LMFDB - Embedded newform 819.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("819.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.a (trivial)

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:	yes
Analytic conductor:	$6.53974792554$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{7} -4.41421 q^{8} +O(q^{10})$	$q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{7} -4.41421 q^{8} -2.00000 q^{11} -1.00000 q^{13} -2.41421 q^{14} +3.00000 q^{16} +0.828427 q^{17} -5.65685 q^{19} +4.82843 q^{22} -1.17157 q^{23} -5.00000 q^{25} +2.41421 q^{26} +3.82843 q^{28} +3.65685 q^{29} +1.65685 q^{31} +1.58579 q^{32} -2.00000 q^{34} -7.65685 q^{37} +13.6569 q^{38} -5.65685 q^{41} +9.65685 q^{43} -7.65685 q^{44} +2.82843 q^{46} -3.17157 q^{47} +1.00000 q^{49} +12.0711 q^{50} -3.82843 q^{52} -3.65685 q^{53} -4.41421 q^{56} -8.82843 q^{58} +10.4853 q^{59} -0.343146 q^{61} -4.00000 q^{62} -9.82843 q^{64} +4.00000 q^{67} +3.17157 q^{68} -14.0000 q^{71} -7.65685 q^{73} +18.4853 q^{74} -21.6569 q^{76} -2.00000 q^{77} -11.3137 q^{79} +13.6569 q^{82} -16.1421 q^{83} -23.3137 q^{86} +8.82843 q^{88} -15.3137 q^{89} -1.00000 q^{91} -4.48528 q^{92} +7.65685 q^{94} -2.00000 q^{97} -2.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 6 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 6 q^{16} - 4 q^{17} + 4 q^{22} - 8 q^{23} - 10 q^{25} + 2 q^{26} + 2 q^{28} - 4 q^{29} - 8 q^{31} + 6 q^{32} - 4 q^{34} - 4 q^{37} + 16 q^{38} + 8 q^{43} - 4 q^{44} - 12 q^{47} + 2 q^{49} + 10 q^{50} - 2 q^{52} + 4 q^{53} - 6 q^{56} - 12 q^{58} + 4 q^{59} - 12 q^{61} - 8 q^{62} - 14 q^{64} + 8 q^{67} + 12 q^{68} - 28 q^{71} - 4 q^{73} + 20 q^{74} - 32 q^{76} - 4 q^{77} + 16 q^{82} - 4 q^{83} - 24 q^{86} + 12 q^{88} - 8 q^{89} - 2 q^{91} + 8 q^{92} + 4 q^{94} - 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 6 * q^8 - 4 * q^11 - 2 * q^13 - 2 * q^14 + 6 * q^16 - 4 * q^17 + 4 * q^22 - 8 * q^23 - 10 * q^25 + 2 * q^26 + 2 * q^28 - 4 * q^29 - 8 * q^31 + 6 * q^32 - 4 * q^34 - 4 * q^37 + 16 * q^38 + 8 * q^43 - 4 * q^44 - 12 * q^47 + 2 * q^49 + 10 * q^50 - 2 * q^52 + 4 * q^53 - 6 * q^56 - 12 * q^58 + 4 * q^59 - 12 * q^61 - 8 * q^62 - 14 * q^64 + 8 * q^67 + 12 * q^68 - 28 * q^71 - 4 * q^73 + 20 * q^74 - 32 * q^76 - 4 * q^77 + 16 * q^82 - 4 * q^83 - 24 * q^86 + 12 * q^88 - 8 * q^89 - 2 * q^91 + 8 * q^92 + 4 * q^94 - 4 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−4.41421	−1.56066
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.41421	−0.645226
$15$	0	0
$16$	3.00000	0.750000
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	4.82843	1.02942
$23$	−1.17157	−0.244290	−0.122145	−	0.992512i	$-0.538977\pi$
$23$	−1.17157	−0.244290	−0.122145	+	0.992512i	$0.538977\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	2.41421	0.473466
$27$	0	0
$28$	3.82843	0.723505
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	1.65685	0.297580	0.148790	−	0.988869i	$-0.452462\pi$
$31$	1.65685	0.297580	0.148790	+	0.988869i	$0.452462\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	13.6569	2.21543
$39$	0	0
$40$	0	0
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	−7.65685	−1.15431
$45$	0	0
$46$	2.82843	0.417029
$47$	−3.17157	−0.462621	−0.231311	−	0.972880i	$-0.574301\pi$
$47$	−3.17157	−0.462621	−0.231311	+	0.972880i	$0.574301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	12.0711	1.70711
$51$	0	0
$52$	−3.82843	−0.530907
$53$	−3.65685	−0.502308	−0.251154	−	0.967947i	$-0.580810\pi$
$53$	−3.65685	−0.502308	−0.251154	+	0.967947i	$0.580810\pi$
$54$	0	0
$55$	0	0
$56$	−4.41421	−0.589874
$57$	0	0
$58$	−8.82843	−1.15923
$59$	10.4853	1.36507	0.682534	−	0.730854i	$-0.260878\pi$
$59$	10.4853	1.36507	0.682534	+	0.730854i	$0.260878\pi$
$60$	0	0
$61$	−0.343146	−0.0439353	−0.0219677	−	0.999759i	$-0.506993\pi$
$61$	−0.343146	−0.0439353	−0.0219677	+	0.999759i	$0.506993\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	−9.82843	−1.22855
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	3.17157	0.384610
$69$	0	0
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	−7.65685	−0.896167	−0.448084	−	0.893992i	$-0.647893\pi$
$73$	−7.65685	−0.896167	−0.448084	+	0.893992i	$0.647893\pi$
$74$	18.4853	2.14887
$75$	0	0
$76$	−21.6569	−2.48421
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	0	0
$82$	13.6569	1.50815
$83$	−16.1421	−1.77183	−0.885915	−	0.463848i	$-0.846468\pi$
$83$	−16.1421	−1.77183	−0.885915	+	0.463848i	$0.846468\pi$
$84$	0	0
$85$	0	0
$86$	−23.3137	−2.51398
$87$	0	0
$88$	8.82843	0.941113
$89$	−15.3137	−1.62325	−0.811625	−	0.584179i	$-0.801417\pi$
$89$	−15.3137	−1.62325	−0.811625	+	0.584179i	$0.801417\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−4.48528	−0.467623
$93$	0	0
$94$	7.65685	0.789744
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−2.41421	−0.243872
$99$	0	0
$100$	−19.1421	−1.91421
$101$	−4.82843	−0.480446	−0.240223	−	0.970718i	$-0.577221\pi$
$101$	−4.82843	−0.480446	−0.240223	+	0.970718i	$0.577221\pi$
$102$	0	0
$103$	13.6569	1.34565	0.672825	−	0.739802i	$-0.265081\pi$
$103$	13.6569	1.34565	0.672825	+	0.739802i	$0.265081\pi$
$104$	4.41421	0.432849
$105$	0	0
$106$	8.82843	0.857493
$107$	−2.82843	−0.273434	−0.136717	−	0.990610i	$-0.543655\pi$
$107$	−2.82843	−0.273434	−0.136717	+	0.990610i	$0.543655\pi$
$108$	0	0
$109$	−11.6569	−1.11652	−0.558262	−	0.829665i	$-0.688532\pi$
$109$	−11.6569	−1.11652	−0.558262	+	0.829665i	$0.688532\pi$
$110$	0	0
$111$	0	0
$112$	3.00000	0.283473
$113$	−0.343146	−0.0322804	−0.0161402	−	0.999870i	$-0.505138\pi$
$113$	−0.343146	−0.0322804	−0.0161402	+	0.999870i	$0.505138\pi$
$114$	0	0
$115$	0	0
$116$	14.0000	1.29987
$117$	0	0
$118$	−25.3137	−2.33032
$119$	0.828427	0.0759418
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0.828427	0.0750023
$123$	0	0
$124$	6.34315	0.569631
$125$	0	0
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	−5.65685	−0.490511
$134$	−9.65685	−0.834225
$135$	0	0
$136$	−3.65685	−0.313573
$137$	22.1421	1.89173	0.945865	−	0.324560i	$-0.105216\pi$
$137$	22.1421	1.89173	0.945865	+	0.324560i	$0.105216\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	0	0
$141$	0	0
$142$	33.7990	2.83635
$143$	2.00000	0.167248
$144$	0	0
$145$	0	0
$146$	18.4853	1.52985
$147$	0	0
$148$	−29.3137	−2.40957
$149$	−4.48528	−0.367449	−0.183724	−	0.982978i	$-0.558815\pi$
$149$	−4.48528	−0.367449	−0.183724	+	0.982978i	$0.558815\pi$
$150$	0	0
$151$	0.686292	0.0558496	0.0279248	−	0.999610i	$-0.491110\pi$
$151$	0.686292	0.0558496	0.0279248	+	0.999610i	$0.491110\pi$
$152$	24.9706	2.02538
$153$	0	0
$154$	4.82843	0.389086
$155$	0	0
$156$	0	0
$157$	9.31371	0.743315	0.371657	−	0.928370i	$-0.378790\pi$
$157$	9.31371	0.743315	0.371657	+	0.928370i	$0.378790\pi$
$158$	27.3137	2.17296
$159$	0	0
$160$	0	0
$161$	−1.17157	−0.0923329
$162$	0	0
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	−21.6569	−1.69112
$165$	0	0
$166$	38.9706	3.02470
$167$	−11.1716	−0.864482	−0.432241	−	0.901758i	$-0.642277\pi$
$167$	−11.1716	−0.864482	−0.432241	+	0.901758i	$0.642277\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	36.9706	2.81898
$173$	14.4853	1.10130	0.550648	−	0.834738i	$-0.314381\pi$
$173$	14.4853	1.10130	0.550648	+	0.834738i	$0.314381\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	−6.00000	−0.452267
$177$	0	0
$178$	36.9706	2.77106
$179$	−10.1421	−0.758059	−0.379029	−	0.925385i	$-0.623742\pi$
$179$	−10.1421	−0.758059	−0.379029	+	0.925385i	$0.623742\pi$
$180$	0	0
$181$	−4.34315	−0.322823	−0.161412	−	0.986887i	$-0.551605\pi$
$181$	−4.34315	−0.322823	−0.161412	+	0.986887i	$0.551605\pi$
$182$	2.41421	0.178953
$183$	0	0
$184$	5.17157	0.381253
$185$	0	0
$186$	0	0
$187$	−1.65685	−0.121161
$188$	−12.1421	−0.885556
$189$	0	0
$190$	0	0
$191$	26.8284	1.94124	0.970618	−	0.240624i	$-0.0773520\pi$
$191$	26.8284	1.94124	0.970618	+	0.240624i	$0.0773520\pi$
$192$	0	0
$193$	26.9706	1.94138	0.970692	−	0.240328i	$-0.0772549\pi$
$193$	26.9706	1.94138	0.970692	+	0.240328i	$0.0772549\pi$
$194$	4.82843	0.346661
$195$	0	0
$196$	3.82843	0.273459
$197$	−2.82843	−0.201517	−0.100759	−	0.994911i	$-0.532127\pi$
$197$	−2.82843	−0.201517	−0.100759	+	0.994911i	$0.532127\pi$
$198$	0	0
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	22.0711	1.56066
$201$	0	0
$202$	11.6569	0.820173
$203$	3.65685	0.256661
$204$	0	0
$205$	0	0
$206$	−32.9706	−2.29717
$207$	0	0
$208$	−3.00000	−0.208013
$209$	11.3137	0.782586
$210$	0	0
$211$	−6.34315	−0.436680	−0.218340	−	0.975873i	$-0.570064\pi$
$211$	−6.34315	−0.436680	−0.218340	+	0.975873i	$0.570064\pi$
$212$	−14.0000	−0.961524
$213$	0	0
$214$	6.82843	0.466782
$215$	0	0
$216$	0	0
$217$	1.65685	0.112475
$218$	28.1421	1.90603
$219$	0	0
$220$	0	0
$221$	−0.828427	−0.0557260
$222$	0	0
$223$	17.6569	1.18239	0.591195	−	0.806529i	$-0.298656\pi$
$223$	17.6569	1.18239	0.591195	+	0.806529i	$0.298656\pi$
$224$	1.58579	0.105955
$225$	0	0
$226$	0.828427	0.0551062
$227$	2.48528	0.164954	0.0824770	−	0.996593i	$-0.473717\pi$
$227$	2.48528	0.164954	0.0824770	+	0.996593i	$0.473717\pi$
$228$	0	0
$229$	−19.6569	−1.29896	−0.649481	−	0.760378i	$-0.725014\pi$
$229$	−19.6569	−1.29896	−0.649481	+	0.760378i	$0.725014\pi$
$230$	0	0
$231$	0	0
$232$	−16.1421	−1.05978
$233$	−1.31371	−0.0860639	−0.0430320	−	0.999074i	$-0.513702\pi$
$233$	−1.31371	−0.0860639	−0.0430320	+	0.999074i	$0.513702\pi$
$234$	0	0
$235$	0	0
$236$	40.1421	2.61303
$237$	0	0
$238$	−2.00000	−0.129641
$239$	12.3431	0.798412	0.399206	−	0.916861i	$-0.369286\pi$
$239$	12.3431	0.798412	0.399206	+	0.916861i	$0.369286\pi$
$240$	0	0
$241$	−23.6569	−1.52387	−0.761936	−	0.647652i	$-0.775751\pi$
$241$	−23.6569	−1.52387	−0.761936	+	0.647652i	$0.775751\pi$
$242$	16.8995	1.08634
$243$	0	0
$244$	−1.31371	−0.0841016
$245$	0	0
$246$	0	0
$247$	5.65685	0.359937
$248$	−7.31371	−0.464421
$249$	0	0
$250$	0	0
$251$	22.6274	1.42823	0.714115	−	0.700028i	$-0.246829\pi$
$251$	22.6274	1.42823	0.714115	+	0.700028i	$0.246829\pi$
$252$	0	0
$253$	2.34315	0.147312
$254$	−38.6274	−2.42370
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−3.17157	−0.197837	−0.0989186	−	0.995096i	$-0.531538\pi$
$257$	−3.17157	−0.197837	−0.0989186	+	0.995096i	$0.531538\pi$
$258$	0	0
$259$	−7.65685	−0.475774
$260$	0	0
$261$	0	0
$262$	40.9706	2.53117
$263$	−4.48528	−0.276574	−0.138287	−	0.990392i	$-0.544160\pi$
$263$	−4.48528	−0.276574	−0.138287	+	0.990392i	$0.544160\pi$
$264$	0	0
$265$	0	0
$266$	13.6569	0.837355
$267$	0	0
$268$	15.3137	0.935434
$269$	−10.4853	−0.639299	−0.319649	−	0.947536i	$-0.603565\pi$
$269$	−10.4853	−0.639299	−0.319649	+	0.947536i	$0.603565\pi$
$270$	0	0
$271$	22.6274	1.37452	0.687259	−	0.726413i	$-0.258814\pi$
$271$	22.6274	1.37452	0.687259	+	0.726413i	$0.258814\pi$
$272$	2.48528	0.150692
$273$	0	0
$274$	−53.4558	−3.22939
$275$	10.0000	0.603023
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	42.6274	2.55662
$279$	0	0
$280$	0	0
$281$	−16.4853	−0.983429	−0.491715	−	0.870756i	$-0.663630\pi$
$281$	−16.4853	−0.983429	−0.491715	+	0.870756i	$0.663630\pi$
$282$	0	0
$283$	−25.6569	−1.52514	−0.762571	−	0.646905i	$-0.776063\pi$
$283$	−25.6569	−1.52514	−0.762571	+	0.646905i	$0.776063\pi$
$284$	−53.5980	−3.18045
$285$	0	0
$286$	−4.82843	−0.285511
$287$	−5.65685	−0.333914
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	0	0
$292$	−29.3137	−1.71546
$293$	7.31371	0.427271	0.213636	−	0.976913i	$-0.431469\pi$
$293$	7.31371	0.427271	0.213636	+	0.976913i	$0.431469\pi$
$294$	0	0
$295$	0	0
$296$	33.7990	1.96453
$297$	0	0
$298$	10.8284	0.627274
$299$	1.17157	0.0677538
$300$	0	0
$301$	9.65685	0.556612
$302$	−1.65685	−0.0953412
$303$	0	0
$304$	−16.9706	−0.973329
$305$	0	0
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	−7.65685	−0.436290
$309$	0	0
$310$	0	0
$311$	31.3137	1.77564	0.887819	−	0.460193i	$-0.152220\pi$
$311$	31.3137	1.77564	0.887819	+	0.460193i	$0.152220\pi$
$312$	0	0
$313$	−10.9706	−0.620093	−0.310046	−	0.950721i	$-0.600345\pi$
$313$	−10.9706	−0.620093	−0.310046	+	0.950721i	$0.600345\pi$
$314$	−22.4853	−1.26892
$315$	0	0
$316$	−43.3137	−2.43659
$317$	0.485281	0.0272561	0.0136281	−	0.999907i	$-0.495662\pi$
$317$	0.485281	0.0272561	0.0136281	+	0.999907i	$0.495662\pi$
$318$	0	0
$319$	−7.31371	−0.409489
$320$	0	0
$321$	0	0
$322$	2.82843	0.157622
$323$	−4.68629	−0.260752
$324$	0	0
$325$	5.00000	0.277350
$326$	27.3137	1.51277
$327$	0	0
$328$	24.9706	1.37877
$329$	−3.17157	−0.174854
$330$	0	0
$331$	26.6274	1.46358	0.731788	−	0.681533i	$-0.238686\pi$
$331$	26.6274	1.46358	0.731788	+	0.681533i	$0.238686\pi$
$332$	−61.7990	−3.39166
$333$	0	0
$334$	26.9706	1.47576
$335$	0	0
$336$	0	0
$337$	17.3137	0.943138	0.471569	−	0.881829i	$-0.343688\pi$
$337$	17.3137	0.943138	0.471569	+	0.881829i	$0.343688\pi$
$338$	−2.41421	−0.131316
$339$	0	0
$340$	0	0
$341$	−3.31371	−0.179447
$342$	0	0
$343$	1.00000	0.0539949
$344$	−42.6274	−2.29832
$345$	0	0
$346$	−34.9706	−1.88003
$347$	1.17157	0.0628933	0.0314467	−	0.999505i	$-0.489989\pi$
$347$	1.17157	0.0628933	0.0314467	+	0.999505i	$0.489989\pi$
$348$	0	0
$349$	−13.3137	−0.712666	−0.356333	−	0.934359i	$-0.615973\pi$
$349$	−13.3137	−0.712666	−0.356333	+	0.934359i	$0.615973\pi$
$350$	12.0711	0.645226
$351$	0	0
$352$	−3.17157	−0.169045
$353$	−11.3137	−0.602168	−0.301084	−	0.953598i	$-0.597348\pi$
$353$	−11.3137	−0.602168	−0.301084	+	0.953598i	$0.597348\pi$
$354$	0	0
$355$	0	0
$356$	−58.6274	−3.10725
$357$	0	0
$358$	24.4853	1.29409
$359$	−25.3137	−1.33601	−0.668003	−	0.744158i	$-0.732851\pi$
$359$	−25.3137	−1.33601	−0.668003	+	0.744158i	$0.732851\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	10.4853	0.551094
$363$	0	0
$364$	−3.82843	−0.200664
$365$	0	0
$366$	0	0
$367$	−22.6274	−1.18114	−0.590571	−	0.806986i	$-0.701097\pi$
$367$	−22.6274	−1.18114	−0.590571	+	0.806986i	$0.701097\pi$
$368$	−3.51472	−0.183217
$369$	0	0
$370$	0	0
$371$	−3.65685	−0.189854
$372$	0	0
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	14.0000	0.721995
$377$	−3.65685	−0.188338
$378$	0	0
$379$	27.3137	1.40301	0.701505	−	0.712664i	$-0.252512\pi$
$379$	27.3137	1.40301	0.701505	+	0.712664i	$0.252512\pi$
$380$	0	0
$381$	0	0
$382$	−64.7696	−3.31390
$383$	−15.1716	−0.775231	−0.387616	−	0.921821i	$-0.626701\pi$
$383$	−15.1716	−0.775231	−0.387616	+	0.921821i	$0.626701\pi$
$384$	0	0
$385$	0	0
$386$	−65.1127	−3.31415
$387$	0	0
$388$	−7.65685	−0.388718
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	−0.970563	−0.0490835
$392$	−4.41421	−0.222951
$393$	0	0
$394$	6.82843	0.344011
$395$	0	0
$396$	0	0
$397$	−20.3431	−1.02099	−0.510497	−	0.859880i	$-0.670538\pi$
$397$	−20.3431	−1.02099	−0.510497	+	0.859880i	$0.670538\pi$
$398$	−40.9706	−2.05367
$399$	0	0
$400$	−15.0000	−0.750000
$401$	35.1127	1.75344	0.876722	−	0.480997i	$-0.159725\pi$
$401$	35.1127	1.75344	0.876722	+	0.480997i	$0.159725\pi$
$402$	0	0
$403$	−1.65685	−0.0825338
$404$	−18.4853	−0.919677
$405$	0	0
$406$	−8.82843	−0.438147
$407$	15.3137	0.759072
$408$	0	0
$409$	9.31371	0.460533	0.230267	−	0.973128i	$-0.426040\pi$
$409$	9.31371	0.460533	0.230267	+	0.973128i	$0.426040\pi$
$410$	0	0
$411$	0	0
$412$	52.2843	2.57586
$413$	10.4853	0.515947
$414$	0	0
$415$	0	0
$416$	−1.58579	−0.0777496
$417$	0	0
$418$	−27.3137	−1.33596
$419$	33.9411	1.65813	0.829066	−	0.559150i	$-0.188873\pi$
$419$	33.9411	1.65813	0.829066	+	0.559150i	$0.188873\pi$
$420$	0	0
$421$	16.6274	0.810371	0.405185	−	0.914235i	$-0.367207\pi$
$421$	16.6274	0.810371	0.405185	+	0.914235i	$0.367207\pi$
$422$	15.3137	0.745460
$423$	0	0
$424$	16.1421	0.783931
$425$	−4.14214	−0.200923
$426$	0	0
$427$	−0.343146	−0.0166060
$428$	−10.8284	−0.523412
$429$	0	0
$430$	0	0
$431$	14.9706	0.721107	0.360553	−	0.932739i	$-0.382588\pi$
$431$	14.9706	0.721107	0.360553	+	0.932739i	$0.382588\pi$
$432$	0	0
$433$	19.6569	0.944648	0.472324	−	0.881425i	$-0.343415\pi$
$433$	19.6569	0.944648	0.472324	+	0.881425i	$0.343415\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−44.6274	−2.13727
$437$	6.62742	0.317032
$438$	0	0
$439$	−3.31371	−0.158155	−0.0790773	−	0.996868i	$-0.525197\pi$
$439$	−3.31371	−0.158155	−0.0790773	+	0.996868i	$0.525197\pi$
$440$	0	0
$441$	0	0
$442$	2.00000	0.0951303
$443$	38.1421	1.81219	0.906094	−	0.423077i	$-0.139050\pi$
$443$	38.1421	1.81219	0.906094	+	0.423077i	$0.139050\pi$
$444$	0	0
$445$	0	0
$446$	−42.6274	−2.01847
$447$	0	0
$448$	−9.82843	−0.464350
$449$	−23.7990	−1.12314	−0.561572	−	0.827428i	$-0.689803\pi$
$449$	−23.7990	−1.12314	−0.561572	+	0.827428i	$0.689803\pi$
$450$	0	0
$451$	11.3137	0.532742
$452$	−1.31371	−0.0617916
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	42.2843	1.97797	0.988987	−	0.148000i	$-0.0472835\pi$
$457$	42.2843	1.97797	0.988987	+	0.148000i	$0.0472835\pi$
$458$	47.4558	2.21747
$459$	0	0
$460$	0	0
$461$	−18.3431	−0.854325	−0.427163	−	0.904175i	$-0.640487\pi$
$461$	−18.3431	−0.854325	−0.427163	+	0.904175i	$0.640487\pi$
$462$	0	0
$463$	27.3137	1.26938	0.634688	−	0.772769i	$-0.281129\pi$
$463$	27.3137	1.26938	0.634688	+	0.772769i	$0.281129\pi$
$464$	10.9706	0.509296
$465$	0	0
$466$	3.17157	0.146920
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	−46.2843	−2.13041
$473$	−19.3137	−0.888045
$474$	0	0
$475$	28.2843	1.29777
$476$	3.17157	0.145369
$477$	0	0
$478$	−29.7990	−1.36297
$479$	6.48528	0.296320	0.148160	−	0.988963i	$-0.452665\pi$
$479$	6.48528	0.296320	0.148160	+	0.988963i	$0.452665\pi$
$480$	0	0
$481$	7.65685	0.349123
$482$	57.1127	2.60141
$483$	0	0
$484$	−26.7990	−1.21814
$485$	0	0
$486$	0	0
$487$	31.3137	1.41896	0.709480	−	0.704726i	$-0.248930\pi$
$487$	31.3137	1.41896	0.709480	+	0.704726i	$0.248930\pi$
$488$	1.51472	0.0685681
$489$	0	0
$490$	0	0
$491$	23.5147	1.06120	0.530602	−	0.847621i	$-0.321966\pi$
$491$	23.5147	1.06120	0.530602	+	0.847621i	$0.321966\pi$
$492$	0	0
$493$	3.02944	0.136439
$494$	−13.6569	−0.614451
$495$	0	0
$496$	4.97056	0.223185
$497$	−14.0000	−0.627986
$498$	0	0
$499$	−7.31371	−0.327407	−0.163703	−	0.986510i	$-0.552344\pi$
$499$	−7.31371	−0.327407	−0.163703	+	0.986510i	$0.552344\pi$
$500$	0	0
$501$	0	0
$502$	−54.6274	−2.43814
$503$	−40.2843	−1.79619	−0.898093	−	0.439805i	$-0.855048\pi$
$503$	−40.2843	−1.79619	−0.898093	+	0.439805i	$0.855048\pi$
$504$	0	0
$505$	0	0
$506$	−5.65685	−0.251478
$507$	0	0
$508$	61.2548	2.71774
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	−7.65685	−0.338719
$512$	31.2426	1.38074
$513$	0	0
$514$	7.65685	0.337729
$515$	0	0
$516$	0	0
$517$	6.34315	0.278971
$518$	18.4853	0.812197
$519$	0	0
$520$	0	0
$521$	5.79899	0.254058	0.127029	−	0.991899i	$-0.459456\pi$
$521$	5.79899	0.254058	0.127029	+	0.991899i	$0.459456\pi$
$522$	0	0
$523$	1.65685	0.0724492	0.0362246	−	0.999344i	$-0.488467\pi$
$523$	1.65685	0.0724492	0.0362246	+	0.999344i	$0.488467\pi$
$524$	−64.9706	−2.83825
$525$	0	0
$526$	10.8284	0.472142
$527$	1.37258	0.0597907
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	0	0
$532$	−21.6569	−0.938944
$533$	5.65685	0.245026
$534$	0	0
$535$	0	0
$536$	−17.6569	−0.762660
$537$	0	0
$538$	25.3137	1.09135
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	5.31371	0.228454	0.114227	−	0.993455i	$-0.463561\pi$
$541$	5.31371	0.228454	0.114227	+	0.993455i	$0.463561\pi$
$542$	−54.6274	−2.34645
$543$	0	0
$544$	1.31371	0.0563248
$545$	0	0
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	84.7696	3.62118
$549$	0	0
$550$	−24.1421	−1.02942
$551$	−20.6863	−0.881266
$552$	0	0
$553$	−11.3137	−0.481108
$554$	24.1421	1.02570
$555$	0	0
$556$	−67.5980	−2.86679
$557$	−44.4853	−1.88490	−0.942451	−	0.334344i	$-0.891485\pi$
$557$	−44.4853	−1.88490	−0.942451	+	0.334344i	$0.891485\pi$
$558$	0	0
$559$	−9.65685	−0.408441
$560$	0	0
$561$	0	0
$562$	39.7990	1.67882
$563$	6.34315	0.267332	0.133666	−	0.991026i	$-0.457325\pi$
$563$	6.34315	0.267332	0.133666	+	0.991026i	$0.457325\pi$
$564$	0	0
$565$	0	0
$566$	61.9411	2.60358
$567$	0	0
$568$	61.7990	2.59303
$569$	−1.31371	−0.0550735	−0.0275368	−	0.999621i	$-0.508766\pi$
$569$	−1.31371	−0.0550735	−0.0275368	+	0.999621i	$0.508766\pi$
$570$	0	0
$571$	10.6274	0.444744	0.222372	−	0.974962i	$-0.428620\pi$
$571$	10.6274	0.444744	0.222372	+	0.974962i	$0.428620\pi$
$572$	7.65685	0.320149
$573$	0	0
$574$	13.6569	0.570026
$575$	5.85786	0.244290
$576$	0	0
$577$	10.9706	0.456711	0.228355	−	0.973578i	$-0.426665\pi$
$577$	10.9706	0.456711	0.228355	+	0.973578i	$0.426665\pi$
$578$	39.3848	1.63819
$579$	0	0
$580$	0	0
$581$	−16.1421	−0.669689
$582$	0	0
$583$	7.31371	0.302903
$584$	33.7990	1.39861
$585$	0	0
$586$	−17.6569	−0.729398
$587$	−25.1127	−1.03651	−0.518256	−	0.855226i	$-0.673419\pi$
$587$	−25.1127	−1.03651	−0.518256	+	0.855226i	$0.673419\pi$
$588$	0	0
$589$	−9.37258	−0.386191
$590$	0	0
$591$	0	0
$592$	−22.9706	−0.944084
$593$	25.6569	1.05360	0.526800	−	0.849989i	$-0.323392\pi$
$593$	25.6569	1.05360	0.526800	+	0.849989i	$0.323392\pi$
$594$	0	0
$595$	0	0
$596$	−17.1716	−0.703375
$597$	0	0
$598$	−2.82843	−0.115663
$599$	20.4853	0.837006	0.418503	−	0.908215i	$-0.362555\pi$
$599$	20.4853	0.837006	0.418503	+	0.908215i	$0.362555\pi$
$600$	0	0
$601$	−10.9706	−0.447499	−0.223749	−	0.974647i	$-0.571830\pi$
$601$	−10.9706	−0.447499	−0.223749	+	0.974647i	$0.571830\pi$
$602$	−23.3137	−0.950196
$603$	0	0
$604$	2.62742	0.106908
$605$	0	0
$606$	0	0
$607$	−22.6274	−0.918419	−0.459209	−	0.888328i	$-0.651867\pi$
$607$	−22.6274	−0.918419	−0.459209	+	0.888328i	$0.651867\pi$
$608$	−8.97056	−0.363804
$609$	0	0
$610$	0	0
$611$	3.17157	0.128308
$612$	0	0
$613$	10.6863	0.431615	0.215808	−	0.976436i	$-0.430762\pi$
$613$	10.6863	0.431615	0.215808	+	0.976436i	$0.430762\pi$
$614$	−9.65685	−0.389719
$615$	0	0
$616$	8.82843	0.355707
$617$	36.7696	1.48029	0.740143	−	0.672449i	$-0.234758\pi$
$617$	36.7696	1.48029	0.740143	+	0.672449i	$0.234758\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	−75.5980	−3.03120
$623$	−15.3137	−0.613531
$624$	0	0
$625$	25.0000	1.00000
$626$	26.4853	1.05856
$627$	0	0
$628$	35.6569	1.42286
$629$	−6.34315	−0.252918
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	49.9411	1.98655
$633$	0	0
$634$	−1.17157	−0.0465291
$635$	0	0
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	17.6569	0.699042
$639$	0	0
$640$	0	0
$641$	19.6569	0.776399	0.388200	−	0.921575i	$-0.373097\pi$
$641$	19.6569	0.776399	0.388200	+	0.921575i	$0.373097\pi$
$642$	0	0
$643$	7.31371	0.288425	0.144212	−	0.989547i	$-0.453935\pi$
$643$	7.31371	0.288425	0.144212	+	0.989547i	$0.453935\pi$
$644$	−4.48528	−0.176745
$645$	0	0
$646$	11.3137	0.445132
$647$	−12.2843	−0.482945	−0.241472	−	0.970408i	$-0.577630\pi$
$647$	−12.2843	−0.482945	−0.241472	+	0.970408i	$0.577630\pi$
$648$	0	0
$649$	−20.9706	−0.823167
$650$	−12.0711	−0.473466
$651$	0	0
$652$	−43.3137	−1.69630
$653$	−41.5980	−1.62785	−0.813927	−	0.580967i	$-0.802675\pi$
$653$	−41.5980	−1.62785	−0.813927	+	0.580967i	$0.802675\pi$
$654$	0	0
$655$	0	0
$656$	−16.9706	−0.662589
$657$	0	0
$658$	7.65685	0.298495
$659$	29.4558	1.14744	0.573718	−	0.819053i	$-0.305500\pi$
$659$	29.4558	1.14744	0.573718	+	0.819053i	$0.305500\pi$
$660$	0	0
$661$	9.31371	0.362261	0.181131	−	0.983459i	$-0.442024\pi$
$661$	9.31371	0.362261	0.181131	+	0.983459i	$0.442024\pi$
$662$	−64.2843	−2.49848
$663$	0	0
$664$	71.2548	2.76522
$665$	0	0
$666$	0	0
$667$	−4.28427	−0.165888
$668$	−42.7696	−1.65480
$669$	0	0
$670$	0	0
$671$	0.686292	0.0264940
$672$	0	0
$673$	0.627417	0.0241851	0.0120926	−	0.999927i	$-0.496151\pi$
$673$	0.627417	0.0241851	0.0120926	+	0.999927i	$0.496151\pi$
$674$	−41.7990	−1.61004
$675$	0	0
$676$	3.82843	0.147247
$677$	−14.4853	−0.556715	−0.278357	−	0.960478i	$-0.589790\pi$
$677$	−14.4853	−0.556715	−0.278357	+	0.960478i	$0.589790\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	8.00000	0.306336
$683$	24.3431	0.931465	0.465732	−	0.884926i	$-0.345791\pi$
$683$	24.3431	0.931465	0.465732	+	0.884926i	$0.345791\pi$
$684$	0	0
$685$	0	0
$686$	−2.41421	−0.0921751
$687$	0	0
$688$	28.9706	1.10449
$689$	3.65685	0.139315
$690$	0	0
$691$	−23.3137	−0.886895	−0.443448	−	0.896300i	$-0.646245\pi$
$691$	−23.3137	−0.886895	−0.443448	+	0.896300i	$0.646245\pi$
$692$	55.4558	2.10811
$693$	0	0
$694$	−2.82843	−0.107366
$695$	0	0
$696$	0	0
$697$	−4.68629	−0.177506
$698$	32.1421	1.21660
$699$	0	0
$700$	−19.1421	−0.723505
$701$	0.627417	0.0236972	0.0118486	−	0.999930i	$-0.496228\pi$
$701$	0.627417	0.0236972	0.0118486	+	0.999930i	$0.496228\pi$
$702$	0	0
$703$	43.3137	1.63361
$704$	19.6569	0.740846
$705$	0	0
$706$	27.3137	1.02796
$707$	−4.82843	−0.181592
$708$	0	0
$709$	2.97056	0.111562	0.0557809	−	0.998443i	$-0.482235\pi$
$709$	2.97056	0.111562	0.0557809	+	0.998443i	$0.482235\pi$
$710$	0	0
$711$	0	0
$712$	67.5980	2.53334
$713$	−1.94113	−0.0726957
$714$	0	0
$715$	0	0
$716$	−38.8284	−1.45109
$717$	0	0
$718$	61.1127	2.28071
$719$	−30.3431	−1.13161	−0.565804	−	0.824540i	$-0.691434\pi$
$719$	−30.3431	−1.13161	−0.565804	+	0.824540i	$0.691434\pi$
$720$	0	0
$721$	13.6569	0.508608
$722$	−31.3848	−1.16802
$723$	0	0
$724$	−16.6274	−0.617953
$725$	−18.2843	−0.679061
$726$	0	0
$727$	−30.6274	−1.13591	−0.567954	−	0.823060i	$-0.692265\pi$
$727$	−30.6274	−1.13591	−0.567954	+	0.823060i	$0.692265\pi$
$728$	4.41421	0.163602
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−18.2843	−0.675345	−0.337672	−	0.941264i	$-0.609640\pi$
$733$	−18.2843	−0.675345	−0.337672	+	0.941264i	$0.609640\pi$
$734$	54.6274	2.01633
$735$	0	0
$736$	−1.85786	−0.0684818
$737$	−8.00000	−0.294684
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	8.82843	0.324102
$743$	24.6274	0.903492	0.451746	−	0.892147i	$-0.350801\pi$
$743$	24.6274	0.903492	0.451746	+	0.892147i	$0.350801\pi$
$744$	0	0
$745$	0	0
$746$	61.1127	2.23749
$747$	0	0
$748$	−6.34315	−0.231928
$749$	−2.82843	−0.103348
$750$	0	0
$751$	28.2843	1.03211	0.516054	−	0.856556i	$-0.327400\pi$
$751$	28.2843	1.03211	0.516054	+	0.856556i	$0.327400\pi$
$752$	−9.51472	−0.346966
$753$	0	0
$754$	8.82843	0.321512
$755$	0	0
$756$	0	0
$757$	−20.6274	−0.749716	−0.374858	−	0.927082i	$-0.622309\pi$
$757$	−20.6274	−0.749716	−0.374858	+	0.927082i	$0.622309\pi$
$758$	−65.9411	−2.39509
$759$	0	0
$760$	0	0
$761$	48.9706	1.77518	0.887591	−	0.460633i	$-0.152378\pi$
$761$	48.9706	1.77518	0.887591	+	0.460633i	$0.152378\pi$
$762$	0	0
$763$	−11.6569	−0.422006
$764$	102.711	3.71594
$765$	0	0
$766$	36.6274	1.32340
$767$	−10.4853	−0.378602
$768$	0	0
$769$	−39.9411	−1.44031	−0.720157	−	0.693811i	$-0.755930\pi$
$769$	−39.9411	−1.44031	−0.720157	+	0.693811i	$0.755930\pi$
$770$	0	0
$771$	0	0
$772$	103.255	3.71622
$773$	−20.0000	−0.719350	−0.359675	−	0.933078i	$-0.617112\pi$
$773$	−20.0000	−0.719350	−0.359675	+	0.933078i	$0.617112\pi$
$774$	0	0
$775$	−8.28427	−0.297580
$776$	8.82843	0.316922
$777$	0	0
$778$	53.1127	1.90418
$779$	32.0000	1.14652
$780$	0	0
$781$	28.0000	1.00192
$782$	2.34315	0.0837907
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	−7.31371	−0.260706	−0.130353	−	0.991468i	$-0.541611\pi$
$787$	−7.31371	−0.260706	−0.130353	+	0.991468i	$0.541611\pi$
$788$	−10.8284	−0.385747
$789$	0	0
$790$	0	0
$791$	−0.343146	−0.0122009
$792$	0	0
$793$	0.343146	0.0121855
$794$	49.1127	1.74294
$795$	0	0
$796$	64.9706	2.30282
$797$	16.1421	0.571784	0.285892	−	0.958262i	$-0.407710\pi$
$797$	16.1421	0.571784	0.285892	+	0.958262i	$0.407710\pi$
$798$	0	0
$799$	−2.62742	−0.0929513
$800$	−7.92893	−0.280330
$801$	0	0
$802$	−84.7696	−2.99332
$803$	15.3137	0.540409
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	21.3137	0.749814
$809$	2.97056	0.104439	0.0522197	−	0.998636i	$-0.483370\pi$
$809$	2.97056	0.104439	0.0522197	+	0.998636i	$0.483370\pi$
$810$	0	0
$811$	−28.2843	−0.993195	−0.496598	−	0.867981i	$-0.665417\pi$
$811$	−28.2843	−0.993195	−0.496598	+	0.867981i	$0.665417\pi$
$812$	14.0000	0.491304
$813$	0	0
$814$	−36.9706	−1.29582
$815$	0	0
$816$	0	0
$817$	−54.6274	−1.91117
$818$	−22.4853	−0.786179
$819$	0	0
$820$	0	0
$821$	21.4558	0.748814	0.374407	−	0.927264i	$-0.377846\pi$
$821$	21.4558	0.748814	0.374407	+	0.927264i	$0.377846\pi$
$822$	0	0
$823$	22.6274	0.788742	0.394371	−	0.918951i	$-0.370962\pi$
$823$	22.6274	0.788742	0.394371	+	0.918951i	$0.370962\pi$
$824$	−60.2843	−2.10010
$825$	0	0
$826$	−25.3137	−0.880777
$827$	22.0000	0.765015	0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000	0.765015	0.382507	+	0.923952i	$0.375061\pi$
$828$	0	0
$829$	−41.3137	−1.43488	−0.717442	−	0.696618i	$-0.754687\pi$
$829$	−41.3137	−1.43488	−0.717442	+	0.696618i	$0.754687\pi$
$830$	0	0
$831$	0	0
$832$	9.82843	0.340739
$833$	0.828427	0.0287033
$834$	0	0
$835$	0	0
$836$	43.3137	1.49804
$837$	0	0
$838$	−81.9411	−2.83061
$839$	49.7990	1.71925	0.859626	−	0.510924i	$-0.170697\pi$
$839$	49.7990	1.71925	0.859626	+	0.510924i	$0.170697\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	−40.1421	−1.38339
$843$	0	0
$844$	−24.2843	−0.835899
$845$	0	0
$846$	0	0
$847$	−7.00000	−0.240523
$848$	−10.9706	−0.376731
$849$	0	0
$850$	10.0000	0.342997
$851$	8.97056	0.307507
$852$	0	0
$853$	6.97056	0.238668	0.119334	−	0.992854i	$-0.461924\pi$
$853$	6.97056	0.238668	0.119334	+	0.992854i	$0.461924\pi$
$854$	0.828427	0.0283482
$855$	0	0
$856$	12.4853	0.426738
$857$	−37.1127	−1.26775	−0.633873	−	0.773437i	$-0.718536\pi$
$857$	−37.1127	−1.26775	−0.633873	+	0.773437i	$0.718536\pi$
$858$	0	0
$859$	−28.9706	−0.988463	−0.494231	−	0.869330i	$-0.664550\pi$
$859$	−28.9706	−0.988463	−0.494231	+	0.869330i	$0.664550\pi$
$860$	0	0
$861$	0	0
$862$	−36.1421	−1.23101
$863$	43.2548	1.47241	0.736206	−	0.676758i	$-0.236616\pi$
$863$	43.2548	1.47241	0.736206	+	0.676758i	$0.236616\pi$
$864$	0	0
$865$	0	0
$866$	−47.4558	−1.61262
$867$	0	0
$868$	6.34315	0.215300
$869$	22.6274	0.767583
$870$	0	0
$871$	−4.00000	−0.135535
$872$	51.4558	1.74251
$873$	0	0
$874$	−16.0000	−0.541208
$875$	0	0
$876$	0	0
$877$	18.6863	0.630991	0.315496	−	0.948927i	$-0.397829\pi$
$877$	18.6863	0.630991	0.315496	+	0.948927i	$0.397829\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	0	0
$881$	−34.7696	−1.17142	−0.585708	−	0.810522i	$-0.699183\pi$
$881$	−34.7696	−1.17142	−0.585708	+	0.810522i	$0.699183\pi$
$882$	0	0
$883$	−8.28427	−0.278788	−0.139394	−	0.990237i	$-0.544515\pi$
$883$	−8.28427	−0.278788	−0.139394	+	0.990237i	$0.544515\pi$
$884$	−3.17157	−0.106672
$885$	0	0
$886$	−92.0833	−3.09360
$887$	−48.9706	−1.64427	−0.822135	−	0.569292i	$-0.807217\pi$
$887$	−48.9706	−1.64427	−0.822135	+	0.569292i	$0.807217\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	67.5980	2.26335
$893$	17.9411	0.600377
$894$	0	0
$895$	0	0
$896$	20.5563	0.686739
$897$	0	0
$898$	57.4558	1.91733
$899$	6.05887	0.202075
$900$	0	0
$901$	−3.02944	−0.100925
$902$	−27.3137	−0.909447
$903$	0	0
$904$	1.51472	0.0503788
$905$	0	0
$906$	0	0
$907$	−49.6569	−1.64883	−0.824414	−	0.565987i	$-0.808495\pi$
$907$	−49.6569	−1.64883	−0.824414	+	0.565987i	$0.808495\pi$
$908$	9.51472	0.315757
$909$	0	0
$910$	0	0
$911$	−7.11270	−0.235654	−0.117827	−	0.993034i	$-0.537593\pi$
$911$	−7.11270	−0.235654	−0.117827	+	0.993034i	$0.537593\pi$
$912$	0	0
$913$	32.2843	1.06845
$914$	−102.083	−3.37661
$915$	0	0
$916$	−75.2548	−2.48649
$917$	−16.9706	−0.560417
$918$	0	0
$919$	−40.9706	−1.35149	−0.675747	−	0.737134i	$-0.736179\pi$
$919$	−40.9706	−1.35149	−0.675747	+	0.737134i	$0.736179\pi$
$920$	0	0
$921$	0	0
$922$	44.2843	1.45842
$923$	14.0000	0.460816
$924$	0	0
$925$	38.2843	1.25878
$926$	−65.9411	−2.16696
$927$	0	0
$928$	5.79899	0.190361
$929$	−0.970563	−0.0318431	−0.0159216	−	0.999873i	$-0.505068\pi$
$929$	−0.970563	−0.0318431	−0.0159216	+	0.999873i	$0.505068\pi$
$930$	0	0
$931$	−5.65685	−0.185396
$932$	−5.02944	−0.164745
$933$	0	0
$934$	−19.3137	−0.631964
$935$	0	0
$936$	0	0
$937$	14.9706	0.489067	0.244533	−	0.969641i	$-0.421365\pi$
$937$	14.9706	0.489067	0.244533	+	0.969641i	$0.421365\pi$
$938$	−9.65685	−0.315307
$939$	0	0
$940$	0	0
$941$	−41.6569	−1.35797	−0.678987	−	0.734150i	$-0.737581\pi$
$941$	−41.6569	−1.35797	−0.678987	+	0.734150i	$0.737581\pi$
$942$	0	0
$943$	6.62742	0.215818
$944$	31.4558	1.02380
$945$	0	0
$946$	46.6274	1.51599
$947$	−6.97056	−0.226513	−0.113256	−	0.993566i	$-0.536128\pi$
$947$	−6.97056	−0.226513	−0.113256	+	0.993566i	$0.536128\pi$
$948$	0	0
$949$	7.65685	0.248552
$950$	−68.2843	−2.21543
$951$	0	0
$952$	−3.65685	−0.118519
$953$	28.3431	0.918125	0.459062	−	0.888404i	$-0.348185\pi$
$953$	28.3431	0.918125	0.459062	+	0.888404i	$0.348185\pi$
$954$	0	0
$955$	0	0
$956$	47.2548	1.52833
$957$	0	0
$958$	−15.6569	−0.505850
$959$	22.1421	0.715007
$960$	0	0
$961$	−28.2548	−0.911446
$962$	−18.4853	−0.595989
$963$	0	0
$964$	−90.5685	−2.91702
$965$	0	0
$966$	0	0
$967$	5.94113	0.191054	0.0955269	−	0.995427i	$-0.469546\pi$
$967$	5.94113	0.191054	0.0955269	+	0.995427i	$0.469546\pi$
$968$	30.8995	0.993147
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−17.6569	−0.566053
$974$	−75.5980	−2.42232
$975$	0	0
$976$	−1.02944	−0.0329515
$977$	−42.1421	−1.34825	−0.674123	−	0.738619i	$-0.735478\pi$
$977$	−42.1421	−1.34825	−0.674123	+	0.738619i	$0.735478\pi$
$978$	0	0
$979$	30.6274	0.978856
$980$	0	0
$981$	0	0
$982$	−56.7696	−1.81159
$983$	−49.1127	−1.56645	−0.783226	−	0.621737i	$-0.786427\pi$
$983$	−49.1127	−1.56645	−0.783226	+	0.621737i	$0.786427\pi$
$984$	0	0
$985$	0	0
$986$	−7.31371	−0.232916
$987$	0	0
$988$	21.6569	0.688996
$989$	−11.3137	−0.359755
$990$	0	0
$991$	−53.6569	−1.70447	−0.852233	−	0.523162i	$-0.824752\pi$
$991$	−53.6569	−1.70447	−0.852233	+	0.523162i	$0.824752\pi$
$992$	2.62742	0.0834206
$993$	0	0
$994$	33.7990	1.07204
$995$	0	0
$996$	0	0
$997$	10.6863	0.338438	0.169219	−	0.985578i	$-0.445875\pi$
$997$	10.6863	0.338438	0.169219	+	0.985578i	$0.445875\pi$
$998$	17.6569	0.558918
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.a.g.1.1		2
3.2	odd	2		273.2.a.c.1.2	✓	2
7.6	odd	2		5733.2.a.n.1.1		2
12.11	even	2		4368.2.a.bj.1.1		2
15.14	odd	2		6825.2.a.m.1.1		2
21.20	even	2		1911.2.a.k.1.2		2
39.38	odd	2		3549.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.c.1.2	✓	2	3.2	odd	2
819.2.a.g.1.1		2	1.1	even	1	trivial
1911.2.a.k.1.2		2	21.20	even	2
3549.2.a.f.1.1		2	39.38	odd	2
4368.2.a.bj.1.1		2	12.11	even	2
5733.2.a.n.1.1		2	7.6	odd	2
6825.2.a.m.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{7} -4.41421 q^{8} -2.00000 q^{11} -1.00000 q^{13} -2.41421 q^{14} +3.00000 q^{16} +0.828427 q^{17} -5.65685 q^{19} +4.82843 q^{22} -1.17157 q^{23} -5.00000 q^{25} +2.41421 q^{26} +3.82843 q^{28} +3.65685 q^{29} +1.65685 q^{31} +1.58579 q^{32} -2.00000 q^{34} -7.65685 q^{37} +13.6569 q^{38} -5.65685 q^{41} +9.65685 q^{43} -7.65685 q^{44} +2.82843 q^{46} -3.17157 q^{47} +1.00000 q^{49} +12.0711 q^{50} -3.82843 q^{52} -3.65685 q^{53} -4.41421 q^{56} -8.82843 q^{58} +10.4853 q^{59} -0.343146 q^{61} -4.00000 q^{62} -9.82843 q^{64} +4.00000 q^{67} +3.17157 q^{68} -14.0000 q^{71} -7.65685 q^{73} +18.4853 q^{74} -21.6569 q^{76} -2.00000 q^{77} -11.3137 q^{79} +13.6569 q^{82} -16.1421 q^{83} -23.3137 q^{86} +8.82843 q^{88} -15.3137 q^{89} -1.00000 q^{91} -4.48528 q^{92} +7.65685 q^{94} -2.00000 q^{97} -2.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 6 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 6 q^{16} - 4 q^{17} + 4 q^{22} - 8 q^{23} - 10 q^{25} + 2 q^{26} + 2 q^{28} - 4 q^{29} - 8 q^{31} + 6 q^{32} - 4 q^{34} - 4 q^{37} + 16 q^{38} + 8 q^{43} - 4 q^{44} - 12 q^{47} + 2 q^{49} + 10 q^{50} - 2 q^{52} + 4 q^{53} - 6 q^{56} - 12 q^{58} + 4 q^{59} - 12 q^{61} - 8 q^{62} - 14 q^{64} + 8 q^{67} + 12 q^{68} - 28 q^{71} - 4 q^{73} + 20 q^{74} - 32 q^{76} - 4 q^{77} + 16 q^{82} - 4 q^{83} - 24 q^{86} + 12 q^{88} - 8 q^{89} - 2 q^{91} + 8 q^{92} + 4 q^{94} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 6 * q^8 - 4 * q^11 - 2 * q^13 - 2 * q^14 + 6 * q^16 - 4 * q^17 + 4 * q^22 - 8 * q^23 - 10 * q^25 + 2 * q^26 + 2 * q^28 - 4 * q^29 - 8 * q^31 + 6 * q^32 - 4 * q^34 - 4 * q^37 + 16 * q^38 + 8 * q^43 - 4 * q^44 - 12 * q^47 + 2 * q^49 + 10 * q^50 - 2 * q^52 + 4 * q^53 - 6 * q^56 - 12 * q^58 + 4 * q^59 - 12 * q^61 - 8 * q^62 - 14 * q^64 + 8 * q^67 + 12 * q^68 - 28 * q^71 - 4 * q^73 + 20 * q^74 - 32 * q^76 - 4 * q^77 + 16 * q^82 - 4 * q^83 - 24 * q^86 + 12 * q^88 - 8 * q^89 - 2 * q^91 + 8 * q^92 + 4 * q^94 - 4 * q^97 - 2 * q^98

Introduction

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