LMFDB - Embedded newform 804.2.s.a.209.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(5,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 5]))

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

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

chi := DirichletCharacter("804.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.s (of order $22$, degree $10$, 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:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label			209.1
Root			$0.723734 - 0.690079i$ of defining polynomial
Character	$\chi$	$=$	804.209
Dual form			804.2.s.a.377.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.30900 + 1.13425i) q^{3} +(2.17449 + 0.312644i) q^{7} +(0.426945 - 2.96946i) q^{9} +O(q^{10})$	$q+(-1.30900 + 1.13425i) q^{3} +(2.17449 + 0.312644i) q^{7} +(0.426945 - 2.96946i) q^{9} +(2.69763 - 1.23197i) q^{13} +(-0.372992 - 2.59422i) q^{19} +(-3.20102 + 2.05717i) q^{21} +(-2.07708 - 4.54816i) q^{25} +(2.80925 + 4.37128i) q^{27} +(9.91965 + 4.53015i) q^{31} +10.4563 q^{37} +(-2.13383 + 4.67243i) q^{39} +(3.26646 + 11.1245i) q^{43} +(-2.08579 - 0.612443i) q^{49} +(3.43074 + 2.97275i) q^{57} +(3.26736 + 5.08410i) q^{61} +(1.85677 - 6.32359i) q^{63} +(1.74779 + 7.99658i) q^{67} +(0.774960 - 0.498036i) q^{73} +(7.87764 + 3.59760i) q^{75} +(9.85470 - 4.50049i) q^{79} +(-8.63544 - 2.53559i) q^{81} +(6.25114 - 1.83550i) q^{91} +(-18.1231 + 5.32143i) q^{93} -19.6545i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^9	$ 20 q + 6 q^{9} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 20 q^{37} - 24 q^{39} + 10 q^{49} - 66 q^{57} - 132 q^{63} - 16 q^{67} + 90 q^{73} + 44 q^{79} - 18 q^{81} + 48 q^{91} - 36 q^{93}+O(q^{100}) $ 20 * q + 6 * q^9 + 16 * q^19 - 12 * q^21 + 10 * q^25 - 20 * q^37 - 24 * q^39 + 10 * q^49 - 66 * q^57 - 132 * q^63 - 16 * q^67 + 90 * q^73 + 44 * q^79 - 18 * q^81 + 48 * q^91 - 36 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{1}{22}\right)$	$1$

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
$3$	−1.30900	+	1.13425i	−0.755750	+	0.654861i
$3$	−1.30900	+	1.13425i	−0.755750	+	0.654861i
$4$		0			0
$5$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$5$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$6$		0			0
$7$	2.17449	+	0.312644i	0.821880	+	0.118169i	0.540410	−	0.841402i	$-0.318269\pi$
$7$	2.17449	+	0.312644i	0.821880	+	0.118169i	0.281470	+	0.959570i	$0.409178\pi$
$8$		0			0
$9$	0.426945	−	2.96946i	0.142315	−	0.989821i
$10$		0			0
$11$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$11$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$12$		0			0
$13$	2.69763	−	1.23197i	0.748188	−	0.341686i	−0.00456441	−	0.999990i	$-0.501453\pi$
$13$	2.69763	−	1.23197i	0.748188	−	0.341686i	0.752753	+	0.658303i	$0.228726\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$17$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$18$		0			0
$19$	−0.372992	−	2.59422i	−0.0855702	−	0.595154i	−0.986816	−	0.161846i	$-0.948255\pi$
$19$	−0.372992	−	2.59422i	−0.0855702	−	0.595154i	0.901246	−	0.433308i	$-0.142654\pi$
$20$		0			0
$21$	−3.20102	+	2.05717i	−0.698519	+	0.448911i
$22$		0			0
$23$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$23$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$24$		0			0
$25$	−2.07708	−	4.54816i	−0.415415	−	0.909632i
$26$		0			0
$27$	2.80925	+	4.37128i	0.540641	+	0.841254i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	9.91965	+	4.53015i	1.78162	+	0.813640i	0.974878	+	0.222738i	$0.0714995\pi$
$31$	9.91965	+	4.53015i	1.78162	+	0.813640i	0.806744	+	0.590901i	$0.201228\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10.4563			1.71900			0.859500	−	0.511136i	$-0.170775\pi$
$37$	10.4563			1.71900			0.859500	+	0.511136i	$0.170775\pi$
$38$		0			0
$39$	−2.13383	+	4.67243i	−0.341686	+	0.748188i
$40$		0			0
$41$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$41$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$42$		0			0
$43$	3.26646	+	11.1245i	0.498130	+	1.69647i	0.697529	+	0.716557i	$0.254283\pi$
$43$	3.26646	+	11.1245i	0.498130	+	1.69647i	−0.199399	+	0.979918i	$0.563899\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$47$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$48$		0			0
$49$	−2.08579	−	0.612443i	−0.297970	−	0.0874919i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$53$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.43074	+	2.97275i	0.454413	+	0.393751i
$58$		0			0
$59$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$59$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$60$		0			0
$61$	3.26736	+	5.08410i	0.418342	+	0.650953i	0.984911	−	0.173061i	$-0.0553659\pi$
$61$	3.26736	+	5.08410i	0.418342	+	0.650953i	−0.566569	+	0.824014i	$0.691730\pi$
$62$		0			0
$63$	1.85677	−	6.32359i	0.233931	−	0.796697i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.74779	+	7.99658i	0.213527	+	0.976937i
$67$	1.74779	+	7.99658i	0.213527	+	0.976937i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$71$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$72$		0			0
$73$	0.774960	−	0.498036i	0.0907022	−	0.0582907i	−0.494504	−	0.869176i	$-0.664650\pi$
$73$	0.774960	−	0.498036i	0.0907022	−	0.0582907i	0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	7.87764	+	3.59760i	0.909632	+	0.415415i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	9.85470	−	4.50049i	1.10874	−	0.506345i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	9.85470	−	4.50049i	1.10874	−	0.506345i	0.883723	+	0.468010i	$0.155029\pi$
$80$		0			0
$81$	−8.63544	−	2.53559i	−0.959493	−	0.281733i
$82$		0			0
$83$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$83$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$89$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$90$		0			0
$91$	6.25114	−	1.83550i	0.655298	−	0.192413i
$92$		0			0
$93$	−18.1231	+	5.32143i	−1.87928	+	0.551807i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	19.6545i		−	1.99561i	−0.0661967	−	0.997807i	$-0.521086\pi$
$97$		−	19.6545i		−	1.99561i	0.0661967	−	0.997807i	$-0.478914\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$101$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$102$		0			0
$103$	2.60679	−	5.70807i	0.256854	−	0.562433i	−0.736644	−	0.676281i	$-0.763591\pi$
$103$	2.60679	−	5.70807i	0.256854	−	0.562433i	0.993498	+	0.113848i	$0.0363178\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$107$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$108$		0			0
$109$	−14.9215	+	6.81440i	−1.42922	+	0.652701i	−0.971639	−	0.236468i	$-0.924010\pi$
$109$	−14.9215	+	6.81440i	−1.42922	+	0.652701i	−0.457578	+	0.889170i	$0.651283\pi$
$110$		0			0
$111$	−13.6872	+	11.8600i	−1.29913	+	1.12571i
$112$		0			0
$113$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$113$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.50654	−	8.53650i	−0.231730	−	0.789200i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.56957	−	10.0060i	−0.415415	−	0.909632i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.162932	+	1.13322i	−0.0144579	+	0.100557i	−0.995773	−	0.0918526i	$-0.970721\pi$
$127$	−0.162932	+	1.13322i	−0.0144579	+	0.100557i	0.981315	+	0.192410i	$0.0616302\pi$
$128$		0			0
$129$	−16.8938	−	10.8570i	−1.48742	−	0.955904i
$130$		0			0
$131$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$131$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$132$		0			0
$133$		−	5.75771i		−	0.499257i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$137$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$138$		0			0
$139$	0.405195	−	0.630496i	0.0343682	−	0.0534780i	−0.823644	−	0.567107i	$-0.808063\pi$
$139$	0.405195	−	0.630496i	0.0343682	−	0.0534780i	0.858013	+	0.513629i	$0.171699\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.42496	−	1.56413i	0.282486	−	0.129007i
$148$		0			0
$149$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$149$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$150$		0			0
$151$	−9.35009	+	2.74544i	−0.760900	+	0.223420i	−0.639089	−	0.769133i	$-0.720688\pi$
$151$	−9.35009	+	2.74544i	−0.760900	+	0.223420i	−0.121811	+	0.992553i	$0.538870\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.1677	−	14.0423i	−0.971091	−	1.12070i	−0.992661	−	0.120931i	$-0.961412\pi$
$157$	−12.1677	−	14.0423i	−0.971091	−	1.12070i	0.0215699	−	0.999767i	$-0.493134\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.844292			−0.0661301			−0.0330650	−	0.999453i	$-0.510527\pi$
$163$	−0.844292			−0.0661301			−0.0330650	+	0.999453i	$0.510527\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$167$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$168$		0			0
$169$	−2.75372	+	3.17796i	−0.211825	+	0.244459i
$170$		0			0
$171$	−7.86268			−0.601274
$172$		0			0
$173$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$173$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$174$		0			0
$175$	−3.09462	−	10.5393i	−0.233931	−	0.796697i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$179$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$180$		0			0
$181$	14.5787	+	16.8248i	1.08363	+	1.25058i	0.966282	+	0.257485i	$0.0828937\pi$
$181$	14.5787	+	16.8248i	1.08363	+	1.25058i	0.117348	+	0.993091i	$0.462561\pi$
$182$		0			0
$183$	−10.0436	−	2.94907i	−0.742445	−	0.218002i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.74203	+	10.3836i	0.344932	+	0.755296i
$190$		0			0
$191$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$191$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$192$		0			0
$193$	11.2769	−	24.6930i	0.811729	−	1.77744i	0.211936	−	0.977284i	$-0.432023\pi$
$193$	11.2769	−	24.6930i	0.811729	−	1.77744i	0.599793	−	0.800155i	$-0.295250\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$197$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$198$		0			0
$199$	3.96229	−	27.5584i	0.280880	−	1.95356i	−0.0197060	−	0.999806i	$-0.506273\pi$
$199$	3.96229	−	27.5584i	0.280880	−	1.95356i	0.300586	−	0.953755i	$-0.402818\pi$
$200$		0			0
$201$	−11.3580	−	8.48506i	−0.801131	−	0.598490i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−17.3996	+	20.0802i	−1.19784	+	1.38238i	−0.293279	+	0.956027i	$0.594746\pi$
$211$	−17.3996	+	20.0802i	−1.19784	+	1.38238i	−0.904558	+	0.426350i	$0.859799\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	20.1539	+	12.9521i	1.36813	+	0.879246i
$218$		0			0
$219$	−0.449521	+	1.53093i	−0.0303758	+	0.103451i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−6.86433	+	7.92186i	−0.459669	+	0.530487i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−6.86433	+	7.92186i	−0.459669	+	0.530487i	0.477840	+	0.878447i	$0.341420\pi$
$224$		0			0
$225$	−14.3924	+	4.22599i	−0.959493	+	0.281733i
$226$		0			0
$227$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$227$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$228$		0			0
$229$	−10.3495	−	4.72645i	−0.683913	−	0.312333i	0.0429870	−	0.999076i	$-0.486313\pi$
$229$	−10.3495	−	4.72645i	−0.683913	−	0.312333i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$233$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−7.79508	+	17.0688i	−0.506345	+	1.10874i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−16.3141	+	10.4845i	−1.05089	+	0.675363i	−0.947656	−	0.319294i	$-0.896554\pi$
$241$	−16.3141	+	10.4845i	−1.05089	+	0.675363i	−0.103230	+	0.994657i	$0.532918\pi$
$242$		0			0
$243$	14.1798	−	6.47568i	0.909632	−	0.415415i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.20218	−	6.53872i	−0.267378	−	0.416049i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$251$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$257$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$258$		0			0
$259$	22.7370	+	3.26909i	1.41281	+	0.203132i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$263$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−22.8085	+	19.7637i	−1.38552	+	1.20056i	−0.431019	+	0.902343i	$0.641846\pi$
$271$	−22.8085	+	19.7637i	−1.38552	+	1.20056i	−0.954499	+	0.298215i	$0.903609\pi$
$272$		0			0
$273$	−6.10080	+	9.49303i	−0.369237	+	0.574544i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.71695	+	18.8968i	−0.163246	+	1.13540i	0.729219	+	0.684281i	$0.239884\pi$
$277$	−2.71695	+	18.8968i	−0.163246	+	1.13540i	−0.892464	+	0.451118i	$0.851025\pi$
$278$		0			0
$279$	17.6873	−	27.5219i	1.05891	−	1.64769i
$280$		0			0
$281$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$281$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$282$		0			0
$283$	−3.03154	−	21.0848i	−0.180206	−	1.25336i	−0.856274	−	0.516522i	$-0.827227\pi$
$283$	−3.03154	−	21.0848i	−0.180206	−	1.25336i	0.676068	−	0.736839i	$-0.263682\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	14.3013	−	9.19089i	0.841254	−	0.540641i
$290$		0			0
$291$	22.2932	+	25.7277i	1.30685	+	1.50818i
$292$		0			0
$293$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$293$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	3.62486	+	25.2114i	0.208933	+	1.45316i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	2.72859	−	5.97477i	0.155729	−	0.340998i	−0.815646	−	0.578551i	$-0.803618\pi$
$307$	2.72859	−	5.97477i	0.155729	−	0.340998i	0.971374	+	0.237553i	$0.0763455\pi$
$308$		0			0
$309$	3.06211	+	10.4286i	0.174197	+	0.593262i
$310$		0			0
$311$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$311$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$312$		0			0
$313$	13.4766	+	11.6776i	0.761744	+	0.660055i	0.946491	−	0.322731i	$-0.104601\pi$
$313$	13.4766	+	11.6776i	0.761744	+	0.660055i	−0.184747	+	0.982786i	$0.559147\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$317$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−11.2064	−	9.71037i	−0.621617	−	0.538634i
$326$		0			0
$327$	11.8029	−	25.8447i	0.652701	−	1.42922i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−10.0090	+	34.0874i	−0.550143	+	1.87361i	−0.0676621	+	0.997708i	$0.521554\pi$
$331$	−10.0090	+	34.0874i	−0.550143	+	1.87361i	−0.482481	+	0.875907i	$0.660264\pi$
$332$		0			0
$333$	4.46425	−	31.0495i	0.244639	−	1.70150i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	24.9151	+	3.58225i	1.35721	+	0.195138i	0.782211	−	0.623014i	$-0.214092\pi$
$337$	24.9151	+	3.58225i	1.35721	+	0.195138i	0.575002	+	0.818152i	$0.305001\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.3323	−	8.37210i	−0.989853	−	0.452051i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$347$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$348$		0			0
$349$	30.9878	+	9.09884i	1.65874	+	0.487050i	0.971034	−	0.238941i	$-0.0768002\pi$
$349$	30.9878	+	9.09884i	1.65874	+	0.487050i	0.687705	+	0.725991i	$0.258618\pi$
$350$		0			0
$351$	12.9636	+	8.33120i	0.691946	+	0.444687i
$352$		0			0
$353$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$353$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$359$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	11.6395	−	3.41768i	0.612607	−	0.179878i
$362$		0			0
$363$	17.3308	+	7.91472i	0.909632	+	0.415415i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−22.5920	−	19.5761i	−1.17929	−	1.02186i	−0.999270	−	0.0382006i	$-0.987837\pi$
$367$	−22.5920	−	19.5761i	−1.17929	−	1.02186i	−0.180023	−	0.983662i	$-0.557617\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	38.5992i		−	1.99859i	−0.0375318	−	0.999295i	$-0.511950\pi$
$373$		−	38.5992i		−	1.99859i	0.0375318	−	0.999295i	$-0.488050\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−10.0828	+	8.73682i	−0.517920	+	0.448780i	−0.874177	−	0.485608i	$-0.838598\pi$
$379$	−10.0828	+	8.73682i	−0.517920	+	0.448780i	0.356257	+	0.934388i	$0.384053\pi$
$380$		0			0
$381$	−1.07208	−	1.66819i	−0.0549243	−	0.0854638i
$382$		0			0
$383$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$383$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	34.4285	−	4.95007i	1.75010	−	0.251626i
$388$		0			0
$389$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$389$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−32.3701	−	20.8030i	−1.62461	−	1.04407i	−0.952911	−	0.303251i	$-0.901928\pi$
$397$	−32.3701	−	20.8030i	−1.62461	−	1.04407i	−0.671700	−	0.740823i	$-0.734436\pi$
$398$		0			0
$399$	6.53069	+	7.53682i	0.326944	+	0.377313i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	32.3406			1.61100
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−8.79714	−	1.26484i	−0.434990	−	0.0625422i	−0.0786585	−	0.996902i	$-0.525064\pi$
$409$	−8.79714	−	1.26484i	−0.434990	−	0.0625422i	−0.356332	+	0.934359i	$0.615973\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.184742	+	1.28491i	0.00904687	+	0.0629224i
$418$		0			0
$419$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$419$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$420$		0			0
$421$	3.18379	+	22.1437i	0.155168	+	1.07922i	0.907384	+	0.420303i	$0.138076\pi$
$421$	3.18379	+	22.1437i	0.155168	+	1.07922i	−0.752216	+	0.658917i	$0.771015\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	5.51531	+	12.0769i	0.266905	+	0.584440i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−17.3628	−	7.92933i	−0.834404	−	0.381059i	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−17.3628	−	7.92933i	−0.834404	−	0.381059i	−0.786347	+	0.617785i	$0.788030\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−35.6494			−1.70145			−0.850725	−	0.525610i	$-0.823837\pi$
$439$	−35.6494			−1.70145			−0.850725	+	0.525610i	$0.823837\pi$
$440$		0			0
$441$	−2.70914	+	5.93220i	−0.129007	+	0.282486i
$442$		0			0
$443$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$443$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$449$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	9.12523	−	14.1991i	0.428741	−	0.667133i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	17.6839	+	38.7224i	0.827218	+	1.81136i	0.498334	+	0.866985i	$0.333945\pi$
$457$	17.6839	+	38.7224i	0.827218	+	1.81136i	0.328884	+	0.944370i	$0.393327\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$461$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$462$		0			0
$463$	13.6346	+	21.2159i	0.633654	+	0.985985i	0.998491	+	0.0549137i	$0.0174884\pi$
$463$	13.6346	+	21.2159i	0.633654	+	0.985985i	−0.364837	+	0.931071i	$0.618875\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$467$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$468$		0			0
$469$	1.30047	+	17.9349i	0.0600501	+	0.828157i
$470$		0			0
$471$	31.8550	+	4.58006i	1.46780	+	0.211038i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−11.0242	+	7.08481i	−0.505824	+	0.325073i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$479$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$480$		0			0
$481$	28.2071	−	12.8818i	1.28614	−	0.587358i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−2.55600	+	8.70495i	−0.115824	+	0.394459i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−2.55600	+	8.70495i	−0.115824	+	0.394459i	0.881092	+	0.472946i	$0.156809\pi$
$488$		0			0
$489$	1.10518	−	0.957640i	0.0499778	−	0.0433060i
$490$		0			0
$491$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$491$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	16.1569i		−	0.723281i	−0.932318	−	0.361640i	$-0.882217\pi$
$499$		−	16.1569i		−	0.723281i	0.932318	−	0.361640i	$-0.117783\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$503$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	7.28335i		−	0.323465i
$508$		0			0
$509$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$509$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$510$		0			0
$511$	1.84085	−	0.840688i	0.0814344	−	0.0371899i
$512$		0			0
$513$	10.2922	−	8.91826i	0.454413	−	0.393751i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$521$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$522$		0			0
$523$	−18.0441	−	39.5112i	−0.789015	−	1.72770i	−0.679449	−	0.733723i	$-0.737781\pi$
$523$	−18.0441	−	39.5112i	−0.789015	−	1.72770i	−0.109567	−	0.993979i	$-0.534946\pi$
$524$		0			0
$525$	16.0051	+	10.2858i	0.698519	+	0.448911i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.27324	+	22.7659i	−0.142315	+	0.989821i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	10.6005	−	16.4947i	0.455750	−	0.709162i	−0.535001	−	0.844851i	$-0.679689\pi$
$541$	10.6005	−	16.4947i	0.455750	−	0.709162i	0.990751	+	0.135690i	$0.0433251\pi$
$542$		0			0
$543$	−38.1671	−	5.48760i	−1.63791	−	0.235495i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.48616	−	10.0927i	0.277328	−	0.431531i	−0.674449	−	0.738321i	$-0.735619\pi$
$547$	6.48616	−	10.0927i	0.277328	−	0.431531i	0.951777	+	0.306791i	$0.0992551\pi$
$548$		0			0
$549$	16.4920	−	7.53166i	0.703863	−	0.321444i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	22.8360	−	6.70526i	0.971086	−	0.285137i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$557$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$558$		0			0
$559$	22.5167	+	25.9857i	0.952357	+	1.09908i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$563$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−17.9849	−	8.21344i	−0.755296	−	0.344932i
$568$		0			0
$569$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$569$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$570$		0			0
$571$	−29.1490	+	33.6397i	−1.21985	+	1.40778i	−0.334790	+	0.942293i	$0.608665\pi$
$571$	−29.1490	+	33.6397i	−1.21985	+	1.40778i	−0.885056	+	0.465484i	$0.845880\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.72246	+	12.6775i	0.154968	+	0.527772i	0.999976	−	0.00698200i	$-0.00222246\pi$
$577$	3.72246	+	12.6775i	0.154968	+	0.527772i	−0.845008	+	0.534754i	$0.820404\pi$
$578$		0			0
$579$	13.2466	+	45.1139i	0.550511	+	1.87487i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$587$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$588$		0			0
$589$	8.05224	−	27.4234i	0.331787	−	1.12996i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$593$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	26.0715	+	40.5680i	1.06704	+	1.66034i
$598$		0			0
$599$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$599$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$600$		0			0
$601$	−0.0855917	+	0.595303i	−0.00349136	+	0.0242829i	−0.991493	−	0.130163i	$-0.958450\pi$
$601$	−0.0855917	+	0.595303i	−0.00349136	+	0.0242829i	0.988001	+	0.154446i	$0.0493591\pi$
$602$		0			0
$603$	24.4918	−	1.77591i	0.997381	−	0.0723206i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−40.0383	−	11.7563i	−1.62511	−	0.477174i	−0.662722	−	0.748866i	$-0.730599\pi$
$607$	−40.0383	−	11.7563i	−1.62511	−	0.477174i	−0.962384	+	0.271691i	$0.912417\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.66422	−	3.07467i	0.107607	−	0.124185i	−0.699391	−	0.714739i	$-0.746545\pi$
$613$	2.66422	−	3.07467i	0.107607	−	0.124185i	0.806998	+	0.590554i	$0.201091\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$617$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$618$		0			0
$619$	39.9775	+	25.6920i	1.60683	+	1.03265i	0.963735	+	0.266860i	$0.0859860\pi$
$619$	39.9775	+	25.6920i	1.60683	+	1.03265i	0.643094	+	0.765787i	$0.277650\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−16.3715	+	18.8937i	−0.654861	+	0.755750i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−15.8034	−	7.21719i	−0.629125	−	0.287312i	0.0752227	−	0.997167i	$-0.476033\pi$
$631$	−15.8034	−	7.21719i	−0.629125	−	0.287312i	−0.704348	+	0.709855i	$0.748761\pi$
$632$		0			0
$633$		−	46.0204i		−	1.82915i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−6.38120	+	0.917478i	−0.252832	+	0.0363518i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	−39.6763	+	25.4984i	−1.56468	+	1.00556i	−0.583579	+	0.812056i	$0.698348\pi$
$643$	−39.6763	+	25.4984i	−1.56468	+	1.00556i	−0.981100	+	0.193502i	$0.938015\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$647$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−41.0723	+	5.90530i	−1.60975	+	0.231447i
$652$		0			0
$653$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$653$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.14804	−	2.51385i	−0.0447892	−	0.0980746i
$658$		0			0
$659$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$659$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$660$		0			0
$661$	−22.3026	−	3.20663i	−0.867472	−	0.124724i	−0.305812	−	0.952092i	$-0.598928\pi$
$661$	−22.3026	−	3.20663i	−0.867472	−	0.124724i	−0.561660	+	0.827368i	$0.689837\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	18.1556i		−	0.701935i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−29.2390	+	25.3358i	−1.12708	+	0.976622i	−0.999884	−	0.0152551i	$-0.995144\pi$
$673$	−29.2390	+	25.3358i	−1.12708	+	0.976622i	−0.127198	+	0.991877i	$0.540599\pi$
$674$		0			0
$675$	14.0463	−	21.8564i	0.540641	−	0.841254i
$676$		0			0
$677$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$677$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$678$		0			0
$679$	6.14487	−	42.7385i	0.235819	−	1.64015i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$683$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	18.9084	−	5.55202i	0.721402	−	0.211823i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−42.5583	+	27.3506i	−1.61900	+	1.04047i	−0.662356	+	0.749189i	$0.730443\pi$
$691$	−42.5583	+	27.3506i	−1.61900	+	1.04047i	−0.956639	+	0.291276i	$0.905920\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$701$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$702$		0			0
$703$	−3.90010	−	27.1258i	−0.147095	−	1.02307i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−21.2106	+	46.4448i	−0.796581	+	1.74427i	−0.139812	+	0.990178i	$0.544650\pi$
$709$	−21.2106	+	46.4448i	−0.796581	+	1.74427i	−0.656770	+	0.754091i	$0.728078\pi$
$710$		0			0
$711$	−9.15664	−	31.1847i	−0.343401	−	1.16952i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$719$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$720$		0			0
$721$	7.45303	−	11.5971i	0.277565	−	0.431900i
$722$		0			0
$723$	9.46313	−	32.2285i	0.351938	−	1.19859i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−13.2391	−	11.4717i	−0.491010	−	0.425462i	0.373840	−	0.927493i	$-0.378041\pi$
$727$	−13.2391	−	11.4717i	−0.491010	−	0.425462i	−0.864850	+	0.502031i	$0.832586\pi$
$728$		0			0
$729$	−11.2162	+	24.5601i	−0.415415	+	0.909632i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	13.1099	−	44.6482i	0.484225	−	1.64912i	−0.248529	−	0.968625i	$-0.579947\pi$
$733$	13.1099	−	44.6482i	0.484225	−	1.64912i	0.732754	−	0.680494i	$-0.238235\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−53.8111	−	7.73687i	−1.97947	−	0.284605i	−0.991602	−	0.129329i	$-0.958718\pi$
$739$	−53.8111	−	7.73687i	−1.97947	−	0.284605i	−0.987871	−	0.155277i	$-0.950373\pi$
$740$		0			0
$741$	12.9172	+	3.79283i	0.474525	+	0.139333i
$742$		0			0
$743$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$743$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−52.5547	−	15.4314i	−1.91775	−	0.563101i	−0.968994	−	0.247084i	$-0.920528\pi$
$751$	−52.5547	−	15.4314i	−1.91775	−	0.563101i	−0.948753	−	0.316017i	$-0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−20.2102	+	17.5123i	−0.734553	+	0.636494i	−0.939606	−	0.342257i	$-0.888809\pi$
$757$	−20.2102	+	17.5123i	−0.734553	+	0.636494i	0.205053	+	0.978751i	$0.434263\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$761$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$762$		0			0
$763$	−34.5771	+	10.1527i	−1.25177	+	0.367554i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−16.0260	−	13.8866i	−0.577911	−	0.500762i	0.316150	−	0.948709i	$-0.397610\pi$
$769$	−16.0260	−	13.8866i	−0.577911	−	0.500762i	−0.894060	+	0.447947i	$0.852155\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$773$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$774$		0			0
$775$		−	54.5256i		−	1.95862i
$776$		0			0
$777$	−33.4707	+	21.5103i	−1.20075	+	0.771678i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−14.7567	−	50.2567i	−0.526020	−	1.79146i	−0.606951	−	0.794739i	$-0.707607\pi$
$787$	−14.7567	−	50.2567i	−0.526020	−	1.79146i	0.0809308	−	0.996720i	$-0.474211\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	15.0776	+	9.68976i	0.535420	+	0.344094i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$797$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$809$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$810$		0			0
$811$	54.6941	+	7.86382i	1.92057	+	0.276136i	0.994791	−	0.101932i	$-0.0325026\pi$
$811$	54.6941	+	7.86382i	1.92057	+	0.276136i	0.925778	+	0.378069i	$0.123412\pi$
$812$		0			0
$813$	7.43926	−	51.7412i	0.260906	−	1.81464i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	27.6411	−	12.6233i	0.967039	−	0.441632i
$818$		0			0
$819$	−2.78156	−	19.3462i	−0.0971957	−	0.676011i
$820$		0			0
$821$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$821$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$822$		0			0
$823$	−2.36264	−	16.4325i	−0.0823566	−	0.572802i	−0.988660	−	0.150174i	$-0.952017\pi$
$823$	−2.36264	−	16.4325i	−0.0823566	−	0.572802i	0.906303	−	0.422628i	$-0.138892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$827$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$828$		0			0
$829$	−22.3893	−	49.0257i	−0.777612	−	1.70273i	−0.709127	−	0.705080i	$-0.750911\pi$
$829$	−22.3893	−	49.0257i	−0.777612	−	1.70273i	−0.0684845	−	0.997652i	$-0.521816\pi$
$830$		0			0
$831$	−17.8773	−	27.8175i	−0.620155	−	0.964980i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	8.06423	+	56.0879i	0.278740	+	1.93868i
$838$		0			0
$839$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$839$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−6.80817	−	23.1865i	−0.233931	−	0.796697i
$848$		0			0
$849$	27.8837	+	24.1614i	0.956967	+	0.829217i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	17.0713	+	5.01257i	0.584509	+	0.171627i	0.560600	−	0.828087i	$-0.310571\pi$
$853$	17.0713	+	5.01257i	0.584509	+	0.171627i	0.0239089	+	0.999714i	$0.492389\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$857$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$858$		0			0
$859$	16.2089	+	35.4925i	0.553040	+	1.21099i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	16.2089	+	35.4925i	0.553040	+	1.21099i	−0.402308	+	0.915505i	$0.631792\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$863$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−8.29558	+	28.2521i	−0.281733	+	0.959493i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	14.5664	+	19.4186i	0.493564	+	0.657974i
$872$		0			0
$873$	−58.3634	−	8.39139i	−1.97530	−	0.284005i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−44.4539	+	28.5688i	−1.50110	+	0.964700i	−0.506357	+	0.862324i	$0.669008\pi$
$877$	−44.4539	+	28.5688i	−1.50110	+	0.964700i	−0.994746	+	0.102376i	$0.967355\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$881$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$882$		0			0
$883$	40.5792	−	18.5319i	1.36560	−	0.623648i	0.408326	−	0.912836i	$-0.366113\pi$
$883$	40.5792	−	18.5319i	1.36560	−	0.623648i	0.957272	+	0.289188i	$0.0933852\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$887$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$888$		0			0
$889$	−0.708590	+	2.41323i	−0.0237653	+	0.0809373i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−33.3410	−	28.8902i	−1.10952	−	0.961405i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−24.0928	+	52.7559i	−0.799989	+	1.75173i	−0.154455	+	0.988000i	$0.549362\pi$
$907$	−24.0928	+	52.7559i	−0.799989	+	1.75173i	−0.645534	+	0.763732i	$0.723365\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$911$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.86644	−	0.268354i	0.0615682	−	0.00885217i	−0.111462	−	0.993769i	$-0.535553\pi$
$919$	1.86644	−	0.268354i	0.0615682	−	0.00885217i	0.173030	+	0.984917i	$0.444644\pi$
$920$		0			0
$921$	3.20518	+	10.9159i	0.105614	+	0.359690i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−21.7184	−	47.5568i	−0.714098	−	1.56366i
$926$		0			0
$927$	−15.8369	−	10.1778i	−0.520154	−	0.334282i
$928$		0			0
$929$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$929$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$930$		0			0
$931$	−0.810827	+	5.63943i	−0.0265738	+	0.184825i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			60.5055i			1.97663i	0.152425	+	0.988315i	$0.451292\pi$
$937$			60.5055i			1.97663i	−0.152425	+	0.988315i	$0.548708\pi$
$938$		0			0
$939$	−30.8862			−1.00793
$940$		0			0
$941$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$941$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$947$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$948$		0			0
$949$	1.47699	−	2.29824i	0.0479452	−	0.0746041i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$953$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	57.5765	+	66.4468i	1.85731	+	2.14345i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	57.6031			1.85239			0.926195	−	0.377044i	$-0.123059\pi$
$967$	57.6031			1.85239			0.926195	+	0.377044i	$0.123059\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$971$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$972$		0			0
$973$	1.07821	−	1.24433i	0.0345660	−	0.0398913i
$974$		0			0
$975$	25.6831			0.822517
$976$		0			0
$977$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$977$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	13.8645	+	47.2181i	0.442659	+	1.50756i
$982$		0			0
$983$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$983$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	17.3752	−	59.1745i	0.551941	−	1.87974i	0.0829100	−	0.996557i	$-0.473579\pi$
$991$	17.3752	−	59.1745i	0.551941	−	1.87974i	0.469031	−	0.883182i	$-0.344603\pi$
$992$		0			0
$993$	−25.5620	−	55.9730i	−0.811186	−	1.77625i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	16.8556	−	36.9086i	0.533821	−	1.16891i	−0.430115	−	0.902774i	$-0.641527\pi$
$997$	16.8556	−	36.9086i	0.533821	−	1.16891i	0.963937	−	0.266132i	$-0.0857457\pi$
$998$		0			0
$999$	29.3743	+	45.7073i	0.929361	+	1.44611i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.209.1	✓	20
3.2	odd	2	CM	804.2.s.a.209.1	✓	20
67.42	odd	22	inner	804.2.s.a.377.1	yes	20
201.176	even	22	inner	804.2.s.a.377.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.209.1	✓	20	1.1	even	1	trivial
804.2.s.a.209.1	✓	20	3.2	odd	2	CM
804.2.s.a.377.1	yes	20	67.42	odd	22	inner
804.2.s.a.377.1	yes	20	201.176	even	22	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:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.s (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.30900 + 1.13425i) q^{3} +(2.17449 + 0.312644i) q^{7} +(0.426945 - 2.96946i) q^{9} +O(q^{10})\)	\(q+(-1.30900 + 1.13425i) q^{3} +(2.17449 + 0.312644i) q^{7} +(0.426945 - 2.96946i) q^{9} +(2.69763 - 1.23197i) q^{13} +(-0.372992 - 2.59422i) q^{19} +(-3.20102 + 2.05717i) q^{21} +(-2.07708 - 4.54816i) q^{25} +(2.80925 + 4.37128i) q^{27} +(9.91965 + 4.53015i) q^{31} +10.4563 q^{37} +(-2.13383 + 4.67243i) q^{39} +(3.26646 + 11.1245i) q^{43} +(-2.08579 - 0.612443i) q^{49} +(3.43074 + 2.97275i) q^{57} +(3.26736 + 5.08410i) q^{61} +(1.85677 - 6.32359i) q^{63} +(1.74779 + 7.99658i) q^{67} +(0.774960 - 0.498036i) q^{73} +(7.87764 + 3.59760i) q^{75} +(9.85470 - 4.50049i) q^{79} +(-8.63544 - 2.53559i) q^{81} +(6.25114 - 1.83550i) q^{91} +(-18.1231 + 5.32143i) q^{93} -19.6545i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 20 * q + 6 * q^9	\( 20 q + 6 q^{9} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 20 q^{37} - 24 q^{39} + 10 q^{49} - 66 q^{57} - 132 q^{63} - 16 q^{67} + 90 q^{73} + 44 q^{79} - 18 q^{81} + 48 q^{91} - 36 q^{93}+O(q^{100}) \) 20 * q + 6 * q^9 + 16 * q^19 - 12 * q^21 + 10 * q^25 - 20 * q^37 - 24 * q^39 + 10 * q^49 - 66 * q^57 - 132 * q^63 - 16 * q^67 + 90 * q^73 + 44 * q^79 - 18 * q^81 + 48 * q^91 - 36 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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