 14 * q^25 - 112 * q^31 - 48 * q^37 + 160 * q^43 + 32 * q^55 + 80 * q^61 + 52 * q^67 + 176 * q^73 - 164 * q^79 + 384 * q^85 - 160 * q^97

Introduction

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Database

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