LMFDB - Embedded newform 4056.2.c.q.337.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 39x^{10} + 601x^{8} + 4599x^{6} + 17849x^{4} + 31203x^{2} + 16129 $ x^12 + 39x^10 + 601x^8 + 4599x^6 + 17849x^4 + 31203*x^2 + 16129
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.12
Root			$0.920510i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.q.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.90568i q^{5} +0.0794899i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.90568i q^{5} +0.0794899i q^{7} +1.00000 q^{9} -1.50475i q^{11} -3.90568i q^{15} +6.52695 q^{17} -0.786340i q^{19} -0.0794899i q^{21} +7.03409 q^{23} -10.2543 q^{25} -1.00000 q^{27} +6.15266 q^{29} +1.43887i q^{31} +1.50475i q^{33} -0.310462 q^{35} -9.23571i q^{37} +7.75379i q^{41} -1.06562 q^{43} +3.90568i q^{45} -12.7564i q^{47} +6.99368 q^{49} -6.52695 q^{51} +2.73092 q^{53} +5.87709 q^{55} +0.786340i q^{57} -10.3775i q^{59} +7.92745 q^{61} +0.0794899i q^{63} -7.80912i q^{67} -7.03409 q^{69} -15.1722i q^{71} -4.16507i q^{73} +10.2543 q^{75} +0.119613 q^{77} +3.83731 q^{79} +1.00000 q^{81} +9.04179i q^{83} +25.4922i q^{85} -6.15266 q^{87} +4.75082i q^{89} -1.43887i q^{93} +3.07119 q^{95} +6.94297i q^{97} -1.50475i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 + 12 * q^9	$ 12 q - 12 q^{3} + 12 q^{9} + 10 q^{17} + 12 q^{23} - 34 q^{25} - 12 q^{27} + 6 q^{29} + 8 q^{35} - 22 q^{43} - 10 q^{49} - 10 q^{51} + 24 q^{53} + 14 q^{55} + 34 q^{61} - 12 q^{69} + 34 q^{75} + 50 q^{77} + 28 q^{79} + 12 q^{81} - 6 q^{87} - 86 q^{95}+O(q^{100}) $ 12 * q - 12 * q^3 + 12 * q^9 + 10 * q^17 + 12 * q^23 - 34 * q^25 - 12 * q^27 + 6 * q^29 + 8 * q^35 - 22 * q^43 - 10 * q^49 - 10 * q^51 + 24 * q^53 + 14 * q^55 + 34 * q^61 - 12 * q^69 + 34 * q^75 + 50 * q^77 + 28 * q^79 + 12 * q^81 - 6 * q^87 - 86 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.90568i	1.74667i	0.487117	+	0.873337i	$0.338049\pi$
$5$	3.90568i	1.74667i	−0.487117	+	0.873337i	$0.661951\pi$
$6$	0	0
$7$	0.0794899i	0.0300444i	0.999887	+	0.0150222i	$0.00478189\pi$
$7$	0.0794899i	0.0300444i	−0.999887	+	0.0150222i	$0.995218\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 1.50475i	− 0.453700i	−0.973930	−	0.226850i	$-0.927157\pi$
$11$	− 1.50475i	− 0.453700i	0.973930	−	0.226850i	$-0.0728427\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 3.90568i	− 1.00844i
$16$	0	0
$17$	6.52695	1.58302	0.791509	−	0.611158i	$-0.209296\pi$
$17$	6.52695	1.58302	0.791509	+	0.611158i	$0.209296\pi$
$18$	0	0
$19$	− 0.786340i	− 0.180399i	−0.995924	−	0.0901994i	$-0.971250\pi$
$19$	− 0.786340i	− 0.180399i	0.995924	−	0.0901994i	$-0.0287504\pi$
$20$	0	0
$21$	− 0.0794899i	− 0.0173461i
$22$	0	0
$23$	7.03409	1.46671	0.733354	−	0.679846i	$-0.237954\pi$
$23$	7.03409	1.46671	0.733354	+	0.679846i	$0.237954\pi$
$24$	0	0
$25$	−10.2543	−2.05087
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.15266	1.14252	0.571260	−	0.820769i	$-0.306455\pi$
$29$	6.15266	1.14252	0.571260	+	0.820769i	$0.306455\pi$
$30$	0	0
$31$	1.43887i	0.258429i	0.991617	+	0.129214i	$0.0412455\pi$
$31$	1.43887i	0.258429i	−0.991617	+	0.129214i	$0.958754\pi$
$32$	0	0
$33$	1.50475i	0.261944i
$34$	0	0
$35$	−0.310462	−0.0524777
$36$	0	0
$37$	− 9.23571i	− 1.51834i	−0.650891	−	0.759171i	$-0.725605\pi$
$37$	− 9.23571i	− 1.51834i	0.650891	−	0.759171i	$-0.274395\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.75379i	1.21094i	0.795869	+	0.605469i	$0.207015\pi$
$41$	7.75379i	1.21094i	−0.795869	+	0.605469i	$0.792985\pi$
$42$	0	0
$43$	−1.06562	−0.162506	−0.0812528	−	0.996694i	$-0.525892\pi$
$43$	−1.06562	−0.162506	−0.0812528	+	0.996694i	$0.525892\pi$
$44$	0	0
$45$	3.90568i	0.582225i
$46$	0	0
$47$	− 12.7564i	− 1.86071i	−0.366659	−	0.930355i	$-0.619498\pi$
$47$	− 12.7564i	− 1.86071i	0.366659	−	0.930355i	$-0.380502\pi$
$48$	0	0
$49$	6.99368	0.999097
$50$	0	0
$51$	−6.52695	−0.913956
$52$	0	0
$53$	2.73092	0.375121	0.187560	−	0.982253i	$-0.439942\pi$
$53$	2.73092	0.375121	0.187560	+	0.982253i	$0.439942\pi$
$54$	0	0
$55$	5.87709	0.792466
$56$	0	0
$57$	0.786340i	0.104153i
$58$	0	0
$59$	− 10.3775i	− 1.35104i	−0.737341	−	0.675520i	$-0.763919\pi$
$59$	− 10.3775i	− 1.35104i	0.737341	−	0.675520i	$-0.236081\pi$
$60$	0	0
$61$	7.92745	1.01501	0.507503	−	0.861650i	$-0.330569\pi$
$61$	7.92745	1.01501	0.507503	+	0.861650i	$0.330569\pi$
$62$	0	0
$63$	0.0794899i	0.0100148i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.80912i	− 0.954036i	−0.878893	−	0.477018i	$-0.841718\pi$
$67$	− 7.80912i	− 0.954036i	0.878893	−	0.477018i	$-0.158282\pi$
$68$	0	0
$69$	−7.03409	−0.846805
$70$	0	0
$71$	− 15.1722i	− 1.80061i	−0.435257	−	0.900306i	$-0.643343\pi$
$71$	− 15.1722i	− 1.80061i	0.435257	−	0.900306i	$-0.356657\pi$
$72$	0	0
$73$	− 4.16507i	− 0.487485i	−0.969840	−	0.243742i	$-0.921625\pi$
$73$	− 4.16507i	− 0.487485i	0.969840	−	0.243742i	$-0.0783751\pi$
$74$	0	0
$75$	10.2543	1.18407
$76$	0	0
$77$	0.119613	0.0136311
$78$	0	0
$79$	3.83731	0.431732	0.215866	−	0.976423i	$-0.430743\pi$
$79$	3.83731	0.431732	0.215866	+	0.976423i	$0.430743\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.04179i	0.992465i	0.868190	+	0.496232i	$0.165284\pi$
$83$	9.04179i	0.992465i	−0.868190	+	0.496232i	$0.834716\pi$
$84$	0	0
$85$	25.4922i	2.76502i
$86$	0	0
$87$	−6.15266	−0.659635
$88$	0	0
$89$	4.75082i	0.503586i	0.967781	+	0.251793i	$0.0810202\pi$
$89$	4.75082i	0.503586i	−0.967781	+	0.251793i	$0.918980\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 1.43887i	− 0.149204i
$94$	0	0
$95$	3.07119	0.315098
$96$	0	0
$97$	6.94297i	0.704952i	0.935821	+	0.352476i	$0.114660\pi$
$97$	6.94297i	0.704952i	−0.935821	+	0.352476i	$0.885340\pi$
$98$	0	0
$99$	− 1.50475i	− 0.151233i
$100$	0	0
$101$	−16.9785	−1.68943	−0.844713	−	0.535219i	$-0.820229\pi$
$101$	−16.9785	−1.68943	−0.844713	+	0.535219i	$0.820229\pi$
$102$	0	0
$103$	0.534038	0.0526203	0.0263102	−	0.999654i	$-0.491624\pi$
$103$	0.534038	0.0526203	0.0263102	+	0.999654i	$0.491624\pi$
$104$	0	0
$105$	0.310462	0.0302980
$106$	0	0
$107$	10.9729	1.06079	0.530396	−	0.847750i	$-0.322043\pi$
$107$	10.9729	1.06079	0.530396	+	0.847750i	$0.322043\pi$
$108$	0	0
$109$	8.24364i	0.789597i	0.918768	+	0.394799i	$0.129186\pi$
$109$	8.24364i	0.789597i	−0.918768	+	0.394799i	$0.870814\pi$
$110$	0	0
$111$	9.23571i	0.876615i
$112$	0	0
$113$	1.35771	0.127723	0.0638613	−	0.997959i	$-0.479658\pi$
$113$	1.35771	0.127723	0.0638613	+	0.997959i	$0.479658\pi$
$114$	0	0
$115$	27.4729i	2.56186i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.518827i	0.0475607i
$120$	0	0
$121$	8.73572	0.794156
$122$	0	0
$123$	− 7.75379i	− 0.699136i
$124$	0	0
$125$	− 20.5218i	− 1.83553i
$126$	0	0
$127$	11.0111	0.977081	0.488541	−	0.872541i	$-0.337529\pi$
$127$	11.0111	0.977081	0.488541	+	0.872541i	$0.337529\pi$
$128$	0	0
$129$	1.06562	0.0938226
$130$	0	0
$131$	−19.8552	−1.73476	−0.867378	−	0.497650i	$-0.834196\pi$
$131$	−19.8552	−1.73476	−0.867378	+	0.497650i	$0.834196\pi$
$132$	0	0
$133$	0.0625061	0.00541996
$134$	0	0
$135$	− 3.90568i	− 0.336148i
$136$	0	0
$137$	− 4.80881i	− 0.410844i	−0.978673	−	0.205422i	$-0.934143\pi$
$137$	− 4.80881i	− 0.410844i	0.978673	−	0.205422i	$-0.0658567\pi$
$138$	0	0
$139$	20.7637	1.76115	0.880575	−	0.473906i	$-0.157156\pi$
$139$	20.7637	1.76115	0.880575	+	0.473906i	$0.157156\pi$
$140$	0	0
$141$	12.7564i	1.07428i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	24.0303i	1.99561i
$146$	0	0
$147$	−6.99368	−0.576829
$148$	0	0
$149$	4.10785i	0.336528i	0.985742	+	0.168264i	$0.0538162\pi$
$149$	4.10785i	0.336528i	−0.985742	+	0.168264i	$0.946184\pi$
$150$	0	0
$151$	12.6178i	1.02683i	0.858142	+	0.513413i	$0.171619\pi$
$151$	12.6178i	1.02683i	−0.858142	+	0.513413i	$0.828381\pi$
$152$	0	0
$153$	6.52695	0.527673
$154$	0	0
$155$	−5.61977	−0.451390
$156$	0	0
$157$	−10.1958	−0.813711	−0.406855	−	0.913493i	$-0.633375\pi$
$157$	−10.1958	−0.813711	−0.406855	+	0.913493i	$0.633375\pi$
$158$	0	0
$159$	−2.73092	−0.216576
$160$	0	0
$161$	0.559139i	0.0440663i
$162$	0	0
$163$	15.8095i	1.23829i	0.785275	+	0.619147i	$0.212522\pi$
$163$	15.8095i	1.23829i	−0.785275	+	0.619147i	$0.787478\pi$
$164$	0	0
$165$	−5.87709	−0.457531
$166$	0	0
$167$	24.8487i	1.92285i	0.275066	+	0.961425i	$0.411300\pi$
$167$	24.8487i	1.92285i	−0.275066	+	0.961425i	$0.588700\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 0.786340i	− 0.0601329i
$172$	0	0
$173$	−12.6367	−0.960752	−0.480376	−	0.877063i	$-0.659500\pi$
$173$	−12.6367	−0.960752	−0.480376	+	0.877063i	$0.659500\pi$
$174$	0	0
$175$	− 0.815117i	− 0.0616170i
$176$	0	0
$177$	10.3775i	0.780024i
$178$	0	0
$179$	−2.34535	−0.175299	−0.0876497	−	0.996151i	$-0.527936\pi$
$179$	−2.34535	−0.175299	−0.0876497	+	0.996151i	$0.527936\pi$
$180$	0	0
$181$	−10.8755	−0.808370	−0.404185	−	0.914677i	$-0.632445\pi$
$181$	−10.8755	−0.808370	−0.404185	+	0.914677i	$0.632445\pi$
$182$	0	0
$183$	−7.92745	−0.586014
$184$	0	0
$185$	36.0717	2.65205
$186$	0	0
$187$	− 9.82145i	− 0.718216i
$188$	0	0
$189$	− 0.0794899i	− 0.00578204i
$190$	0	0
$191$	25.0561	1.81300	0.906499	−	0.422207i	$-0.138745\pi$
$191$	25.0561	1.81300	0.906499	+	0.422207i	$0.138745\pi$
$192$	0	0
$193$	− 2.03469i	− 0.146460i	−0.997315	−	0.0732300i	$-0.976669\pi$
$193$	− 2.03469i	− 0.146460i	0.997315	−	0.0732300i	$-0.0233307\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.93816i	− 0.209336i	−0.994507	−	0.104668i	$-0.966622\pi$
$197$	− 2.93816i	− 0.209336i	0.994507	−	0.104668i	$-0.0333779\pi$
$198$	0	0
$199$	−21.3170	−1.51112	−0.755561	−	0.655078i	$-0.772636\pi$
$199$	−21.3170	−1.51112	−0.755561	+	0.655078i	$0.772636\pi$
$200$	0	0
$201$	7.80912i	0.550813i
$202$	0	0
$203$	0.489074i	0.0343263i
$204$	0	0
$205$	−30.2838	−2.11511
$206$	0	0
$207$	7.03409	0.488903
$208$	0	0
$209$	−1.18325	−0.0818469
$210$	0	0
$211$	9.26059	0.637525	0.318763	−	0.947835i	$-0.396733\pi$
$211$	9.26059	0.637525	0.318763	+	0.947835i	$0.396733\pi$
$212$	0	0
$213$	15.1722i	1.03958i
$214$	0	0
$215$	− 4.16197i	− 0.283844i
$216$	0	0
$217$	−0.114376	−0.00776432
$218$	0	0
$219$	4.16507i	0.281449i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.1700i	1.28372i	0.766821	+	0.641861i	$0.221837\pi$
$223$	19.1700i	1.28372i	−0.766821	+	0.641861i	$0.778163\pi$
$224$	0	0
$225$	−10.2543	−0.683623
$226$	0	0
$227$	− 20.6095i	− 1.36790i	−0.729528	−	0.683951i	$-0.760260\pi$
$227$	− 20.6095i	− 1.36790i	0.729528	−	0.683951i	$-0.239740\pi$
$228$	0	0
$229$	− 11.2937i	− 0.746309i	−0.927769	−	0.373154i	$-0.878276\pi$
$229$	− 11.2937i	− 0.746309i	0.927769	−	0.373154i	$-0.121724\pi$
$230$	0	0
$231$	−0.119613	−0.00786993
$232$	0	0
$233$	13.2668	0.869137	0.434569	−	0.900639i	$-0.356901\pi$
$233$	13.2668	0.869137	0.434569	+	0.900639i	$0.356901\pi$
$234$	0	0
$235$	49.8224	3.25005
$236$	0	0
$237$	−3.83731	−0.249260
$238$	0	0
$239$	0.412559i	0.0266862i	0.999911	+	0.0133431i	$0.00424737\pi$
$239$	0.412559i	0.0266862i	−0.999911	+	0.0133431i	$0.995753\pi$
$240$	0	0
$241$	20.8992i	1.34624i	0.739535	+	0.673118i	$0.235045\pi$
$241$	20.8992i	1.34624i	−0.739535	+	0.673118i	$0.764955\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	27.3151i	1.74510i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 9.04179i	− 0.573000i
$250$	0	0
$251$	−5.10303	−0.322100	−0.161050	−	0.986946i	$-0.551488\pi$
$251$	−5.10303	−0.322100	−0.161050	+	0.986946i	$0.551488\pi$
$252$	0	0
$253$	− 10.5846i	− 0.665446i
$254$	0	0
$255$	− 25.4922i	− 1.59638i
$256$	0	0
$257$	26.7241	1.66700	0.833501	−	0.552518i	$-0.186333\pi$
$257$	26.7241	1.66700	0.833501	+	0.552518i	$0.186333\pi$
$258$	0	0
$259$	0.734146	0.0456176
$260$	0	0
$261$	6.15266	0.380840
$262$	0	0
$263$	−9.89772	−0.610319	−0.305160	−	0.952301i	$-0.598710\pi$
$263$	−9.89772	−0.610319	−0.305160	+	0.952301i	$0.598710\pi$
$264$	0	0
$265$	10.6661i	0.655214i
$266$	0	0
$267$	− 4.75082i	− 0.290745i
$268$	0	0
$269$	−18.9609	−1.15607	−0.578035	−	0.816012i	$-0.696180\pi$
$269$	−18.9609	−1.15607	−0.578035	+	0.816012i	$0.696180\pi$
$270$	0	0
$271$	− 23.7343i	− 1.44176i	−0.693062	−	0.720878i	$-0.743739\pi$
$271$	− 23.7343i	− 1.44176i	0.693062	−	0.720878i	$-0.256261\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15.4303i	0.930480i
$276$	0	0
$277$	−16.1577	−0.970822	−0.485411	−	0.874286i	$-0.661330\pi$
$277$	−16.1577	−0.970822	−0.485411	+	0.874286i	$0.661330\pi$
$278$	0	0
$279$	1.43887i	0.0861429i
$280$	0	0
$281$	− 4.20881i	− 0.251077i	−0.992089	−	0.125538i	$-0.959934\pi$
$281$	− 4.20881i	− 0.251077i	0.992089	−	0.125538i	$-0.0400658\pi$
$282$	0	0
$283$	−3.57052	−0.212245	−0.106123	−	0.994353i	$-0.533844\pi$
$283$	−3.57052	−0.212245	−0.106123	+	0.994353i	$0.533844\pi$
$284$	0	0
$285$	−3.07119	−0.181922
$286$	0	0
$287$	−0.616348	−0.0363819
$288$	0	0
$289$	25.6011	1.50595
$290$	0	0
$291$	− 6.94297i	− 0.407004i
$292$	0	0
$293$	15.0734i	0.880599i	0.897851	+	0.440300i	$0.145128\pi$
$293$	15.0734i	0.880599i	−0.897851	+	0.440300i	$0.854872\pi$
$294$	0	0
$295$	40.5314	2.35983
$296$	0	0
$297$	1.50475i	0.0873146i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 0.0847060i	− 0.00488238i
$302$	0	0
$303$	16.9785	0.975391
$304$	0	0
$305$	30.9621i	1.77288i
$306$	0	0
$307$	22.3150i	1.27359i	0.771035	+	0.636793i	$0.219739\pi$
$307$	22.3150i	1.27359i	−0.771035	+	0.636793i	$0.780261\pi$
$308$	0	0
$309$	−0.534038	−0.0303804
$310$	0	0
$311$	−2.15132	−0.121990	−0.0609951	−	0.998138i	$-0.519427\pi$
$311$	−2.15132	−0.121990	−0.0609951	+	0.998138i	$0.519427\pi$
$312$	0	0
$313$	−26.0404	−1.47189	−0.735945	−	0.677042i	$-0.763262\pi$
$313$	−26.0404	−1.47189	−0.735945	+	0.677042i	$0.763262\pi$
$314$	0	0
$315$	−0.310462	−0.0174926
$316$	0	0
$317$	13.3249i	0.748403i	0.927347	+	0.374201i	$0.122083\pi$
$317$	13.3249i	0.748403i	−0.927347	+	0.374201i	$0.877917\pi$
$318$	0	0
$319$	− 9.25824i	− 0.518362i
$320$	0	0
$321$	−10.9729	−0.612448
$322$	0	0
$323$	− 5.13240i	− 0.285574i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.24364i	− 0.455874i
$328$	0	0
$329$	1.01400	0.0559038
$330$	0	0
$331$	9.11986i	0.501273i	0.968081	+	0.250637i	$0.0806399\pi$
$331$	9.11986i	0.501273i	−0.968081	+	0.250637i	$0.919360\pi$
$332$	0	0
$333$	− 9.23571i	− 0.506114i
$334$	0	0
$335$	30.5000	1.66639
$336$	0	0
$337$	−11.8690	−0.646546	−0.323273	−	0.946306i	$-0.604783\pi$
$337$	−11.8690	−0.646546	−0.323273	+	0.946306i	$0.604783\pi$
$338$	0	0
$339$	−1.35771	−0.0737406
$340$	0	0
$341$	2.16514	0.117249
$342$	0	0
$343$	1.11236i	0.0600616i
$344$	0	0
$345$	− 27.4729i	− 1.47909i
$346$	0	0
$347$	−1.20818	−0.0648584	−0.0324292	−	0.999474i	$-0.510324\pi$
$347$	−1.20818	−0.0648584	−0.0324292	+	0.999474i	$0.510324\pi$
$348$	0	0
$349$	5.45450i	0.291973i	0.989287	+	0.145986i	$0.0466356\pi$
$349$	5.45450i	0.291973i	−0.989287	+	0.145986i	$0.953364\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 35.5083i	− 1.88991i	−0.327195	−	0.944957i	$-0.606103\pi$
$353$	− 35.5083i	− 1.88991i	0.327195	−	0.944957i	$-0.393897\pi$
$354$	0	0
$355$	59.2579	3.14508
$356$	0	0
$357$	− 0.518827i	− 0.0274592i
$358$	0	0
$359$	18.4695i	0.974782i	0.873184	+	0.487391i	$0.162051\pi$
$359$	18.4695i	0.974782i	−0.873184	+	0.487391i	$0.837949\pi$
$360$	0	0
$361$	18.3817	0.967456
$362$	0	0
$363$	−8.73572	−0.458506
$364$	0	0
$365$	16.2674	0.851477
$366$	0	0
$367$	18.7044	0.976362	0.488181	−	0.872742i	$-0.337661\pi$
$367$	18.7044	0.976362	0.488181	+	0.872742i	$0.337661\pi$
$368$	0	0
$369$	7.75379i	0.403646i
$370$	0	0
$371$	0.217081i	0.0112703i
$372$	0	0
$373$	12.3213	0.637973	0.318987	−	0.947759i	$-0.396657\pi$
$373$	12.3213	0.637973	0.318987	+	0.947759i	$0.396657\pi$
$374$	0	0
$375$	20.5218i	1.05974i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 14.9123i	− 0.765991i	−0.923750	−	0.382996i	$-0.874892\pi$
$379$	− 14.9123i	− 0.765991i	0.923750	−	0.382996i	$-0.125108\pi$
$380$	0	0
$381$	−11.0111	−0.564118
$382$	0	0
$383$	5.27199i	0.269386i	0.990887	+	0.134693i	$0.0430048\pi$
$383$	5.27199i	0.269386i	−0.990887	+	0.134693i	$0.956995\pi$
$384$	0	0
$385$	0.467169i	0.0238091i
$386$	0	0
$387$	−1.06562	−0.0541685
$388$	0	0
$389$	2.66665	0.135205	0.0676023	−	0.997712i	$-0.478465\pi$
$389$	2.66665	0.135205	0.0676023	+	0.997712i	$0.478465\pi$
$390$	0	0
$391$	45.9111	2.32183
$392$	0	0
$393$	19.8552	1.00156
$394$	0	0
$395$	14.9873i	0.754094i
$396$	0	0
$397$	− 12.7244i	− 0.638618i	−0.947651	−	0.319309i	$-0.896549\pi$
$397$	− 12.7244i	− 0.638618i	0.947651	−	0.319309i	$-0.103451\pi$
$398$	0	0
$399$	−0.0625061	−0.00312922
$400$	0	0
$401$	− 23.5633i	− 1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$401$	− 23.5633i	− 1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.90568i	0.194075i
$406$	0	0
$407$	−13.8975	−0.688872
$408$	0	0
$409$	29.2475i	1.44620i	0.690745	+	0.723098i	$0.257283\pi$
$409$	29.2475i	1.44620i	−0.690745	+	0.723098i	$0.742717\pi$
$410$	0	0
$411$	4.80881i	0.237201i
$412$	0	0
$413$	0.824910	0.0405911
$414$	0	0
$415$	−35.3143	−1.73351
$416$	0	0
$417$	−20.7637	−1.01680
$418$	0	0
$419$	−23.3488	−1.14066	−0.570331	−	0.821415i	$-0.693185\pi$
$419$	−23.3488	−1.14066	−0.570331	+	0.821415i	$0.693185\pi$
$420$	0	0
$421$	6.92772i	0.337636i	0.985647	+	0.168818i	$0.0539950\pi$
$421$	6.92772i	0.337636i	−0.985647	+	0.168818i	$0.946005\pi$
$422$	0	0
$423$	− 12.7564i	− 0.620237i
$424$	0	0
$425$	−66.9296	−3.24656
$426$	0	0
$427$	0.630152i	0.0304952i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 20.3178i	− 0.978675i	−0.872095	−	0.489337i	$-0.837239\pi$
$431$	− 20.3178i	− 0.978675i	0.872095	−	0.489337i	$-0.162761\pi$
$432$	0	0
$433$	35.6153	1.71156	0.855780	−	0.517340i	$-0.173078\pi$
$433$	35.6153	1.71156	0.855780	+	0.517340i	$0.173078\pi$
$434$	0	0
$435$	− 24.0303i	− 1.15217i
$436$	0	0
$437$	− 5.53118i	− 0.264592i
$438$	0	0
$439$	−28.4092	−1.35590	−0.677948	−	0.735110i	$-0.737131\pi$
$439$	−28.4092	−1.35590	−0.677948	+	0.735110i	$0.737131\pi$
$440$	0	0
$441$	6.99368	0.333032
$442$	0	0
$443$	24.8982	1.18295	0.591474	−	0.806324i	$-0.298546\pi$
$443$	24.8982	1.18295	0.591474	+	0.806324i	$0.298546\pi$
$444$	0	0
$445$	−18.5552	−0.879600
$446$	0	0
$447$	− 4.10785i	− 0.194295i
$448$	0	0
$449$	− 7.33361i	− 0.346094i	−0.984914	−	0.173047i	$-0.944639\pi$
$449$	− 7.33361i	− 0.346094i	0.984914	−	0.173047i	$-0.0553613\pi$
$450$	0	0
$451$	11.6675	0.549403
$452$	0	0
$453$	− 12.6178i	− 0.592838i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.8511i	0.741481i	0.928736	+	0.370741i	$0.120896\pi$
$457$	15.8511i	0.741481i	−0.928736	+	0.370741i	$0.879104\pi$
$458$	0	0
$459$	−6.52695	−0.304652
$460$	0	0
$461$	− 10.5739i	− 0.492474i	−0.969210	−	0.246237i	$-0.920806\pi$
$461$	− 10.5739i	− 0.492474i	0.969210	−	0.246237i	$-0.0791941\pi$
$462$	0	0
$463$	− 5.34050i	− 0.248194i	−0.992270	−	0.124097i	$-0.960397\pi$
$463$	− 5.34050i	− 0.248194i	0.992270	−	0.124097i	$-0.0396034\pi$
$464$	0	0
$465$	5.61977	0.260610
$466$	0	0
$467$	−25.7751	−1.19273	−0.596364	−	0.802714i	$-0.703389\pi$
$467$	−25.7751	−1.19273	−0.596364	+	0.802714i	$0.703389\pi$
$468$	0	0
$469$	0.620746	0.0286634
$470$	0	0
$471$	10.1958	0.469796
$472$	0	0
$473$	1.60350i	0.0737288i
$474$	0	0
$475$	8.06340i	0.369974i
$476$	0	0
$477$	2.73092	0.125040
$478$	0	0
$479$	− 17.2136i	− 0.786511i	−0.919429	−	0.393255i	$-0.871349\pi$
$479$	− 17.2136i	− 0.786511i	0.919429	−	0.393255i	$-0.128651\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 0.559139i	− 0.0254417i
$484$	0	0
$485$	−27.1170	−1.23132
$486$	0	0
$487$	− 17.0725i	− 0.773628i	−0.922158	−	0.386814i	$-0.873575\pi$
$487$	− 17.0725i	− 0.773628i	0.922158	−	0.386814i	$-0.126425\pi$
$488$	0	0
$489$	− 15.8095i	− 0.714930i
$490$	0	0
$491$	24.7127	1.11527	0.557634	−	0.830087i	$-0.311709\pi$
$491$	24.7127	1.11527	0.557634	+	0.830087i	$0.311709\pi$
$492$	0	0
$493$	40.1581	1.80863
$494$	0	0
$495$	5.87709	0.264155
$496$	0	0
$497$	1.20604	0.0540982
$498$	0	0
$499$	0.887509i	0.0397303i	0.999803	+	0.0198652i	$0.00632370\pi$
$499$	0.887509i	0.0397303i	−0.999803	+	0.0198652i	$0.993676\pi$
$500$	0	0
$501$	− 24.8487i	− 1.11016i
$502$	0	0
$503$	7.24671	0.323115	0.161557	−	0.986863i	$-0.448348\pi$
$503$	7.24671	0.323115	0.161557	+	0.986863i	$0.448348\pi$
$504$	0	0
$505$	− 66.3127i	− 2.95088i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 35.6937i	− 1.58210i	−0.611754	−	0.791048i	$-0.709536\pi$
$509$	− 35.6937i	− 1.58210i	0.611754	−	0.791048i	$-0.290464\pi$
$510$	0	0
$511$	0.331081	0.0146462
$512$	0	0
$513$	0.786340i	0.0347178i
$514$	0	0
$515$	2.08578i	0.0919106i
$516$	0	0
$517$	−19.1952	−0.844205
$518$	0	0
$519$	12.6367	0.554690
$520$	0	0
$521$	−3.56585	−0.156223	−0.0781113	−	0.996945i	$-0.524889\pi$
$521$	−3.56585	−0.156223	−0.0781113	+	0.996945i	$0.524889\pi$
$522$	0	0
$523$	−8.08344	−0.353464	−0.176732	−	0.984259i	$-0.556553\pi$
$523$	−8.08344	−0.353464	−0.176732	+	0.984259i	$0.556553\pi$
$524$	0	0
$525$	0.815117i	0.0355746i
$526$	0	0
$527$	9.39143i	0.409097i
$528$	0	0
$529$	26.4784	1.15123
$530$	0	0
$531$	− 10.3775i	− 0.450347i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	42.8567i	1.85286i
$536$	0	0
$537$	2.34535	0.101209
$538$	0	0
$539$	− 10.5238i	− 0.453291i
$540$	0	0
$541$	14.2925i	0.614483i	0.951632	+	0.307241i	$0.0994059\pi$
$541$	14.2925i	0.614483i	−0.951632	+	0.307241i	$0.900594\pi$
$542$	0	0
$543$	10.8755	0.466712
$544$	0	0
$545$	−32.1970	−1.37917
$546$	0	0
$547$	−25.0004	−1.06894	−0.534471	−	0.845187i	$-0.679489\pi$
$547$	−25.0004	−1.06894	−0.534471	+	0.845187i	$0.679489\pi$
$548$	0	0
$549$	7.92745	0.338335
$550$	0	0
$551$	− 4.83808i	− 0.206109i
$552$	0	0
$553$	0.305028i	0.0129711i
$554$	0	0
$555$	−36.0717	−1.53116
$556$	0	0
$557$	2.39071i	0.101298i	0.998717	+	0.0506488i	$0.0161289\pi$
$557$	2.39071i	0.101298i	−0.998717	+	0.0506488i	$0.983871\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	9.82145i	0.414662i
$562$	0	0
$563$	−37.8778	−1.59636	−0.798178	−	0.602421i	$-0.794203\pi$
$563$	−37.8778	−1.59636	−0.798178	+	0.602421i	$0.794203\pi$
$564$	0	0
$565$	5.30278i	0.223090i
$566$	0	0
$567$	0.0794899i	0.00333826i
$568$	0	0
$569$	6.58037	0.275863	0.137932	−	0.990442i	$-0.455955\pi$
$569$	6.58037	0.275863	0.137932	+	0.990442i	$0.455955\pi$
$570$	0	0
$571$	21.5937	0.903670	0.451835	−	0.892102i	$-0.350770\pi$
$571$	21.5937	0.903670	0.451835	+	0.892102i	$0.350770\pi$
$572$	0	0
$573$	−25.0561	−1.04674
$574$	0	0
$575$	−72.1300	−3.00803
$576$	0	0
$577$	20.0143i	0.833208i	0.909088	+	0.416604i	$0.136780\pi$
$577$	20.0143i	0.833208i	−0.909088	+	0.416604i	$0.863220\pi$
$578$	0	0
$579$	2.03469i	0.0845588i
$580$	0	0
$581$	−0.718730	−0.0298180
$582$	0	0
$583$	− 4.10936i	− 0.170192i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.9326i	0.657610i	0.944398	+	0.328805i	$0.106646\pi$
$587$	15.9326i	0.657610i	−0.944398	+	0.328805i	$0.893354\pi$
$588$	0	0
$589$	1.13144	0.0466202
$590$	0	0
$591$	2.93816i	0.120860i
$592$	0	0
$593$	12.7746i	0.524592i	0.964987	+	0.262296i	$0.0844796\pi$
$593$	12.7746i	0.524592i	−0.964987	+	0.262296i	$0.915520\pi$
$594$	0	0
$595$	−2.02637	−0.0830731
$596$	0	0
$597$	21.3170	0.872446
$598$	0	0
$599$	−20.4940	−0.837362	−0.418681	−	0.908133i	$-0.637507\pi$
$599$	−20.4940	−0.837362	−0.418681	+	0.908133i	$0.637507\pi$
$600$	0	0
$601$	−3.59245	−0.146539	−0.0732695	−	0.997312i	$-0.523343\pi$
$601$	−3.59245	−0.146539	−0.0732695	+	0.997312i	$0.523343\pi$
$602$	0	0
$603$	− 7.80912i	− 0.318012i
$604$	0	0
$605$	34.1189i	1.38713i
$606$	0	0
$607$	−13.0627	−0.530197	−0.265099	−	0.964221i	$-0.585404\pi$
$607$	−13.0627	−0.530197	−0.265099	+	0.964221i	$0.585404\pi$
$608$	0	0
$609$	− 0.489074i	− 0.0198183i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.56139i	0.345791i	0.984940	+	0.172896i	$0.0553123\pi$
$613$	8.56139i	0.345791i	−0.984940	+	0.172896i	$0.944688\pi$
$614$	0	0
$615$	30.2838	1.22116
$616$	0	0
$617$	9.50428i	0.382628i	0.981529	+	0.191314i	$0.0612749\pi$
$617$	9.50428i	0.382628i	−0.981529	+	0.191314i	$0.938725\pi$
$618$	0	0
$619$	− 0.765606i	− 0.0307723i	−0.999882	−	0.0153861i	$-0.995102\pi$
$619$	− 0.765606i	− 0.0307723i	0.999882	−	0.0153861i	$-0.00489776\pi$
$620$	0	0
$621$	−7.03409	−0.282268
$622$	0	0
$623$	−0.377642	−0.0151299
$624$	0	0
$625$	28.8799	1.15520
$626$	0	0
$627$	1.18325	0.0472544
$628$	0	0
$629$	− 60.2810i	− 2.40356i
$630$	0	0
$631$	− 15.2955i	− 0.608902i	−0.952528	−	0.304451i	$-0.901527\pi$
$631$	− 15.2955i	− 0.608902i	0.952528	−	0.304451i	$-0.0984731\pi$
$632$	0	0
$633$	−9.26059	−0.368075
$634$	0	0
$635$	43.0060i	1.70664i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 15.1722i	− 0.600204i
$640$	0	0
$641$	16.7020	0.659690	0.329845	−	0.944035i	$-0.393003\pi$
$641$	16.7020	0.659690	0.329845	+	0.944035i	$0.393003\pi$
$642$	0	0
$643$	− 12.3219i	− 0.485927i	−0.970035	−	0.242963i	$-0.921881\pi$
$643$	− 12.3219i	− 0.485927i	0.970035	−	0.242963i	$-0.0781194\pi$
$644$	0	0
$645$	4.16197i	0.163878i
$646$	0	0
$647$	6.92612	0.272294	0.136147	−	0.990689i	$-0.456528\pi$
$647$	6.92612	0.272294	0.136147	+	0.990689i	$0.456528\pi$
$648$	0	0
$649$	−15.6156	−0.612967
$650$	0	0
$651$	0.114376	0.00448273
$652$	0	0
$653$	−6.42407	−0.251393	−0.125697	−	0.992069i	$-0.540117\pi$
$653$	−6.42407	−0.251393	−0.125697	+	0.992069i	$0.540117\pi$
$654$	0	0
$655$	− 77.5480i	− 3.03005i
$656$	0	0
$657$	− 4.16507i	− 0.162495i
$658$	0	0
$659$	14.0188	0.546096	0.273048	−	0.962000i	$-0.411968\pi$
$659$	14.0188	0.546096	0.273048	+	0.962000i	$0.411968\pi$
$660$	0	0
$661$	− 17.4663i	− 0.679359i	−0.940541	−	0.339680i	$-0.889681\pi$
$661$	− 17.4663i	− 0.679359i	0.940541	−	0.339680i	$-0.110319\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.244129i	0.00946691i
$666$	0	0
$667$	43.2784	1.67574
$668$	0	0
$669$	− 19.1700i	− 0.741157i
$670$	0	0
$671$	− 11.9289i	− 0.460508i
$672$	0	0
$673$	−32.6688	−1.25929	−0.629644	−	0.776883i	$-0.716799\pi$
$673$	−32.6688	−1.25929	−0.629644	+	0.776883i	$0.716799\pi$
$674$	0	0
$675$	10.2543	0.394690
$676$	0	0
$677$	−49.3618	−1.89713	−0.948565	−	0.316583i	$-0.897465\pi$
$677$	−49.3618	−1.89713	−0.948565	+	0.316583i	$0.897465\pi$
$678$	0	0
$679$	−0.551896	−0.0211798
$680$	0	0
$681$	20.6095i	0.789759i
$682$	0	0
$683$	− 44.9159i	− 1.71866i	−0.511422	−	0.859330i	$-0.670881\pi$
$683$	− 44.9159i	− 1.71866i	0.511422	−	0.859330i	$-0.329119\pi$
$684$	0	0
$685$	18.7817	0.717611
$686$	0	0
$687$	11.2937i	0.430882i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	21.5931i	0.821440i	0.911762	+	0.410720i	$0.134723\pi$
$691$	21.5931i	0.821440i	−0.911762	+	0.410720i	$0.865277\pi$
$692$	0	0
$693$	0.119613	0.00454371
$694$	0	0
$695$	81.0962i	3.07616i
$696$	0	0
$697$	50.6086i	1.91694i
$698$	0	0
$699$	−13.2668	−0.501797
$700$	0	0
$701$	31.8311	1.20224	0.601121	−	0.799158i	$-0.294721\pi$
$701$	31.8311	1.20224	0.601121	+	0.799158i	$0.294721\pi$
$702$	0	0
$703$	−7.26241	−0.273907
$704$	0	0
$705$	−49.8224	−1.87642
$706$	0	0
$707$	− 1.34962i	− 0.0507577i
$708$	0	0
$709$	48.5378i	1.82288i	0.411436	+	0.911439i	$0.365027\pi$
$709$	48.5378i	1.82288i	−0.411436	+	0.911439i	$0.634973\pi$
$710$	0	0
$711$	3.83731	0.143911
$712$	0	0
$713$	10.1211i	0.379039i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.412559i	− 0.0154073i
$718$	0	0
$719$	−5.13015	−0.191322	−0.0956611	−	0.995414i	$-0.530497\pi$
$719$	−5.13015	−0.191322	−0.0956611	+	0.995414i	$0.530497\pi$
$720$	0	0
$721$	0.0424506i	0.00158094i
$722$	0	0
$723$	− 20.8992i	− 0.777249i
$724$	0	0
$725$	−63.0915	−2.34316
$726$	0	0
$727$	45.8944	1.70213	0.851065	−	0.525061i	$-0.175957\pi$
$727$	45.8944	1.70213	0.851065	+	0.525061i	$0.175957\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.95525	−0.257249
$732$	0	0
$733$	− 19.2958i	− 0.712706i	−0.934351	−	0.356353i	$-0.884020\pi$
$733$	− 19.2958i	− 0.712706i	0.934351	−	0.356353i	$-0.115980\pi$
$734$	0	0
$735$	− 27.3151i	− 1.00753i
$736$	0	0
$737$	−11.7508	−0.432846
$738$	0	0
$739$	− 24.7702i	− 0.911187i	−0.890188	−	0.455594i	$-0.849427\pi$
$739$	− 24.7702i	− 0.911187i	0.890188	−	0.455594i	$-0.150573\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 29.1083i	− 1.06788i	−0.845523	−	0.533939i	$-0.820711\pi$
$743$	− 29.1083i	− 1.06788i	0.845523	−	0.533939i	$-0.179289\pi$
$744$	0	0
$745$	−16.0440	−0.587805
$746$	0	0
$747$	9.04179i	0.330822i
$748$	0	0
$749$	0.872235i	0.0318708i
$750$	0	0
$751$	−27.5195	−1.00420	−0.502101	−	0.864809i	$-0.667439\pi$
$751$	−27.5195	−1.00420	−0.502101	+	0.864809i	$0.667439\pi$
$752$	0	0
$753$	5.10303	0.185965
$754$	0	0
$755$	−49.2813	−1.79353
$756$	0	0
$757$	5.71110	0.207574	0.103787	−	0.994600i	$-0.466904\pi$
$757$	5.71110	0.207574	0.103787	+	0.994600i	$0.466904\pi$
$758$	0	0
$759$	10.5846i	0.384195i
$760$	0	0
$761$	40.1705i	1.45618i	0.685482	+	0.728089i	$0.259591\pi$
$761$	40.1705i	1.45618i	−0.685482	+	0.728089i	$0.740409\pi$
$762$	0	0
$763$	−0.655286	−0.0237229
$764$	0	0
$765$	25.4922i	0.921672i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 15.8754i	− 0.572483i	−0.958158	−	0.286241i	$-0.907594\pi$
$769$	− 15.8754i	− 0.572483i	0.958158	−	0.286241i	$-0.0924059\pi$
$770$	0	0
$771$	−26.7241	−0.962444
$772$	0	0
$773$	− 1.19739i	− 0.0430670i	−0.999768	−	0.0215335i	$-0.993145\pi$
$773$	− 1.19739i	− 0.0430670i	0.999768	−	0.0215335i	$-0.00685486\pi$
$774$	0	0
$775$	− 14.7547i	− 0.530003i
$776$	0	0
$777$	−0.734146	−0.0263373
$778$	0	0
$779$	6.09711	0.218452
$780$	0	0
$781$	−22.8305	−0.816938
$782$	0	0
$783$	−6.15266	−0.219878
$784$	0	0
$785$	− 39.8214i	− 1.42129i
$786$	0	0
$787$	32.2576i	1.14986i	0.818203	+	0.574929i	$0.194970\pi$
$787$	32.2576i	1.14986i	−0.818203	+	0.574929i	$0.805030\pi$
$788$	0	0
$789$	9.89772	0.352368
$790$	0	0
$791$	0.107924i	0.00383734i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 10.6661i	− 0.378288i
$796$	0	0
$797$	28.2439	1.00045	0.500226	−	0.865895i	$-0.333250\pi$
$797$	28.2439	1.00045	0.500226	+	0.865895i	$0.333250\pi$
$798$	0	0
$799$	− 83.2603i	− 2.94554i
$800$	0	0
$801$	4.75082i	0.167862i
$802$	0	0
$803$	−6.26740	−0.221172
$804$	0	0
$805$	−2.18382	−0.0769695
$806$	0	0
$807$	18.9609	0.667457
$808$	0	0
$809$	−37.2877	−1.31096	−0.655482	−	0.755210i	$-0.727535\pi$
$809$	−37.2877	−1.31096	−0.655482	+	0.755210i	$0.727535\pi$
$810$	0	0
$811$	6.09538i	0.214038i	0.994257	+	0.107019i	$0.0341305\pi$
$811$	6.09538i	0.214038i	−0.994257	+	0.107019i	$0.965869\pi$
$812$	0	0
$813$	23.7343i	0.832398i
$814$	0	0
$815$	−61.7468	−2.16290
$816$	0	0
$817$	0.837940i	0.0293158i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.6514i	0.581138i	0.956854	+	0.290569i	$0.0938446\pi$
$821$	16.6514i	0.581138i	−0.956854	+	0.290569i	$0.906155\pi$
$822$	0	0
$823$	−32.1734	−1.12150	−0.560748	−	0.827987i	$-0.689486\pi$
$823$	−32.1734	−1.12150	−0.560748	+	0.827987i	$0.689486\pi$
$824$	0	0
$825$	− 15.4303i	− 0.537213i
$826$	0	0
$827$	53.7727i	1.86986i	0.354831	+	0.934930i	$0.384538\pi$
$827$	53.7727i	1.86986i	−0.354831	+	0.934930i	$0.615462\pi$
$828$	0	0
$829$	−47.5426	−1.65122	−0.825611	−	0.564240i	$-0.809169\pi$
$829$	−47.5426	−1.65122	−0.825611	+	0.564240i	$0.809169\pi$
$830$	0	0
$831$	16.1577	0.560504
$832$	0	0
$833$	45.6474	1.58159
$834$	0	0
$835$	−97.0511	−3.35859
$836$	0	0
$837$	− 1.43887i	− 0.0497346i
$838$	0	0
$839$	− 29.9379i	− 1.03357i	−0.856115	−	0.516785i	$-0.827129\pi$
$839$	− 29.9379i	− 1.03357i	0.856115	−	0.516785i	$-0.172871\pi$
$840$	0	0
$841$	8.85524	0.305353
$842$	0	0
$843$	4.20881i	0.144959i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.694401i	0.0238599i
$848$	0	0
$849$	3.57052	0.122540
$850$	0	0
$851$	− 64.9648i	− 2.22697i
$852$	0	0
$853$	38.2677i	1.31026i	0.755516	+	0.655130i	$0.227386\pi$
$853$	38.2677i	1.31026i	−0.755516	+	0.655130i	$0.772614\pi$
$854$	0	0
$855$	3.07119	0.105033
$856$	0	0
$857$	10.6995	0.365486	0.182743	−	0.983161i	$-0.441502\pi$
$857$	10.6995	0.365486	0.182743	+	0.983161i	$0.441502\pi$
$858$	0	0
$859$	−22.6031	−0.771207	−0.385604	−	0.922665i	$-0.626007\pi$
$859$	−22.6031	−0.771207	−0.385604	+	0.922665i	$0.626007\pi$
$860$	0	0
$861$	0.616348	0.0210051
$862$	0	0
$863$	20.7503i	0.706349i	0.935557	+	0.353174i	$0.114898\pi$
$863$	20.7503i	0.706349i	−0.935557	+	0.353174i	$0.885102\pi$
$864$	0	0
$865$	− 49.3550i	− 1.67812i
$866$	0	0
$867$	−25.6011	−0.869458
$868$	0	0
$869$	− 5.77421i	− 0.195877i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.94297i	0.234984i
$874$	0	0
$875$	1.63128	0.0551472
$876$	0	0
$877$	20.8423i	0.703793i	0.936039	+	0.351897i	$0.114463\pi$
$877$	20.8423i	0.703793i	−0.936039	+	0.351897i	$0.885537\pi$
$878$	0	0
$879$	− 15.0734i	− 0.508414i
$880$	0	0
$881$	9.03338	0.304342	0.152171	−	0.988354i	$-0.451373\pi$
$881$	9.03338	0.304342	0.152171	+	0.988354i	$0.451373\pi$
$882$	0	0
$883$	−21.4380	−0.721447	−0.360724	−	0.932673i	$-0.617470\pi$
$883$	−21.4380	−0.721447	−0.360724	+	0.932673i	$0.617470\pi$
$884$	0	0
$885$	−40.5314	−1.36245
$886$	0	0
$887$	5.41708	0.181888	0.0909438	−	0.995856i	$-0.471012\pi$
$887$	5.41708	0.181888	0.0909438	+	0.995856i	$0.471012\pi$
$888$	0	0
$889$	0.875275i	0.0293558i
$890$	0	0
$891$	− 1.50475i	− 0.0504111i
$892$	0	0
$893$	−10.0309	−0.335670
$894$	0	0
$895$	− 9.16018i	− 0.306191i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.85288i	0.295260i
$900$	0	0
$901$	17.8246	0.593823
$902$	0	0
$903$	0.0847060i	0.00281884i
$904$	0	0
$905$	− 42.4762i	− 1.41196i
$906$	0	0
$907$	−13.5292	−0.449229	−0.224615	−	0.974448i	$-0.572112\pi$
$907$	−13.5292	−0.449229	−0.224615	+	0.974448i	$0.572112\pi$
$908$	0	0
$909$	−16.9785	−0.563142
$910$	0	0
$911$	−25.7848	−0.854290	−0.427145	−	0.904183i	$-0.640481\pi$
$911$	−25.7848	−0.854290	−0.427145	+	0.904183i	$0.640481\pi$
$912$	0	0
$913$	13.6057	0.450281
$914$	0	0
$915$	− 30.9621i	− 1.02358i
$916$	0	0
$917$	− 1.57829i	− 0.0521196i
$918$	0	0
$919$	−28.3412	−0.934889	−0.467445	−	0.884022i	$-0.654825\pi$
$919$	−28.3412	−0.934889	−0.467445	+	0.884022i	$0.654825\pi$
$920$	0	0
$921$	− 22.3150i	− 0.735305i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	94.7062i	3.11392i
$926$	0	0
$927$	0.534038	0.0175401
$928$	0	0
$929$	− 27.2671i	− 0.894606i	−0.894382	−	0.447303i	$-0.852385\pi$
$929$	− 27.2671i	− 0.894606i	0.894382	−	0.447303i	$-0.147615\pi$
$930$	0	0
$931$	− 5.49941i	− 0.180236i
$932$	0	0
$933$	2.15132	0.0704310
$934$	0	0
$935$	38.3595	1.25449
$936$	0	0
$937$	57.0811	1.86476	0.932379	−	0.361483i	$-0.117729\pi$
$937$	57.0811	1.86476	0.932379	+	0.361483i	$0.117729\pi$
$938$	0	0
$939$	26.0404	0.849796
$940$	0	0
$941$	− 33.2566i	− 1.08413i	−0.840335	−	0.542067i	$-0.817642\pi$
$941$	− 33.2566i	− 1.08413i	0.840335	−	0.542067i	$-0.182358\pi$
$942$	0	0
$943$	54.5408i	1.77609i
$944$	0	0
$945$	0.310462	0.0100993
$946$	0	0
$947$	− 38.9304i	− 1.26507i	−0.774533	−	0.632534i	$-0.782015\pi$
$947$	− 38.9304i	− 1.26507i	0.774533	−	0.632534i	$-0.217985\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 13.3249i	− 0.432090i
$952$	0	0
$953$	27.3688	0.886562	0.443281	−	0.896383i	$-0.353814\pi$
$953$	27.3688	0.886562	0.443281	+	0.896383i	$0.353814\pi$
$954$	0	0
$955$	97.8613i	3.16672i
$956$	0	0
$957$	9.25824i	0.299276i
$958$	0	0
$959$	0.382251	0.0123435
$960$	0	0
$961$	28.9297	0.933215
$962$	0	0
$963$	10.9729	0.353597
$964$	0	0
$965$	7.94684	0.255818
$966$	0	0
$967$	− 48.2047i	− 1.55016i	−0.631865	−	0.775079i	$-0.717710\pi$
$967$	− 48.2047i	− 1.55016i	0.631865	−	0.775079i	$-0.282290\pi$
$968$	0	0
$969$	5.13240i	0.164877i
$970$	0	0
$971$	−40.7160	−1.30664	−0.653319	−	0.757083i	$-0.726624\pi$
$971$	−40.7160	−1.30664	−0.653319	+	0.757083i	$0.726624\pi$
$972$	0	0
$973$	1.65050i	0.0529126i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 17.2632i	− 0.552297i	−0.961115	−	0.276149i	$-0.910942\pi$
$977$	− 17.2632i	− 0.552297i	0.961115	−	0.276149i	$-0.0890582\pi$
$978$	0	0
$979$	7.14881	0.228477
$980$	0	0
$981$	8.24364i	0.263199i
$982$	0	0
$983$	50.0600i	1.59667i	0.602216	+	0.798333i	$0.294285\pi$
$983$	50.0600i	1.59667i	−0.602216	+	0.798333i	$0.705715\pi$
$984$	0	0
$985$	11.4755	0.365641
$986$	0	0
$987$	−1.01400	−0.0322761
$988$	0	0
$989$	−7.49567	−0.238348
$990$	0	0
$991$	0.122255	0.00388355	0.00194177	−	0.999998i	$-0.499382\pi$
$991$	0.122255	0.00388355	0.00194177	+	0.999998i	$0.499382\pi$
$992$	0	0
$993$	− 9.11986i	− 0.289410i
$994$	0	0
$995$	− 83.2574i	− 2.63944i
$996$	0	0
$997$	−4.95155	−0.156817	−0.0784086	−	0.996921i	$-0.524984\pi$
$997$	−4.95155	−0.156817	−0.0784086	+	0.996921i	$0.524984\pi$
$998$	0	0
$999$	9.23571i	0.292205i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.q.337.12		12
13.5	odd	4		4056.2.a.bf.1.6	✓	6
13.8	odd	4		4056.2.a.bg.1.1	yes	6
13.12	even	2	inner	4056.2.c.q.337.1		12
52.31	even	4		8112.2.a.cv.1.6		6
52.47	even	4		8112.2.a.cw.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bf.1.6	✓	6	13.5	odd	4
4056.2.a.bg.1.1	yes	6	13.8	odd	4
4056.2.c.q.337.1		12	13.12	even	2	inner
4056.2.c.q.337.12		12	1.1	even	1	trivial
8112.2.a.cv.1.6		6	52.31	even	4
8112.2.a.cw.1.1		6	52.47	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.90568i q^{5} +0.0794899i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.90568i q^{5} +0.0794899i q^{7} +1.00000 q^{9} -1.50475i q^{11} -3.90568i q^{15} +6.52695 q^{17} -0.786340i q^{19} -0.0794899i q^{21} +7.03409 q^{23} -10.2543 q^{25} -1.00000 q^{27} +6.15266 q^{29} +1.43887i q^{31} +1.50475i q^{33} -0.310462 q^{35} -9.23571i q^{37} +7.75379i q^{41} -1.06562 q^{43} +3.90568i q^{45} -12.7564i q^{47} +6.99368 q^{49} -6.52695 q^{51} +2.73092 q^{53} +5.87709 q^{55} +0.786340i q^{57} -10.3775i q^{59} +7.92745 q^{61} +0.0794899i q^{63} -7.80912i q^{67} -7.03409 q^{69} -15.1722i q^{71} -4.16507i q^{73} +10.2543 q^{75} +0.119613 q^{77} +3.83731 q^{79} +1.00000 q^{81} +9.04179i q^{83} +25.4922i q^{85} -6.15266 q^{87} +4.75082i q^{89} -1.43887i q^{93} +3.07119 q^{95} +6.94297i q^{97} -1.50475i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 + 12 * q^9	\( 12 q - 12 q^{3} + 12 q^{9} + 10 q^{17} + 12 q^{23} - 34 q^{25} - 12 q^{27} + 6 q^{29} + 8 q^{35} - 22 q^{43} - 10 q^{49} - 10 q^{51} + 24 q^{53} + 14 q^{55} + 34 q^{61} - 12 q^{69} + 34 q^{75} + 50 q^{77} + 28 q^{79} + 12 q^{81} - 6 q^{87} - 86 q^{95}+O(q^{100}) \) 12 * q - 12 * q^3 + 12 * q^9 + 10 * q^17 + 12 * q^23 - 34 * q^25 - 12 * q^27 + 6 * q^29 + 8 * q^35 - 22 * q^43 - 10 * q^49 - 10 * q^51 + 24 * q^53 + 14 * q^55 + 34 * q^61 - 12 * q^69 + 34 * q^75 + 50 * q^77 + 28 * q^79 + 12 * q^81 - 6 * q^87 - 86 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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