LMFDB - Embedded newform 4050.2.c.y.649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4050,2,Mod(649,4050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4050.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.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:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 450)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	4050.649
Dual form			4050.2.c.y.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.449490i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +0.449490i q^{7} -1.00000i q^{8} -4.89898 q^{11} -0.449490i q^{13} -0.449490 q^{14} +1.00000 q^{16} -4.89898i q^{17} -7.44949 q^{19} -4.89898i q^{22} -2.44949i q^{23} +0.449490 q^{26} -0.449490i q^{28} +2.44949 q^{29} +4.44949 q^{31} +1.00000i q^{32} +4.89898 q^{34} +11.3485i q^{37} -7.44949i q^{38} +9.00000 q^{41} +2.55051i q^{43} +4.89898 q^{44} +2.44949 q^{46} +10.8990i q^{47} +6.79796 q^{49} +0.449490i q^{52} -3.55051i q^{53} +0.449490 q^{56} +2.44949i q^{58} +5.44949 q^{59} +8.00000 q^{61} +4.44949i q^{62} -1.00000 q^{64} -0.348469i q^{67} +4.89898i q^{68} +13.3485 q^{71} -1.00000i q^{73} -11.3485 q^{74} +7.44949 q^{76} -2.20204i q^{77} -16.6969 q^{79} +9.00000i q^{82} -5.44949i q^{83} -2.55051 q^{86} +4.89898i q^{88} +9.00000 q^{89} +0.202041 q^{91} +2.44949i q^{92} -10.8990 q^{94} -8.79796i q^{97} +6.79796i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 8 q^{14} + 4 q^{16} - 20 q^{19} - 8 q^{26} + 8 q^{31} + 36 q^{41} - 12 q^{49} - 8 q^{56} + 12 q^{59} + 32 q^{61} - 4 q^{64} + 24 q^{71} - 16 q^{74} + 20 q^{76} - 8 q^{79} - 20 q^{86} + 36 q^{89} + 40 q^{91} - 24 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^14 + 4 * q^16 - 20 * q^19 - 8 * q^26 + 8 * q^31 + 36 * q^41 - 12 * q^49 - 8 * q^56 + 12 * q^59 + 32 * q^61 - 4 * q^64 + 24 * q^71 - 16 * q^74 + 20 * q^76 - 8 * q^79 - 20 * q^86 + 36 * q^89 + 40 * q^91 - 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times$.

$n$	$2351$	$3727$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.449490i	0.169891i	0.996386	+	0.0849456i	$0.0270716\pi$
$7$	0.449490i	0.169891i	−0.996386	+	0.0849456i	$0.972928\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	− 0.449490i	− 0.124666i	−0.998055	−	0.0623330i	$-0.980146\pi$
$13$	− 0.449490i	− 0.124666i	0.998055	−	0.0623330i	$-0.0198541\pi$
$14$	−0.449490	−0.120131
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.89898i	− 1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$	− 4.89898i	− 1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$	0	0
$19$	−7.44949	−1.70903	−0.854515	−	0.519427i	$-0.826146\pi$
$19$	−7.44949	−1.70903	−0.854515	+	0.519427i	$0.826146\pi$
$20$	0	0
$21$	0	0
$22$	− 4.89898i	− 1.04447i
$23$	− 2.44949i	− 0.510754i	−0.966842	−	0.255377i	$-0.917800\pi$
$23$	− 2.44949i	− 0.510754i	0.966842	−	0.255377i	$-0.0821996\pi$
$24$	0	0
$25$	0	0
$26$	0.449490	0.0881522
$27$	0	0
$28$	− 0.449490i	− 0.0849456i
$29$	2.44949	0.454859	0.227429	−	0.973795i	$-0.426968\pi$
$29$	2.44949	0.454859	0.227429	+	0.973795i	$0.426968\pi$
$30$	0	0
$31$	4.44949	0.799152	0.399576	−	0.916700i	$-0.369157\pi$
$31$	4.44949	0.799152	0.399576	+	0.916700i	$0.369157\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	4.89898	0.840168
$35$	0	0
$36$	0	0
$37$	11.3485i	1.86568i	0.360295	+	0.932838i	$0.382676\pi$
$37$	11.3485i	1.86568i	−0.360295	+	0.932838i	$0.617324\pi$
$38$	− 7.44949i	− 1.20847i
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	2.55051i	0.388949i	0.980908	+	0.194475i	$0.0623002\pi$
$43$	2.55051i	0.388949i	−0.980908	+	0.194475i	$0.937700\pi$
$44$	4.89898	0.738549
$45$	0	0
$46$	2.44949	0.361158
$47$	10.8990i	1.58978i	0.606754	+	0.794890i	$0.292471\pi$
$47$	10.8990i	1.58978i	−0.606754	+	0.794890i	$0.707529\pi$
$48$	0	0
$49$	6.79796	0.971137
$50$	0	0
$51$	0	0
$52$	0.449490i	0.0623330i
$53$	− 3.55051i	− 0.487700i	−0.969813	−	0.243850i	$-0.921590\pi$
$53$	− 3.55051i	− 0.487700i	0.969813	−	0.243850i	$-0.0784105\pi$
$54$	0	0
$55$	0	0
$56$	0.449490	0.0600656
$57$	0	0
$58$	2.44949i	0.321634i
$59$	5.44949	0.709463	0.354732	−	0.934968i	$-0.384572\pi$
$59$	5.44949	0.709463	0.354732	+	0.934968i	$0.384572\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.44949i	0.565086i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 0.348469i	− 0.0425723i	−0.999773	−	0.0212861i	$-0.993224\pi$
$67$	− 0.348469i	− 0.0425723i	0.999773	−	0.0212861i	$-0.00677610\pi$
$68$	4.89898i	0.594089i
$69$	0	0
$70$	0	0
$71$	13.3485	1.58417	0.792086	−	0.610410i	$-0.208995\pi$
$71$	13.3485	1.58417	0.792086	+	0.610410i	$0.208995\pi$
$72$	0	0
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	−11.3485	−1.31923
$75$	0	0
$76$	7.44949	0.854515
$77$	− 2.20204i	− 0.250946i
$78$	0	0
$79$	−16.6969	−1.87855	−0.939276	−	0.343162i	$-0.888502\pi$
$79$	−16.6969	−1.87855	−0.939276	+	0.343162i	$0.888502\pi$
$80$	0	0
$81$	0	0
$82$	9.00000i	0.993884i
$83$	− 5.44949i	− 0.598159i	−0.954228	−	0.299080i	$-0.903320\pi$
$83$	− 5.44949i	− 0.598159i	0.954228	−	0.299080i	$-0.0966796\pi$
$84$	0	0
$85$	0	0
$86$	−2.55051	−0.275029
$87$	0	0
$88$	4.89898i	0.522233i
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	0.202041	0.0211797
$92$	2.44949i	0.255377i
$93$	0	0
$94$	−10.8990	−1.12414
$95$	0	0
$96$	0	0
$97$	− 8.79796i	− 0.893297i	−0.894709	−	0.446649i	$-0.852617\pi$
$97$	− 8.79796i	− 0.893297i	0.894709	−	0.446649i	$-0.147383\pi$
$98$	6.79796i	0.686698i
$99$	0	0
$100$	0	0
$101$	8.44949	0.840756	0.420378	−	0.907349i	$-0.361898\pi$
$101$	8.44949	0.840756	0.420378	+	0.907349i	$0.361898\pi$
$102$	0	0
$103$	16.6969i	1.64520i	0.568622	+	0.822599i	$0.307477\pi$
$103$	16.6969i	1.64520i	−0.568622	+	0.822599i	$0.692523\pi$
$104$	−0.449490	−0.0440761
$105$	0	0
$106$	3.55051	0.344856
$107$	− 9.24745i	− 0.893985i	−0.894538	−	0.446992i	$-0.852495\pi$
$107$	− 9.24745i	− 0.893985i	0.894538	−	0.446992i	$-0.147505\pi$
$108$	0	0
$109$	−5.55051	−0.531642	−0.265821	−	0.964022i	$-0.585643\pi$
$109$	−5.55051	−0.531642	−0.265821	+	0.964022i	$0.585643\pi$
$110$	0	0
$111$	0	0
$112$	0.449490i	0.0424728i
$113$	− 4.10102i	− 0.385792i	−0.981219	−	0.192896i	$-0.938212\pi$
$113$	− 4.10102i	− 0.385792i	0.981219	−	0.192896i	$-0.0617879\pi$
$114$	0	0
$115$	0	0
$116$	−2.44949	−0.227429
$117$	0	0
$118$	5.44949i	0.501666i
$119$	2.20204	0.201861
$120$	0	0
$121$	13.0000	1.18182
$122$	8.00000i	0.724286i
$123$	0	0
$124$	−4.44949	−0.399576
$125$	0	0
$126$	0	0
$127$	− 3.34847i	− 0.297129i	−0.988903	−	0.148564i	$-0.952535\pi$
$127$	− 3.34847i	− 0.297129i	0.988903	−	0.148564i	$-0.0474652\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−3.79796	−0.331829	−0.165915	−	0.986140i	$-0.553058\pi$
$131$	−3.79796	−0.331829	−0.165915	+	0.986140i	$0.553058\pi$
$132$	0	0
$133$	− 3.34847i	− 0.290349i
$134$	0.348469	0.0301032
$135$	0	0
$136$	−4.89898	−0.420084
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	13.3485i	1.12018i
$143$	2.20204i	0.184144i
$144$	0	0
$145$	0	0
$146$	1.00000	0.0827606
$147$	0	0
$148$	− 11.3485i	− 0.932838i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	7.44949i	0.604233i
$153$	0	0
$154$	2.20204	0.177446
$155$	0	0
$156$	0	0
$157$	19.7980i	1.58005i	0.613075	+	0.790025i	$0.289932\pi$
$157$	19.7980i	1.58005i	−0.613075	+	0.790025i	$0.710068\pi$
$158$	− 16.6969i	− 1.32834i
$159$	0	0
$160$	0	0
$161$	1.10102	0.0867726
$162$	0	0
$163$	7.44949i	0.583489i	0.956496	+	0.291745i	$0.0942357\pi$
$163$	7.44949i	0.583489i	−0.956496	+	0.291745i	$0.905764\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	5.44949	0.422962
$167$	− 19.5959i	− 1.51638i	−0.652035	−	0.758189i	$-0.726085\pi$
$167$	− 19.5959i	− 1.51638i	0.652035	−	0.758189i	$-0.273915\pi$
$168$	0	0
$169$	12.7980	0.984458
$170$	0	0
$171$	0	0
$172$	− 2.55051i	− 0.194475i
$173$	9.79796i	0.744925i	0.928047	+	0.372463i	$0.121486\pi$
$173$	9.79796i	0.744925i	−0.928047	+	0.372463i	$0.878514\pi$
$174$	0	0
$175$	0	0
$176$	−4.89898	−0.369274
$177$	0	0
$178$	9.00000i	0.674579i
$179$	−9.24745	−0.691187	−0.345593	−	0.938384i	$-0.612322\pi$
$179$	−9.24745	−0.691187	−0.345593	+	0.938384i	$0.612322\pi$
$180$	0	0
$181$	17.7980	1.32291	0.661456	−	0.749984i	$-0.269939\pi$
$181$	17.7980	1.32291	0.661456	+	0.749984i	$0.269939\pi$
$182$	0.202041i	0.0149763i
$183$	0	0
$184$	−2.44949	−0.180579
$185$	0	0
$186$	0	0
$187$	24.0000i	1.75505i
$188$	− 10.8990i	− 0.794890i
$189$	0	0
$190$	0	0
$191$	−1.10102	−0.0796670	−0.0398335	−	0.999206i	$-0.512683\pi$
$191$	−1.10102	−0.0796670	−0.0398335	+	0.999206i	$0.512683\pi$
$192$	0	0
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	8.79796	0.631657
$195$	0	0
$196$	−6.79796	−0.485568
$197$	0.247449i	0.0176300i	0.999961	+	0.00881500i	$0.00280594\pi$
$197$	0.247449i	0.0176300i	−0.999961	+	0.00881500i	$0.997194\pi$
$198$	0	0
$199$	13.7980	0.978111	0.489056	−	0.872253i	$-0.337341\pi$
$199$	13.7980	0.978111	0.489056	+	0.872253i	$0.337341\pi$
$200$	0	0
$201$	0	0
$202$	8.44949i	0.594504i
$203$	1.10102i	0.0772765i
$204$	0	0
$205$	0	0
$206$	−16.6969	−1.16333
$207$	0	0
$208$	− 0.449490i	− 0.0311665i
$209$	36.4949	2.52440
$210$	0	0
$211$	−3.44949	−0.237473	−0.118736	−	0.992926i	$-0.537884\pi$
$211$	−3.44949	−0.237473	−0.118736	+	0.992926i	$0.537884\pi$
$212$	3.55051i	0.243850i
$213$	0	0
$214$	9.24745	0.632143
$215$	0	0
$216$	0	0
$217$	2.00000i	0.135769i
$218$	− 5.55051i	− 0.375928i
$219$	0	0
$220$	0	0
$221$	−2.20204	−0.148125
$222$	0	0
$223$	17.7980i	1.19184i	0.803044	+	0.595920i	$0.203212\pi$
$223$	17.7980i	1.19184i	−0.803044	+	0.595920i	$0.796788\pi$
$224$	−0.449490	−0.0300328
$225$	0	0
$226$	4.10102	0.272796
$227$	− 13.6515i	− 0.906084i	−0.891489	−	0.453042i	$-0.850339\pi$
$227$	− 13.6515i	− 0.906084i	0.891489	−	0.453042i	$-0.149661\pi$
$228$	0	0
$229$	−13.1464	−0.868740	−0.434370	−	0.900734i	$-0.643029\pi$
$229$	−13.1464	−0.868740	−0.434370	+	0.900734i	$0.643029\pi$
$230$	0	0
$231$	0	0
$232$	− 2.44949i	− 0.160817i
$233$	− 23.6969i	− 1.55244i	−0.630463	−	0.776219i	$-0.717135\pi$
$233$	− 23.6969i	− 1.55244i	0.630463	−	0.776219i	$-0.282865\pi$
$234$	0	0
$235$	0	0
$236$	−5.44949	−0.354732
$237$	0	0
$238$	2.20204i	0.142737i
$239$	−14.4495	−0.934660	−0.467330	−	0.884083i	$-0.654784\pi$
$239$	−14.4495	−0.934660	−0.467330	+	0.884083i	$0.654784\pi$
$240$	0	0
$241$	−3.20204	−0.206262	−0.103131	−	0.994668i	$-0.532886\pi$
$241$	−3.20204	−0.206262	−0.103131	+	0.994668i	$0.532886\pi$
$242$	13.0000i	0.835672i
$243$	0	0
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	3.34847i	0.213058i
$248$	− 4.44949i	− 0.282543i
$249$	0	0
$250$	0	0
$251$	−0.550510	−0.0347479	−0.0173739	−	0.999849i	$-0.505531\pi$
$251$	−0.550510	−0.0347479	−0.0173739	+	0.999849i	$0.505531\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	3.34847	0.210102
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.7980i	0.798315i	0.916882	+	0.399157i	$0.130697\pi$
$257$	12.7980i	0.798315i	−0.916882	+	0.399157i	$0.869303\pi$
$258$	0	0
$259$	−5.10102	−0.316962
$260$	0	0
$261$	0	0
$262$	− 3.79796i	− 0.234639i
$263$	20.4495i	1.26097i	0.776202	+	0.630485i	$0.217144\pi$
$263$	20.4495i	1.26097i	−0.776202	+	0.630485i	$0.782856\pi$
$264$	0	0
$265$	0	0
$266$	3.34847	0.205308
$267$	0	0
$268$	0.348469i	0.0212861i
$269$	−14.4495	−0.881001	−0.440500	−	0.897752i	$-0.645199\pi$
$269$	−14.4495	−0.881001	−0.440500	+	0.897752i	$0.645199\pi$
$270$	0	0
$271$	15.3485	0.932353	0.466177	−	0.884692i	$-0.345631\pi$
$271$	15.3485	0.932353	0.466177	+	0.884692i	$0.345631\pi$
$272$	− 4.89898i	− 0.297044i
$273$	0	0
$274$	3.00000	0.181237
$275$	0	0
$276$	0	0
$277$	1.55051i	0.0931611i	0.998915	+	0.0465806i	$0.0148324\pi$
$277$	1.55051i	0.0931611i	−0.998915	+	0.0465806i	$0.985168\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	19.1010	1.13947	0.569736	−	0.821828i	$-0.307046\pi$
$281$	19.1010	1.13947	0.569736	+	0.821828i	$0.307046\pi$
$282$	0	0
$283$	− 13.2474i	− 0.787479i	−0.919222	−	0.393740i	$-0.871181\pi$
$283$	− 13.2474i	− 0.787479i	0.919222	−	0.393740i	$-0.128819\pi$
$284$	−13.3485	−0.792086
$285$	0	0
$286$	−2.20204	−0.130209
$287$	4.04541i	0.238793i
$288$	0	0
$289$	−7.00000	−0.411765
$290$	0	0
$291$	0	0
$292$	1.00000i	0.0585206i
$293$	16.0454i	0.937383i	0.883362	+	0.468691i	$0.155274\pi$
$293$	16.0454i	0.937383i	−0.883362	+	0.468691i	$0.844726\pi$
$294$	0	0
$295$	0	0
$296$	11.3485	0.659616
$297$	0	0
$298$	6.00000i	0.347571i
$299$	−1.10102	−0.0636737
$300$	0	0
$301$	−1.14643	−0.0660790
$302$	20.0000i	1.15087i
$303$	0	0
$304$	−7.44949	−0.427258
$305$	0	0
$306$	0	0
$307$	− 22.6969i	− 1.29538i	−0.761903	−	0.647691i	$-0.775735\pi$
$307$	− 22.6969i	− 1.29538i	0.761903	−	0.647691i	$-0.224265\pi$
$308$	2.20204i	0.125473i
$309$	0	0
$310$	0	0
$311$	−1.10102	−0.0624331	−0.0312166	−	0.999513i	$-0.509938\pi$
$311$	−1.10102	−0.0624331	−0.0312166	+	0.999513i	$0.509938\pi$
$312$	0	0
$313$	− 5.89898i	− 0.333430i	−0.986005	−	0.166715i	$-0.946684\pi$
$313$	− 5.89898i	− 0.333430i	0.986005	−	0.166715i	$-0.0533160\pi$
$314$	−19.7980	−1.11726
$315$	0	0
$316$	16.6969	0.939276
$317$	− 17.1464i	− 0.963039i	−0.876435	−	0.481520i	$-0.840085\pi$
$317$	− 17.1464i	− 0.963039i	0.876435	−	0.481520i	$-0.159915\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	1.10102i	0.0613575i
$323$	36.4949i	2.03063i
$324$	0	0
$325$	0	0
$326$	−7.44949	−0.412589
$327$	0	0
$328$	− 9.00000i	− 0.496942i
$329$	−4.89898	−0.270089
$330$	0	0
$331$	6.34847	0.348943	0.174472	−	0.984662i	$-0.444178\pi$
$331$	6.34847	0.348943	0.174472	+	0.984662i	$0.444178\pi$
$332$	5.44949i	0.299080i
$333$	0	0
$334$	19.5959	1.07224
$335$	0	0
$336$	0	0
$337$	20.8990i	1.13844i	0.822185	+	0.569220i	$0.192755\pi$
$337$	20.8990i	1.13844i	−0.822185	+	0.569220i	$0.807245\pi$
$338$	12.7980i	0.696117i
$339$	0	0
$340$	0	0
$341$	−21.7980	−1.18043
$342$	0	0
$343$	6.20204i	0.334879i
$344$	2.55051	0.137514
$345$	0	0
$346$	−9.79796	−0.526742
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	− 4.89898i	− 0.261116i
$353$	− 9.00000i	− 0.479022i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	− 9.00000i	− 0.479022i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$	0	0
$355$	0	0
$356$	−9.00000	−0.476999
$357$	0	0
$358$	− 9.24745i	− 0.488743i
$359$	14.2020	0.749555	0.374778	−	0.927115i	$-0.377719\pi$
$359$	14.2020	0.749555	0.374778	+	0.927115i	$0.377719\pi$
$360$	0	0
$361$	36.4949	1.92078
$362$	17.7980i	0.935440i
$363$	0	0
$364$	−0.202041	−0.0105898
$365$	0	0
$366$	0	0
$367$	12.6969i	0.662775i	0.943495	+	0.331387i	$0.107517\pi$
$367$	12.6969i	0.662775i	−0.943495	+	0.331387i	$0.892483\pi$
$368$	− 2.44949i	− 0.127688i
$369$	0	0
$370$	0	0
$371$	1.59592	0.0828559
$372$	0	0
$373$	− 23.5959i	− 1.22175i	−0.791727	−	0.610875i	$-0.790818\pi$
$373$	− 23.5959i	− 1.22175i	0.791727	−	0.610875i	$-0.209182\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	10.8990	0.562072
$377$	− 1.10102i	− 0.0567054i
$378$	0	0
$379$	8.89898	0.457110	0.228555	−	0.973531i	$-0.426600\pi$
$379$	8.89898	0.457110	0.228555	+	0.973531i	$0.426600\pi$
$380$	0	0
$381$	0	0
$382$	− 1.10102i	− 0.0563331i
$383$	21.5505i	1.10118i	0.834776	+	0.550590i	$0.185597\pi$
$383$	21.5505i	1.10118i	−0.834776	+	0.550590i	$0.814403\pi$
$384$	0	0
$385$	0	0
$386$	−20.0000	−1.01797
$387$	0	0
$388$	8.79796i	0.446649i
$389$	−25.5959	−1.29776	−0.648882	−	0.760889i	$-0.724763\pi$
$389$	−25.5959	−1.29776	−0.648882	+	0.760889i	$0.724763\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	− 6.79796i	− 0.343349i
$393$	0	0
$394$	−0.247449	−0.0124663
$395$	0	0
$396$	0	0
$397$	17.5959i	0.883114i	0.897233	+	0.441557i	$0.145574\pi$
$397$	17.5959i	0.883114i	−0.897233	+	0.441557i	$0.854426\pi$
$398$	13.7980i	0.691629i
$399$	0	0
$400$	0	0
$401$	38.6969	1.93243	0.966216	−	0.257732i	$-0.0829752\pi$
$401$	38.6969	1.93243	0.966216	+	0.257732i	$0.0829752\pi$
$402$	0	0
$403$	− 2.00000i	− 0.0996271i
$404$	−8.44949	−0.420378
$405$	0	0
$406$	−1.10102	−0.0546427
$407$	− 55.5959i	− 2.75579i
$408$	0	0
$409$	−0.101021	−0.00499514	−0.00249757	−	0.999997i	$-0.500795\pi$
$409$	−0.101021	−0.00499514	−0.00249757	+	0.999997i	$0.500795\pi$
$410$	0	0
$411$	0	0
$412$	− 16.6969i	− 0.822599i
$413$	2.44949i	0.120532i
$414$	0	0
$415$	0	0
$416$	0.449490	0.0220380
$417$	0	0
$418$	36.4949i	1.78502i
$419$	26.1464	1.27734	0.638668	−	0.769482i	$-0.279486\pi$
$419$	26.1464	1.27734	0.638668	+	0.769482i	$0.279486\pi$
$420$	0	0
$421$	−26.0454	−1.26938	−0.634688	−	0.772769i	$-0.718871\pi$
$421$	−26.0454	−1.26938	−0.634688	+	0.772769i	$0.718871\pi$
$422$	− 3.44949i	− 0.167919i
$423$	0	0
$424$	−3.55051	−0.172428
$425$	0	0
$426$	0	0
$427$	3.59592i	0.174019i
$428$	9.24745i	0.446992i
$429$	0	0
$430$	0	0
$431$	25.3485	1.22099	0.610496	−	0.792019i	$-0.290970\pi$
$431$	25.3485	1.22099	0.610496	+	0.792019i	$0.290970\pi$
$432$	0	0
$433$	9.59592i	0.461150i	0.973055	+	0.230575i	$0.0740608\pi$
$433$	9.59592i	0.461150i	−0.973055	+	0.230575i	$0.925939\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	5.55051	0.265821
$437$	18.2474i	0.872894i
$438$	0	0
$439$	−3.34847	−0.159814	−0.0799069	−	0.996802i	$-0.525462\pi$
$439$	−3.34847	−0.159814	−0.0799069	+	0.996802i	$0.525462\pi$
$440$	0	0
$441$	0	0
$442$	− 2.20204i	− 0.104740i
$443$	8.20204i	0.389691i	0.980834	+	0.194845i	$0.0624205\pi$
$443$	8.20204i	0.389691i	−0.980834	+	0.194845i	$0.937579\pi$
$444$	0	0
$445$	0	0
$446$	−17.7980	−0.842758
$447$	0	0
$448$	− 0.449490i	− 0.0212364i
$449$	28.5959	1.34952	0.674762	−	0.738035i	$-0.264246\pi$
$449$	28.5959	1.34952	0.674762	+	0.738035i	$0.264246\pi$
$450$	0	0
$451$	−44.0908	−2.07616
$452$	4.10102i	0.192896i
$453$	0	0
$454$	13.6515	0.640698
$455$	0	0
$456$	0	0
$457$	− 13.6969i	− 0.640716i	−0.947297	−	0.320358i	$-0.896197\pi$
$457$	− 13.6969i	− 0.640716i	0.947297	−	0.320358i	$-0.103803\pi$
$458$	− 13.1464i	− 0.614292i
$459$	0	0
$460$	0	0
$461$	4.65153	0.216643	0.108322	−	0.994116i	$-0.465452\pi$
$461$	4.65153	0.216643	0.108322	+	0.994116i	$0.465452\pi$
$462$	0	0
$463$	− 5.34847i	− 0.248564i	−0.992247	−	0.124282i	$-0.960337\pi$
$463$	− 5.34847i	− 0.248564i	0.992247	−	0.124282i	$-0.0396628\pi$
$464$	2.44949	0.113715
$465$	0	0
$466$	23.6969	1.09774
$467$	10.3485i	0.478870i	0.970912	+	0.239435i	$0.0769622\pi$
$467$	10.3485i	0.478870i	−0.970912	+	0.239435i	$0.923038\pi$
$468$	0	0
$469$	0.156633	0.00723266
$470$	0	0
$471$	0	0
$472$	− 5.44949i	− 0.250833i
$473$	− 12.4949i	− 0.574516i
$474$	0	0
$475$	0	0
$476$	−2.20204	−0.100930
$477$	0	0
$478$	− 14.4495i	− 0.660904i
$479$	−0.247449	−0.0113062	−0.00565311	−	0.999984i	$-0.501799\pi$
$479$	−0.247449	−0.0113062	−0.00565311	+	0.999984i	$0.501799\pi$
$480$	0	0
$481$	5.10102	0.232587
$482$	− 3.20204i	− 0.145849i
$483$	0	0
$484$	−13.0000	−0.590909
$485$	0	0
$486$	0	0
$487$	24.4495i	1.10791i	0.832546	+	0.553956i	$0.186882\pi$
$487$	24.4495i	1.10791i	−0.832546	+	0.553956i	$0.813118\pi$
$488$	− 8.00000i	− 0.362143i
$489$	0	0
$490$	0	0
$491$	−27.2474	−1.22966	−0.614830	−	0.788660i	$-0.710775\pi$
$491$	−27.2474	−1.22966	−0.614830	+	0.788660i	$0.710775\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	−3.34847	−0.150655
$495$	0	0
$496$	4.44949	0.199788
$497$	6.00000i	0.269137i
$498$	0	0
$499$	8.34847	0.373729	0.186864	−	0.982386i	$-0.440167\pi$
$499$	8.34847	0.373729	0.186864	+	0.982386i	$0.440167\pi$
$500$	0	0
$501$	0	0
$502$	− 0.550510i	− 0.0245705i
$503$	21.5505i	0.960890i	0.877025	+	0.480445i	$0.159525\pi$
$503$	21.5505i	0.960890i	−0.877025	+	0.480445i	$0.840475\pi$
$504$	0	0
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	3.34847i	0.148564i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0.449490	0.0198843
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−12.7980	−0.564494
$515$	0	0
$516$	0	0
$517$	− 53.3939i	− 2.34826i
$518$	− 5.10102i	− 0.224126i
$519$	0	0
$520$	0	0
$521$	−29.3939	−1.28777	−0.643885	−	0.765123i	$-0.722678\pi$
$521$	−29.3939	−1.28777	−0.643885	+	0.765123i	$0.722678\pi$
$522$	0	0
$523$	− 20.3485i	− 0.889776i	−0.895586	−	0.444888i	$-0.853243\pi$
$523$	− 20.3485i	− 0.889776i	0.895586	−	0.444888i	$-0.146757\pi$
$524$	3.79796	0.165915
$525$	0	0
$526$	−20.4495	−0.891640
$527$	− 21.7980i	− 0.949534i
$528$	0	0
$529$	17.0000	0.739130
$530$	0	0
$531$	0	0
$532$	3.34847i	0.145175i
$533$	− 4.04541i	− 0.175226i
$534$	0	0
$535$	0	0
$536$	−0.348469	−0.0150516
$537$	0	0
$538$	− 14.4495i	− 0.622962i
$539$	−33.3031	−1.43446
$540$	0	0
$541$	−37.7980	−1.62506	−0.812531	−	0.582919i	$-0.801911\pi$
$541$	−37.7980	−1.62506	−0.812531	+	0.582919i	$0.801911\pi$
$542$	15.3485i	0.659273i
$543$	0	0
$544$	4.89898	0.210042
$545$	0	0
$546$	0	0
$547$	− 15.6515i	− 0.669211i	−0.942358	−	0.334606i	$-0.891397\pi$
$547$	− 15.6515i	− 0.669211i	0.942358	−	0.334606i	$-0.108603\pi$
$548$	3.00000i	0.128154i
$549$	0	0
$550$	0	0
$551$	−18.2474	−0.777367
$552$	0	0
$553$	− 7.50510i	− 0.319149i
$554$	−1.55051	−0.0658749
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 41.3939i	− 1.75391i	−0.480568	−	0.876957i	$-0.659570\pi$
$557$	− 41.3939i	− 1.75391i	0.480568	−	0.876957i	$-0.340430\pi$
$558$	0	0
$559$	1.14643	0.0484887
$560$	0	0
$561$	0	0
$562$	19.1010i	0.805728i
$563$	− 23.9444i	− 1.00914i	−0.863372	−	0.504568i	$-0.831652\pi$
$563$	− 23.9444i	− 1.00914i	0.863372	−	0.504568i	$-0.168348\pi$
$564$	0	0
$565$	0	0
$566$	13.2474	0.556832
$567$	0	0
$568$	− 13.3485i	− 0.560089i
$569$	−33.7980	−1.41688	−0.708442	−	0.705769i	$-0.750602\pi$
$569$	−33.7980	−1.41688	−0.708442	+	0.705769i	$0.750602\pi$
$570$	0	0
$571$	25.9444	1.08574	0.542869	−	0.839817i	$-0.317338\pi$
$571$	25.9444	1.08574	0.542869	+	0.839817i	$0.317338\pi$
$572$	− 2.20204i	− 0.0920720i
$573$	0	0
$574$	−4.04541	−0.168852
$575$	0	0
$576$	0	0
$577$	− 40.3939i	− 1.68162i	−0.541331	−	0.840810i	$-0.682079\pi$
$577$	− 40.3939i	− 1.68162i	0.541331	−	0.840810i	$-0.317921\pi$
$578$	− 7.00000i	− 0.291162i
$579$	0	0
$580$	0	0
$581$	2.44949	0.101622
$582$	0	0
$583$	17.3939i	0.720381i
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	−16.0454	−0.662830
$587$	26.6969i	1.10190i	0.834538	+	0.550950i	$0.185735\pi$
$587$	26.6969i	1.10190i	−0.834538	+	0.550950i	$0.814265\pi$
$588$	0	0
$589$	−33.1464	−1.36577
$590$	0	0
$591$	0	0
$592$	11.3485i	0.466419i
$593$	− 7.89898i	− 0.324372i	−0.986760	−	0.162186i	$-0.948146\pi$
$593$	− 7.89898i	− 0.324372i	0.986760	−	0.162186i	$-0.0518545\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	− 1.10102i	− 0.0450241i
$599$	22.6515	0.925516	0.462758	−	0.886485i	$-0.346860\pi$
$599$	22.6515	0.925516	0.462758	+	0.886485i	$0.346860\pi$
$600$	0	0
$601$	−16.4949	−0.672841	−0.336420	−	0.941712i	$-0.609216\pi$
$601$	−16.4949	−0.672841	−0.336420	+	0.941712i	$0.609216\pi$
$602$	− 1.14643i	− 0.0467249i
$603$	0	0
$604$	−20.0000	−0.813788
$605$	0	0
$606$	0	0
$607$	− 3.34847i	− 0.135910i	−0.997688	−	0.0679551i	$-0.978353\pi$
$607$	− 3.34847i	− 0.135910i	0.997688	−	0.0679551i	$-0.0216475\pi$
$608$	− 7.44949i	− 0.302117i
$609$	0	0
$610$	0	0
$611$	4.89898	0.198191
$612$	0	0
$613$	12.0454i	0.486509i	0.969962	+	0.243255i	$0.0782151\pi$
$613$	12.0454i	0.486509i	−0.969962	+	0.243255i	$0.921785\pi$
$614$	22.6969	0.915974
$615$	0	0
$616$	−2.20204	−0.0887228
$617$	− 44.3939i	− 1.78723i	−0.448834	−	0.893615i	$-0.648161\pi$
$617$	− 44.3939i	− 1.78723i	0.448834	−	0.893615i	$-0.351839\pi$
$618$	0	0
$619$	31.7423	1.27583	0.637916	−	0.770106i	$-0.279797\pi$
$619$	31.7423	1.27583	0.637916	+	0.770106i	$0.279797\pi$
$620$	0	0
$621$	0	0
$622$	− 1.10102i	− 0.0441469i
$623$	4.04541i	0.162076i
$624$	0	0
$625$	0	0
$626$	5.89898	0.235771
$627$	0	0
$628$	− 19.7980i	− 0.790025i
$629$	55.5959	2.21675
$630$	0	0
$631$	−6.20204	−0.246899	−0.123450	−	0.992351i	$-0.539396\pi$
$631$	−6.20204	−0.246899	−0.123450	+	0.992351i	$0.539396\pi$
$632$	16.6969i	0.664169i
$633$	0	0
$634$	17.1464	0.680972
$635$	0	0
$636$	0	0
$637$	− 3.05561i	− 0.121068i
$638$	− 12.0000i	− 0.475085i
$639$	0	0
$640$	0	0
$641$	−14.3939	−0.568524	−0.284262	−	0.958747i	$-0.591749\pi$
$641$	−14.3939	−0.568524	−0.284262	+	0.958747i	$0.591749\pi$
$642$	0	0
$643$	− 17.6515i	− 0.696108i	−0.937474	−	0.348054i	$-0.886843\pi$
$643$	− 17.6515i	− 0.696108i	0.937474	−	0.348054i	$-0.113157\pi$
$644$	−1.10102	−0.0433863
$645$	0	0
$646$	−36.4949	−1.43587
$647$	− 24.2474i	− 0.953266i	−0.879102	−	0.476633i	$-0.841857\pi$
$647$	− 24.2474i	− 0.953266i	0.879102	−	0.476633i	$-0.158143\pi$
$648$	0	0
$649$	−26.6969	−1.04795
$650$	0	0
$651$	0	0
$652$	− 7.44949i	− 0.291745i
$653$	18.2474i	0.714078i	0.934090	+	0.357039i	$0.116214\pi$
$653$	18.2474i	0.714078i	−0.934090	+	0.357039i	$0.883786\pi$
$654$	0	0
$655$	0	0
$656$	9.00000	0.351391
$657$	0	0
$658$	− 4.89898i	− 0.190982i
$659$	−9.85357	−0.383841	−0.191920	−	0.981411i	$-0.561472\pi$
$659$	−9.85357	−0.383841	−0.191920	+	0.981411i	$0.561472\pi$
$660$	0	0
$661$	7.39388	0.287588	0.143794	−	0.989608i	$-0.454070\pi$
$661$	7.39388	0.287588	0.143794	+	0.989608i	$0.454070\pi$
$662$	6.34847i	0.246740i
$663$	0	0
$664$	−5.44949	−0.211481
$665$	0	0
$666$	0	0
$667$	− 6.00000i	− 0.232321i
$668$	19.5959i	0.758189i
$669$	0	0
$670$	0	0
$671$	−39.1918	−1.51298
$672$	0	0
$673$	22.6969i	0.874903i	0.899242	+	0.437451i	$0.144119\pi$
$673$	22.6969i	0.874903i	−0.899242	+	0.437451i	$0.855881\pi$
$674$	−20.8990	−0.804999
$675$	0	0
$676$	−12.7980	−0.492229
$677$	25.8434i	0.993241i	0.867968	+	0.496621i	$0.165426\pi$
$677$	25.8434i	0.993241i	−0.867968	+	0.496621i	$0.834574\pi$
$678$	0	0
$679$	3.95459	0.151763
$680$	0	0
$681$	0	0
$682$	− 21.7980i	− 0.834687i
$683$	41.9444i	1.60496i	0.596681	+	0.802479i	$0.296486\pi$
$683$	41.9444i	1.60496i	−0.596681	+	0.802479i	$0.703514\pi$
$684$	0	0
$685$	0	0
$686$	−6.20204	−0.236795
$687$	0	0
$688$	2.55051i	0.0972373i
$689$	−1.59592	−0.0607996
$690$	0	0
$691$	39.0454	1.48536	0.742679	−	0.669648i	$-0.233555\pi$
$691$	39.0454	1.48536	0.742679	+	0.669648i	$0.233555\pi$
$692$	− 9.79796i	− 0.372463i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	− 44.0908i	− 1.67006i
$698$	− 14.0000i	− 0.529908i
$699$	0	0
$700$	0	0
$701$	33.7980	1.27653	0.638266	−	0.769816i	$-0.279652\pi$
$701$	33.7980	1.27653	0.638266	+	0.769816i	$0.279652\pi$
$702$	0	0
$703$	− 84.5403i	− 3.18850i
$704$	4.89898	0.184637
$705$	0	0
$706$	9.00000	0.338719
$707$	3.79796i	0.142837i
$708$	0	0
$709$	−4.44949	−0.167104	−0.0835520	−	0.996503i	$-0.526626\pi$
$709$	−4.44949	−0.167104	−0.0835520	+	0.996503i	$0.526626\pi$
$710$	0	0
$711$	0	0
$712$	− 9.00000i	− 0.337289i
$713$	− 10.8990i	− 0.408170i
$714$	0	0
$715$	0	0
$716$	9.24745	0.345593
$717$	0	0
$718$	14.2020i	0.530015i
$719$	−7.95459	−0.296656	−0.148328	−	0.988938i	$-0.547389\pi$
$719$	−7.95459	−0.296656	−0.148328	+	0.988938i	$0.547389\pi$
$720$	0	0
$721$	−7.50510	−0.279505
$722$	36.4949i	1.35820i
$723$	0	0
$724$	−17.7980	−0.661456
$725$	0	0
$726$	0	0
$727$	16.0000i	0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$	16.0000i	0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$	− 0.202041i	− 0.00748814i
$729$	0	0
$730$	0	0
$731$	12.4949	0.462140
$732$	0	0
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	−12.6969	−0.468653
$735$	0	0
$736$	2.44949	0.0902894
$737$	1.70714i	0.0628834i
$738$	0	0
$739$	−3.04541	−0.112027	−0.0560136	−	0.998430i	$-0.517839\pi$
$739$	−3.04541	−0.112027	−0.0560136	+	0.998430i	$0.517839\pi$
$740$	0	0
$741$	0	0
$742$	1.59592i	0.0585880i
$743$	49.3485i	1.81042i	0.424965	+	0.905210i	$0.360286\pi$
$743$	49.3485i	1.81042i	−0.424965	+	0.905210i	$0.639714\pi$
$744$	0	0
$745$	0	0
$746$	23.5959	0.863908
$747$	0	0
$748$	− 24.0000i	− 0.877527i
$749$	4.15663	0.151880
$750$	0	0
$751$	−5.95459	−0.217286	−0.108643	−	0.994081i	$-0.534651\pi$
$751$	−5.95459	−0.217286	−0.108643	+	0.994081i	$0.534651\pi$
$752$	10.8990i	0.397445i
$753$	0	0
$754$	1.10102	0.0400968
$755$	0	0
$756$	0	0
$757$	38.0454i	1.38278i	0.722480	+	0.691392i	$0.243002\pi$
$757$	38.0454i	1.38278i	−0.722480	+	0.691392i	$0.756998\pi$
$758$	8.89898i	0.323225i
$759$	0	0
$760$	0	0
$761$	13.8990	0.503838	0.251919	−	0.967748i	$-0.418938\pi$
$761$	13.8990	0.503838	0.251919	+	0.967748i	$0.418938\pi$
$762$	0	0
$763$	− 2.49490i	− 0.0903214i
$764$	1.10102	0.0398335
$765$	0	0
$766$	−21.5505	−0.778652
$767$	− 2.44949i	− 0.0884459i
$768$	0	0
$769$	28.1918	1.01662	0.508312	−	0.861173i	$-0.330270\pi$
$769$	28.1918	1.01662	0.508312	+	0.861173i	$0.330270\pi$
$770$	0	0
$771$	0	0
$772$	− 20.0000i	− 0.719816i
$773$	− 10.4041i	− 0.374209i	−0.982340	−	0.187104i	$-0.940090\pi$
$773$	− 10.4041i	− 0.374209i	0.982340	−	0.187104i	$-0.0599103\pi$
$774$	0	0
$775$	0	0
$776$	−8.79796	−0.315828
$777$	0	0
$778$	− 25.5959i	− 0.917658i
$779$	−67.0454	−2.40215
$780$	0	0
$781$	−65.3939	−2.33998
$782$	− 12.0000i	− 0.429119i
$783$	0	0
$784$	6.79796	0.242784
$785$	0	0
$786$	0	0
$787$	− 34.6969i	− 1.23681i	−0.785859	−	0.618406i	$-0.787779\pi$
$787$	− 34.6969i	− 1.23681i	0.785859	−	0.618406i	$-0.212221\pi$
$788$	− 0.247449i	− 0.00881500i
$789$	0	0
$790$	0	0
$791$	1.84337	0.0655426
$792$	0	0
$793$	− 3.59592i	− 0.127695i
$794$	−17.5959	−0.624456
$795$	0	0
$796$	−13.7980	−0.489056
$797$	30.2474i	1.07142i	0.844402	+	0.535710i	$0.179956\pi$
$797$	30.2474i	1.07142i	−0.844402	+	0.535710i	$0.820044\pi$
$798$	0	0
$799$	53.3939	1.88894
$800$	0	0
$801$	0	0
$802$	38.6969i	1.36644i
$803$	4.89898i	0.172881i
$804$	0	0
$805$	0	0
$806$	2.00000	0.0704470
$807$	0	0
$808$	− 8.44949i	− 0.297252i
$809$	6.30306	0.221604	0.110802	−	0.993843i	$-0.464658\pi$
$809$	6.30306	0.221604	0.110802	+	0.993843i	$0.464658\pi$
$810$	0	0
$811$	−28.5505	−1.00254	−0.501272	−	0.865290i	$-0.667134\pi$
$811$	−28.5505	−1.00254	−0.501272	+	0.865290i	$0.667134\pi$
$812$	− 1.10102i	− 0.0386382i
$813$	0	0
$814$	55.5959	1.94864
$815$	0	0
$816$	0	0
$817$	− 19.0000i	− 0.664726i
$818$	− 0.101021i	− 0.00353210i
$819$	0	0
$820$	0	0
$821$	27.1918	0.949002	0.474501	−	0.880255i	$-0.342629\pi$
$821$	27.1918	0.949002	0.474501	+	0.880255i	$0.342629\pi$
$822$	0	0
$823$	− 17.1010i	− 0.596104i	−0.954550	−	0.298052i	$-0.903663\pi$
$823$	− 17.1010i	− 0.596104i	0.954550	−	0.298052i	$-0.0963369\pi$
$824$	16.6969	0.581665
$825$	0	0
$826$	−2.44949	−0.0852286
$827$	35.9444i	1.24991i	0.780661	+	0.624954i	$0.214882\pi$
$827$	35.9444i	1.24991i	−0.780661	+	0.624954i	$0.785118\pi$
$828$	0	0
$829$	46.7423	1.62343	0.811714	−	0.584055i	$-0.198535\pi$
$829$	46.7423	1.62343	0.811714	+	0.584055i	$0.198535\pi$
$830$	0	0
$831$	0	0
$832$	0.449490i	0.0155833i
$833$	− 33.3031i	− 1.15388i
$834$	0	0
$835$	0	0
$836$	−36.4949	−1.26220
$837$	0	0
$838$	26.1464i	0.903213i
$839$	−37.3485	−1.28941	−0.644706	−	0.764430i	$-0.723020\pi$
$839$	−37.3485	−1.28941	−0.644706	+	0.764430i	$0.723020\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	− 26.0454i	− 0.897584i
$843$	0	0
$844$	3.44949	0.118736
$845$	0	0
$846$	0	0
$847$	5.84337i	0.200780i
$848$	− 3.55051i	− 0.121925i
$849$	0	0
$850$	0	0
$851$	27.7980	0.952902
$852$	0	0
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	−3.59592	−0.123050
$855$	0	0
$856$	−9.24745	−0.316071
$857$	49.8990i	1.70452i	0.523121	+	0.852258i	$0.324768\pi$
$857$	49.8990i	1.70452i	−0.523121	+	0.852258i	$0.675232\pi$
$858$	0	0
$859$	20.3485	0.694281	0.347140	−	0.937813i	$-0.387153\pi$
$859$	20.3485	0.694281	0.347140	+	0.937813i	$0.387153\pi$
$860$	0	0
$861$	0	0
$862$	25.3485i	0.863372i
$863$	− 43.8434i	− 1.49245i	−0.665696	−	0.746223i	$-0.731865\pi$
$863$	− 43.8434i	− 1.49245i	0.665696	−	0.746223i	$-0.268135\pi$
$864$	0	0
$865$	0	0
$866$	−9.59592	−0.326083
$867$	0	0
$868$	− 2.00000i	− 0.0678844i
$869$	81.7980	2.77481
$870$	0	0
$871$	−0.156633	−0.00530732
$872$	5.55051i	0.187964i
$873$	0	0
$874$	−18.2474	−0.617229
$875$	0	0
$876$	0	0
$877$	31.7980i	1.07374i	0.843665	+	0.536870i	$0.180394\pi$
$877$	31.7980i	1.07374i	−0.843665	+	0.536870i	$0.819606\pi$
$878$	− 3.34847i	− 0.113005i
$879$	0	0
$880$	0	0
$881$	38.6969	1.30373	0.651866	−	0.758334i	$-0.273986\pi$
$881$	38.6969	1.30373	0.651866	+	0.758334i	$0.273986\pi$
$882$	0	0
$883$	47.7980i	1.60853i	0.594271	+	0.804265i	$0.297441\pi$
$883$	47.7980i	1.60853i	−0.594271	+	0.804265i	$0.702559\pi$
$884$	2.20204	0.0740627
$885$	0	0
$886$	−8.20204	−0.275553
$887$	− 5.50510i	− 0.184843i	−0.995720	−	0.0924216i	$-0.970539\pi$
$887$	− 5.50510i	− 0.184843i	0.995720	−	0.0924216i	$-0.0294608\pi$
$888$	0	0
$889$	1.50510	0.0504795
$890$	0	0
$891$	0	0
$892$	− 17.7980i	− 0.595920i
$893$	− 81.1918i	− 2.71698i
$894$	0	0
$895$	0	0
$896$	0.449490	0.0150164
$897$	0	0
$898$	28.5959i	0.954258i
$899$	10.8990	0.363501
$900$	0	0
$901$	−17.3939	−0.579474
$902$	− 44.0908i	− 1.46806i
$903$	0	0
$904$	−4.10102	−0.136398
$905$	0	0
$906$	0	0
$907$	− 36.3485i	− 1.20693i	−0.797389	−	0.603466i	$-0.793786\pi$
$907$	− 36.3485i	− 1.20693i	0.797389	−	0.603466i	$-0.206214\pi$
$908$	13.6515i	0.453042i
$909$	0	0
$910$	0	0
$911$	−7.34847	−0.243466	−0.121733	−	0.992563i	$-0.538845\pi$
$911$	−7.34847	−0.243466	−0.121733	+	0.992563i	$0.538845\pi$
$912$	0	0
$913$	26.6969i	0.883540i
$914$	13.6969	0.453054
$915$	0	0
$916$	13.1464	0.434370
$917$	− 1.70714i	− 0.0563748i
$918$	0	0
$919$	−3.34847	−0.110456	−0.0552279	−	0.998474i	$-0.517589\pi$
$919$	−3.34847	−0.110456	−0.0552279	+	0.998474i	$0.517589\pi$
$920$	0	0
$921$	0	0
$922$	4.65153i	0.153190i
$923$	− 6.00000i	− 0.197492i
$924$	0	0
$925$	0	0
$926$	5.34847	0.175762
$927$	0	0
$928$	2.44949i	0.0804084i
$929$	51.1918	1.67955	0.839775	−	0.542935i	$-0.182687\pi$
$929$	51.1918	1.67955	0.839775	+	0.542935i	$0.182687\pi$
$930$	0	0
$931$	−50.6413	−1.65970
$932$	23.6969i	0.776219i
$933$	0	0
$934$	−10.3485	−0.338612
$935$	0	0
$936$	0	0
$937$	− 7.20204i	− 0.235280i	−0.993056	−	0.117640i	$-0.962467\pi$
$937$	− 7.20204i	− 0.235280i	0.993056	−	0.117640i	$-0.0375330\pi$
$938$	0.156633i	0.00511426i
$939$	0	0
$940$	0	0
$941$	53.6413	1.74866	0.874329	−	0.485334i	$-0.161302\pi$
$941$	53.6413	1.74866	0.874329	+	0.485334i	$0.161302\pi$
$942$	0	0
$943$	− 22.0454i	− 0.717897i
$944$	5.44949	0.177366
$945$	0	0
$946$	12.4949	0.406244
$947$	− 26.1464i	− 0.849645i	−0.905277	−	0.424822i	$-0.860337\pi$
$947$	− 26.1464i	− 0.849645i	0.905277	−	0.424822i	$-0.139663\pi$
$948$	0	0
$949$	−0.449490	−0.0145911
$950$	0	0
$951$	0	0
$952$	− 2.20204i	− 0.0713686i
$953$	2.20204i	0.0713311i	0.999364	+	0.0356656i	$0.0113551\pi$
$953$	2.20204i	0.0713311i	−0.999364	+	0.0356656i	$0.988645\pi$
$954$	0	0
$955$	0	0
$956$	14.4495	0.467330
$957$	0	0
$958$	− 0.247449i	− 0.00799471i
$959$	1.34847	0.0435443
$960$	0	0
$961$	−11.2020	−0.361356
$962$	5.10102i	0.164464i
$963$	0	0
$964$	3.20204	0.103131
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	− 13.0000i	− 0.417836i
$969$	0	0
$970$	0	0
$971$	−40.8434	−1.31073	−0.655363	−	0.755314i	$-0.727484\pi$
$971$	−40.8434	−1.31073	−0.655363	+	0.755314i	$0.727484\pi$
$972$	0	0
$973$	1.79796i	0.0576399i
$974$	−24.4495	−0.783412
$975$	0	0
$976$	8.00000	0.256074
$977$	9.00000i	0.287936i	0.989582	+	0.143968i	$0.0459862\pi$
$977$	9.00000i	0.287936i	−0.989582	+	0.143968i	$0.954014\pi$
$978$	0	0
$979$	−44.0908	−1.40915
$980$	0	0
$981$	0	0
$982$	− 27.2474i	− 0.869501i
$983$	− 13.1010i	− 0.417858i	−0.977931	−	0.208929i	$-0.933002\pi$
$983$	− 13.1010i	− 0.417858i	0.977931	−	0.208929i	$-0.0669977\pi$
$984$	0	0
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	− 3.34847i	− 0.106529i
$989$	6.24745	0.198657
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	4.44949i	0.141271i
$993$	0	0
$994$	−6.00000	−0.190308
$995$	0	0
$996$	0	0
$997$	8.04541i	0.254801i	0.991851	+	0.127400i	$0.0406633\pi$
$997$	8.04541i	0.254801i	−0.991851	+	0.127400i	$0.959337\pi$
$998$	8.34847i	0.264266i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.c.y.649.4		4
3.2	odd	2		4050.2.c.w.649.2		4
5.2	odd	4		4050.2.a.br.1.1		2
5.3	odd	4		4050.2.a.bu.1.2		2
5.4	even	2	inner	4050.2.c.y.649.1		4
9.2	odd	6		1350.2.j.g.199.2		8
9.4	even	3		450.2.j.f.349.1		8
9.5	odd	6		1350.2.j.g.1099.3		8
9.7	even	3		450.2.j.f.49.4		8
15.2	even	4		4050.2.a.by.1.1		2
15.8	even	4		4050.2.a.bl.1.2		2
15.14	odd	2		4050.2.c.w.649.3		4
45.2	even	12		1350.2.e.k.901.2		4
45.4	even	6		450.2.j.f.349.4		8
45.7	odd	12		450.2.e.m.301.1	yes	4
45.13	odd	12		450.2.e.l.151.1	✓	4
45.14	odd	6		1350.2.j.g.1099.2		8
45.22	odd	12		450.2.e.m.151.2	yes	4
45.23	even	12		1350.2.e.n.451.1		4
45.29	odd	6		1350.2.j.g.199.3		8
45.32	even	12		1350.2.e.k.451.2		4
45.34	even	6		450.2.j.f.49.1		8
45.38	even	12		1350.2.e.n.901.1		4
45.43	odd	12		450.2.e.l.301.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.e.l.151.1	✓	4	45.13	odd	12
450.2.e.l.301.2	yes	4	45.43	odd	12
450.2.e.m.151.2	yes	4	45.22	odd	12
450.2.e.m.301.1	yes	4	45.7	odd	12
450.2.j.f.49.1		8	45.34	even	6
450.2.j.f.49.4		8	9.7	even	3
450.2.j.f.349.1		8	9.4	even	3
450.2.j.f.349.4		8	45.4	even	6
1350.2.e.k.451.2		4	45.32	even	12
1350.2.e.k.901.2		4	45.2	even	12
1350.2.e.n.451.1		4	45.23	even	12
1350.2.e.n.901.1		4	45.38	even	12
1350.2.j.g.199.2		8	9.2	odd	6
1350.2.j.g.199.3		8	45.29	odd	6
1350.2.j.g.1099.2		8	45.14	odd	6
1350.2.j.g.1099.3		8	9.5	odd	6
4050.2.a.bl.1.2		2	15.8	even	4
4050.2.a.br.1.1		2	5.2	odd	4
4050.2.a.bu.1.2		2	5.3	odd	4
4050.2.a.by.1.1		2	15.2	even	4
4050.2.c.w.649.2		4	3.2	odd	2
4050.2.c.w.649.3		4	15.14	odd	2
4050.2.c.y.649.1		4	5.4	even	2	inner
4050.2.c.y.649.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.449490i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.449490i q^{7} -1.00000i q^{8} -4.89898 q^{11} -0.449490i q^{13} -0.449490 q^{14} +1.00000 q^{16} -4.89898i q^{17} -7.44949 q^{19} -4.89898i q^{22} -2.44949i q^{23} +0.449490 q^{26} -0.449490i q^{28} +2.44949 q^{29} +4.44949 q^{31} +1.00000i q^{32} +4.89898 q^{34} +11.3485i q^{37} -7.44949i q^{38} +9.00000 q^{41} +2.55051i q^{43} +4.89898 q^{44} +2.44949 q^{46} +10.8990i q^{47} +6.79796 q^{49} +0.449490i q^{52} -3.55051i q^{53} +0.449490 q^{56} +2.44949i q^{58} +5.44949 q^{59} +8.00000 q^{61} +4.44949i q^{62} -1.00000 q^{64} -0.348469i q^{67} +4.89898i q^{68} +13.3485 q^{71} -1.00000i q^{73} -11.3485 q^{74} +7.44949 q^{76} -2.20204i q^{77} -16.6969 q^{79} +9.00000i q^{82} -5.44949i q^{83} -2.55051 q^{86} +4.89898i q^{88} +9.00000 q^{89} +0.202041 q^{91} +2.44949i q^{92} -10.8990 q^{94} -8.79796i q^{97} +6.79796i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 8 q^{14} + 4 q^{16} - 20 q^{19} - 8 q^{26} + 8 q^{31} + 36 q^{41} - 12 q^{49} - 8 q^{56} + 12 q^{59} + 32 q^{61} - 4 q^{64} + 24 q^{71} - 16 q^{74} + 20 q^{76} - 8 q^{79} - 20 q^{86} + 36 q^{89} + 40 q^{91} - 24 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^14 + 4 * q^16 - 20 * q^19 - 8 * q^26 + 8 * q^31 + 36 * q^41 - 12 * q^49 - 8 * q^56 + 12 * q^59 + 32 * q^61 - 4 * q^64 + 24 * q^71 - 16 * q^74 + 20 * q^76 - 8 * q^79 - 20 * q^86 + 36 * q^89 + 40 * q^91 - 24 * q^94

Introduction

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