LMFDB - Embedded newform 3584.2.b.i.1793.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.b (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:	$28.6183840844$
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} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + \cdots + 8 $ x^12 - 4x^11 + 4x^10 + 8x^9 - 24x^8 + 24x^7 + 176x^6 + 160x^5 + 145x^4 + 108x^3 + 68x^2 + 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.5
Root			$2.93456 - 1.21553i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.i.1793.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.01685i q^{3} -1.29145i q^{5} -1.00000 q^{7} +1.96601 q^{9} +O(q^{10})$	$q-1.01685i q^{3} -1.29145i q^{5} -1.00000 q^{7} +1.96601 q^{9} +3.99188i q^{11} +0.534938i q^{13} -1.31322 q^{15} -6.47379 q^{17} +3.19046i q^{19} +1.01685i q^{21} -2.28447 q^{23} +3.33215 q^{25} -5.04971i q^{27} -6.09287i q^{29} +5.70453 q^{31} +4.05916 q^{33} +1.29145i q^{35} -6.44008i q^{37} +0.543954 q^{39} +4.27923 q^{41} -5.11578i q^{43} -2.53901i q^{45} +7.83993 q^{47} +1.00000 q^{49} +6.58291i q^{51} -13.5814i q^{53} +5.15533 q^{55} +3.24424 q^{57} -9.71935i q^{59} -2.06814i q^{61} -1.96601 q^{63} +0.690848 q^{65} +9.06175i q^{67} +2.32298i q^{69} +16.4535 q^{71} -8.18408 q^{73} -3.38831i q^{75} -3.99188i q^{77} +0.960761 q^{79} +0.763200 q^{81} -5.40599i q^{83} +8.36061i q^{85} -6.19556 q^{87} -4.97125 q^{89} -0.534938i q^{91} -5.80068i q^{93} +4.12034 q^{95} +6.97532 q^{97} +7.84806i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 20 * q^9	$ 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) $ 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.01685i	− 0.587081i	−0.955947	−	0.293541i	$-0.905166\pi$
$3$	− 1.01685i	− 0.587081i	0.955947	−	0.293541i	$-0.0948336\pi$
$4$	0	0
$5$	− 1.29145i	− 0.577556i	−0.957396	−	0.288778i	$-0.906751\pi$
$5$	− 1.29145i	− 0.577556i	0.957396	−	0.288778i	$-0.0932489\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.96601	0.655335
$10$	0	0
$11$	3.99188i	1.20360i	0.798648	+	0.601798i	$0.205549\pi$
$11$	3.99188i	1.20360i	−0.798648	+	0.601798i	$0.794451\pi$
$12$	0	0
$13$	0.534938i	0.148365i	0.997245	+	0.0741825i	$0.0236347\pi$
$13$	0.534938i	0.148365i	−0.997245	+	0.0741825i	$0.976365\pi$
$14$	0	0
$15$	−1.31322	−0.339072
$16$	0	0
$17$	−6.47379	−1.57013	−0.785063	−	0.619416i	$-0.787369\pi$
$17$	−6.47379	−1.57013	−0.785063	+	0.619416i	$0.787369\pi$
$18$	0	0
$19$	3.19046i	0.731942i	0.930626	+	0.365971i	$0.119263\pi$
$19$	3.19046i	0.731942i	−0.930626	+	0.365971i	$0.880737\pi$
$20$	0	0
$21$	1.01685i	0.221896i
$22$	0	0
$23$	−2.28447	−0.476346	−0.238173	−	0.971223i	$-0.576548\pi$
$23$	−2.28447	−0.476346	−0.238173	+	0.971223i	$0.576548\pi$
$24$	0	0
$25$	3.33215	0.666429
$26$	0	0
$27$	− 5.04971i	− 0.971817i
$28$	0	0
$29$	− 6.09287i	− 1.13142i	−0.824605	−	0.565709i	$-0.808603\pi$
$29$	− 6.09287i	− 1.13142i	0.824605	−	0.565709i	$-0.191397\pi$
$30$	0	0
$31$	5.70453	1.02456	0.512282	−	0.858818i	$-0.328800\pi$
$31$	5.70453	1.02456	0.512282	+	0.858818i	$0.328800\pi$
$32$	0	0
$33$	4.05916	0.706609
$34$	0	0
$35$	1.29145i	0.218296i
$36$	0	0
$37$	− 6.44008i	− 1.05874i	−0.848390	−	0.529372i	$-0.822428\pi$
$37$	− 6.44008i	− 1.05874i	0.848390	−	0.529372i	$-0.177572\pi$
$38$	0	0
$39$	0.543954	0.0871024
$40$	0	0
$41$	4.27923	0.668303	0.334152	−	0.942519i	$-0.391550\pi$
$41$	4.27923	0.668303	0.334152	+	0.942519i	$0.391550\pi$
$42$	0	0
$43$	− 5.11578i	− 0.780149i	−0.920783	−	0.390074i	$-0.872449\pi$
$43$	− 5.11578i	− 0.780149i	0.920783	−	0.390074i	$-0.127551\pi$
$44$	0	0
$45$	− 2.53901i	− 0.378493i
$46$	0	0
$47$	7.83993	1.14357	0.571786	−	0.820403i	$-0.306251\pi$
$47$	7.83993	1.14357	0.571786	+	0.820403i	$0.306251\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.58291i	0.921792i
$52$	0	0
$53$	− 13.5814i	− 1.86556i	−0.360452	−	0.932778i	$-0.617378\pi$
$53$	− 13.5814i	− 1.86556i	0.360452	−	0.932778i	$-0.382622\pi$
$54$	0	0
$55$	5.15533	0.695144
$56$	0	0
$57$	3.24424	0.429710
$58$	0	0
$59$	− 9.71935i	− 1.26535i	−0.774417	−	0.632676i	$-0.781957\pi$
$59$	− 9.71935i	− 1.26535i	0.774417	−	0.632676i	$-0.218043\pi$
$60$	0	0
$61$	− 2.06814i	− 0.264798i	−0.991196	−	0.132399i	$-0.957732\pi$
$61$	− 2.06814i	− 0.264798i	0.991196	−	0.132399i	$-0.0422680\pi$
$62$	0	0
$63$	−1.96601	−0.247694
$64$	0	0
$65$	0.690848	0.0856891
$66$	0	0
$67$	9.06175i	1.10707i	0.832826	+	0.553535i	$0.186721\pi$
$67$	9.06175i	1.10707i	−0.832826	+	0.553535i	$0.813279\pi$
$68$	0	0
$69$	2.32298i	0.279654i
$70$	0	0
$71$	16.4535	1.95267	0.976334	−	0.216266i	$-0.0693880\pi$
$71$	16.4535	1.95267	0.976334	+	0.216266i	$0.0693880\pi$
$72$	0	0
$73$	−8.18408	−0.957874	−0.478937	−	0.877849i	$-0.658978\pi$
$73$	−8.18408	−0.957874	−0.478937	+	0.877849i	$0.658978\pi$
$74$	0	0
$75$	− 3.38831i	− 0.391248i
$76$	0	0
$77$	− 3.99188i	− 0.454917i
$78$	0	0
$79$	0.960761	0.108094	0.0540470	−	0.998538i	$-0.482788\pi$
$79$	0.960761	0.108094	0.0540470	+	0.998538i	$0.482788\pi$
$80$	0	0
$81$	0.763200	0.0848000
$82$	0	0
$83$	− 5.40599i	− 0.593384i	−0.954973	−	0.296692i	$-0.904116\pi$
$83$	− 5.40599i	− 0.593384i	0.954973	−	0.296692i	$-0.0958835\pi$
$84$	0	0
$85$	8.36061i	0.906835i
$86$	0	0
$87$	−6.19556	−0.664234
$88$	0	0
$89$	−4.97125	−0.526952	−0.263476	−	0.964666i	$-0.584869\pi$
$89$	−4.97125	−0.526952	−0.263476	+	0.964666i	$0.584869\pi$
$90$	0	0
$91$	− 0.534938i	− 0.0560767i
$92$	0	0
$93$	− 5.80068i	− 0.601502i
$94$	0	0
$95$	4.12034	0.422738
$96$	0	0
$97$	6.97532	0.708237	0.354118	−	0.935201i	$-0.384781\pi$
$97$	6.97532	0.708237	0.354118	+	0.935201i	$0.384781\pi$
$98$	0	0
$99$	7.84806i	0.788759i
$100$	0	0
$101$	− 6.19179i	− 0.616106i	−0.951369	−	0.308053i	$-0.900323\pi$
$101$	− 6.19179i	− 0.616106i	0.951369	−	0.308053i	$-0.0996775\pi$
$102$	0	0
$103$	−9.38281	−0.924516	−0.462258	−	0.886745i	$-0.652961\pi$
$103$	−9.38281	−0.924516	−0.462258	+	0.886745i	$0.652961\pi$
$104$	0	0
$105$	1.31322	0.128157
$106$	0	0
$107$	2.14532i	0.207396i	0.994609	+	0.103698i	$0.0330675\pi$
$107$	2.14532i	0.207396i	−0.994609	+	0.103698i	$0.966933\pi$
$108$	0	0
$109$	8.50343i	0.814481i	0.913321	+	0.407241i	$0.133509\pi$
$109$	8.50343i	0.814481i	−0.913321	+	0.407241i	$0.866491\pi$
$110$	0	0
$111$	−6.54863	−0.621569
$112$	0	0
$113$	−13.0194	−1.22476	−0.612382	−	0.790562i	$-0.709788\pi$
$113$	−13.0194	−1.22476	−0.612382	+	0.790562i	$0.709788\pi$
$114$	0	0
$115$	2.95029i	0.275116i
$116$	0	0
$117$	1.05169i	0.0972289i
$118$	0	0
$119$	6.47379	0.593452
$120$	0	0
$121$	−4.93509	−0.448644
$122$	0	0
$123$	− 4.35135i	− 0.392348i
$124$	0	0
$125$	− 10.7606i	− 0.962456i
$126$	0	0
$127$	−2.47162	−0.219321	−0.109660	−	0.993969i	$-0.534976\pi$
$127$	−2.47162	−0.219321	−0.109660	+	0.993969i	$0.534976\pi$
$128$	0	0
$129$	−5.20200	−0.458011
$130$	0	0
$131$	3.65319i	0.319181i	0.987183	+	0.159590i	$0.0510173\pi$
$131$	3.65319i	0.319181i	−0.987183	+	0.159590i	$0.948983\pi$
$132$	0	0
$133$	− 3.19046i	− 0.276648i
$134$	0	0
$135$	−6.52147	−0.561279
$136$	0	0
$137$	13.4016	1.14498	0.572490	−	0.819912i	$-0.305978\pi$
$137$	13.4016	1.14498	0.572490	+	0.819912i	$0.305978\pi$
$138$	0	0
$139$	− 16.3563i	− 1.38732i	−0.720302	−	0.693661i	$-0.755997\pi$
$139$	− 16.3563i	− 1.38732i	0.720302	−	0.693661i	$-0.244003\pi$
$140$	0	0
$141$	− 7.97207i	− 0.671370i
$142$	0	0
$143$	−2.13541	−0.178572
$144$	0	0
$145$	−7.86866	−0.653457
$146$	0	0
$147$	− 1.01685i	− 0.0838688i
$148$	0	0
$149$	1.92053i	0.157336i	0.996901	+	0.0786678i	$0.0250666\pi$
$149$	1.92053i	0.157336i	−0.996901	+	0.0786678i	$0.974933\pi$
$150$	0	0
$151$	4.47078	0.363827	0.181913	−	0.983315i	$-0.441771\pi$
$151$	4.47078	0.363827	0.181913	+	0.983315i	$0.441771\pi$
$152$	0	0
$153$	−12.7275	−1.02896
$154$	0	0
$155$	− 7.36714i	− 0.591743i
$156$	0	0
$157$	− 4.92407i	− 0.392984i	−0.980505	−	0.196492i	$-0.937045\pi$
$157$	− 4.92407i	− 0.392984i	0.980505	−	0.196492i	$-0.0629549\pi$
$158$	0	0
$159$	−13.8104	−1.09523
$160$	0	0
$161$	2.28447	0.180042
$162$	0	0
$163$	11.7644i	0.921458i	0.887541	+	0.460729i	$0.152412\pi$
$163$	11.7644i	0.921458i	−0.887541	+	0.460729i	$0.847588\pi$
$164$	0	0
$165$	− 5.24222i	− 0.408106i
$166$	0	0
$167$	−11.5691	−0.895245	−0.447623	−	0.894223i	$-0.647729\pi$
$167$	−11.5691	−0.895245	−0.447623	+	0.894223i	$0.647729\pi$
$168$	0	0
$169$	12.7138	0.977988
$170$	0	0
$171$	6.27247i	0.479668i
$172$	0	0
$173$	− 25.6008i	− 1.94640i	−0.229968	−	0.973198i	$-0.573862\pi$
$173$	− 25.6008i	− 1.94640i	0.229968	−	0.973198i	$-0.426138\pi$
$174$	0	0
$175$	−3.33215	−0.251887
$176$	0	0
$177$	−9.88316	−0.742864
$178$	0	0
$179$	10.2520i	0.766267i	0.923693	+	0.383134i	$0.125155\pi$
$179$	10.2520i	0.766267i	−0.923693	+	0.383134i	$0.874845\pi$
$180$	0	0
$181$	− 26.2524i	− 1.95133i	−0.219270	−	0.975664i	$-0.570368\pi$
$181$	− 26.2524i	− 1.95133i	0.219270	−	0.975664i	$-0.429632\pi$
$182$	0	0
$183$	−2.10300	−0.155458
$184$	0	0
$185$	−8.31708	−0.611484
$186$	0	0
$187$	− 25.8426i	− 1.88980i
$188$	0	0
$189$	5.04971i	0.367312i
$190$	0	0
$191$	8.18715	0.592401	0.296201	−	0.955126i	$-0.404280\pi$
$191$	8.18715	0.592401	0.296201	+	0.955126i	$0.404280\pi$
$192$	0	0
$193$	25.3226	1.82276	0.911382	−	0.411561i	$-0.135016\pi$
$193$	25.3226	1.82276	0.911382	+	0.411561i	$0.135016\pi$
$194$	0	0
$195$	− 0.702492i	− 0.0503065i
$196$	0	0
$197$	− 20.8467i	− 1.48526i	−0.669699	−	0.742632i	$-0.733577\pi$
$197$	− 20.8467i	− 1.48526i	0.669699	−	0.742632i	$-0.266423\pi$
$198$	0	0
$199$	20.2055	1.43233	0.716164	−	0.697932i	$-0.245896\pi$
$199$	20.2055	1.43233	0.716164	+	0.697932i	$0.245896\pi$
$200$	0	0
$201$	9.21449	0.649940
$202$	0	0
$203$	6.09287i	0.427636i
$204$	0	0
$205$	− 5.52643i	− 0.385982i
$206$	0	0
$207$	−4.49129	−0.312166
$208$	0	0
$209$	−12.7359	−0.880963
$210$	0	0
$211$	− 0.231640i	− 0.0159468i	−0.999968	−	0.00797339i	$-0.997462\pi$
$211$	− 0.231640i	− 0.0159468i	0.999968	−	0.00797339i	$-0.00253804\pi$
$212$	0	0
$213$	− 16.7308i	− 1.14638i
$214$	0	0
$215$	−6.60679	−0.450580
$216$	0	0
$217$	−5.70453	−0.387249
$218$	0	0
$219$	8.32202i	0.562350i
$220$	0	0
$221$	− 3.46308i	− 0.232952i
$222$	0	0
$223$	−22.3886	−1.49925	−0.749625	−	0.661863i	$-0.769766\pi$
$223$	−22.3886	−1.49925	−0.749625	+	0.661863i	$0.769766\pi$
$224$	0	0
$225$	6.55102	0.436735
$226$	0	0
$227$	17.1167i	1.13608i	0.823002	+	0.568038i	$0.192297\pi$
$227$	17.1167i	1.13608i	−0.823002	+	0.568038i	$0.807703\pi$
$228$	0	0
$229$	15.0638i	0.995445i	0.867336	+	0.497723i	$0.165830\pi$
$229$	15.0638i	0.995445i	−0.867336	+	0.497723i	$0.834170\pi$
$230$	0	0
$231$	−4.05916	−0.267073
$232$	0	0
$233$	−11.8566	−0.776752	−0.388376	−	0.921501i	$-0.626964\pi$
$233$	−11.8566	−0.776752	−0.388376	+	0.921501i	$0.626964\pi$
$234$	0	0
$235$	− 10.1249i	− 0.660477i
$236$	0	0
$237$	− 0.976955i	− 0.0634600i
$238$	0	0
$239$	8.55444	0.553341	0.276670	−	0.960965i	$-0.410769\pi$
$239$	8.55444	0.553341	0.276670	+	0.960965i	$0.410769\pi$
$240$	0	0
$241$	−14.3170	−0.922236	−0.461118	−	0.887339i	$-0.652552\pi$
$241$	−14.3170	−0.922236	−0.461118	+	0.887339i	$0.652552\pi$
$242$	0	0
$243$	− 15.9252i	− 1.02160i
$244$	0	0
$245$	− 1.29145i	− 0.0825080i
$246$	0	0
$247$	−1.70670	−0.108595
$248$	0	0
$249$	−5.49710	−0.348365
$250$	0	0
$251$	0.346212i	0.0218527i	0.999940	+	0.0109264i	$0.00347804\pi$
$251$	0.346212i	0.0218527i	−0.999940	+	0.0109264i	$0.996522\pi$
$252$	0	0
$253$	− 9.11934i	− 0.573328i
$254$	0	0
$255$	8.50153	0.532386
$256$	0	0
$257$	21.7703	1.35799	0.678996	−	0.734142i	$-0.262415\pi$
$257$	21.7703	1.35799	0.678996	+	0.734142i	$0.262415\pi$
$258$	0	0
$259$	6.44008i	0.400167i
$260$	0	0
$261$	− 11.9786i	− 0.741458i
$262$	0	0
$263$	−16.0380	−0.988946	−0.494473	−	0.869193i	$-0.664639\pi$
$263$	−16.0380	−0.988946	−0.494473	+	0.869193i	$0.664639\pi$
$264$	0	0
$265$	−17.5398	−1.07746
$266$	0	0
$267$	5.05504i	0.309363i
$268$	0	0
$269$	6.98157i	0.425674i	0.977088	+	0.212837i	$0.0682703\pi$
$269$	6.98157i	0.425674i	−0.977088	+	0.212837i	$0.931730\pi$
$270$	0	0
$271$	−3.13231	−0.190275	−0.0951373	−	0.995464i	$-0.530329\pi$
$271$	−3.13231	−0.190275	−0.0951373	+	0.995464i	$0.530329\pi$
$272$	0	0
$273$	−0.543954	−0.0329216
$274$	0	0
$275$	13.3015i	0.802112i
$276$	0	0
$277$	− 2.07134i	− 0.124454i	−0.998062	−	0.0622272i	$-0.980180\pi$
$277$	− 2.07134i	− 0.124454i	0.998062	−	0.0622272i	$-0.0198204\pi$
$278$	0	0
$279$	11.2151	0.671433
$280$	0	0
$281$	−10.1923	−0.608024	−0.304012	−	0.952668i	$-0.598326\pi$
$281$	−10.1923	−0.608024	−0.304012	+	0.952668i	$0.598326\pi$
$282$	0	0
$283$	14.8446i	0.882421i	0.897404	+	0.441210i	$0.145451\pi$
$283$	14.8446i	0.882421i	−0.897404	+	0.441210i	$0.854549\pi$
$284$	0	0
$285$	− 4.18978i	− 0.248181i
$286$	0	0
$287$	−4.27923	−0.252595
$288$	0	0
$289$	24.9100	1.46529
$290$	0	0
$291$	− 7.09289i	− 0.415792i
$292$	0	0
$293$	10.2362i	0.598007i	0.954252	+	0.299004i	$0.0966543\pi$
$293$	10.2362i	0.598007i	−0.954252	+	0.299004i	$0.903346\pi$
$294$	0	0
$295$	−12.5521	−0.730811
$296$	0	0
$297$	20.1578	1.16967
$298$	0	0
$299$	− 1.22205i	− 0.0706730i
$300$	0	0
$301$	5.11578i	0.294869i
$302$	0	0
$303$	−6.29615	−0.361705
$304$	0	0
$305$	−2.67091	−0.152936
$306$	0	0
$307$	19.6132i	1.11938i	0.828701	+	0.559691i	$0.189080\pi$
$307$	19.6132i	1.11938i	−0.828701	+	0.559691i	$0.810920\pi$
$308$	0	0
$309$	9.54096i	0.542766i
$310$	0	0
$311$	−9.88322	−0.560426	−0.280213	−	0.959938i	$-0.590405\pi$
$311$	−9.88322	−0.560426	−0.280213	+	0.959938i	$0.590405\pi$
$312$	0	0
$313$	−0.573015	−0.0323887	−0.0161944	−	0.999869i	$-0.505155\pi$
$313$	−0.573015	−0.0323887	−0.0161944	+	0.999869i	$0.505155\pi$
$314$	0	0
$315$	2.53901i	0.143057i
$316$	0	0
$317$	19.7491i	1.10922i	0.832110	+	0.554611i	$0.187133\pi$
$317$	19.7491i	1.10922i	−0.832110	+	0.554611i	$0.812867\pi$
$318$	0	0
$319$	24.3220	1.36177
$320$	0	0
$321$	2.18148	0.121758
$322$	0	0
$323$	− 20.6544i	− 1.14924i
$324$	0	0
$325$	1.78249i	0.0988748i
$326$	0	0
$327$	8.64676	0.478167
$328$	0	0
$329$	−7.83993	−0.432230
$330$	0	0
$331$	− 22.8436i	− 1.25560i	−0.778376	−	0.627799i	$-0.783956\pi$
$331$	− 22.8436i	− 1.25560i	0.778376	−	0.627799i	$-0.216044\pi$
$332$	0	0
$333$	− 12.6612i	− 0.693832i
$334$	0	0
$335$	11.7028	0.639395
$336$	0	0
$337$	−2.69392	−0.146747	−0.0733736	−	0.997305i	$-0.523377\pi$
$337$	−2.69392	−0.146747	−0.0733736	+	0.997305i	$0.523377\pi$
$338$	0	0
$339$	13.2389i	0.719036i
$340$	0	0
$341$	22.7718i	1.23316i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.00002	0.161516
$346$	0	0
$347$	− 6.08073i	− 0.326431i	−0.986590	−	0.163215i	$-0.947813\pi$
$347$	− 6.08073i	− 0.326431i	0.986590	−	0.163215i	$-0.0521865\pi$
$348$	0	0
$349$	17.0896i	0.914787i	0.889264	+	0.457394i	$0.151217\pi$
$349$	17.0896i	0.914787i	−0.889264	+	0.457394i	$0.848783\pi$
$350$	0	0
$351$	2.70128	0.144184
$352$	0	0
$353$	28.3967	1.51140	0.755702	−	0.654916i	$-0.227296\pi$
$353$	28.3967	1.51140	0.755702	+	0.654916i	$0.227296\pi$
$354$	0	0
$355$	− 21.2489i	− 1.12778i
$356$	0	0
$357$	− 6.58291i	− 0.348404i
$358$	0	0
$359$	−5.18843	−0.273835	−0.136917	−	0.990582i	$-0.543719\pi$
$359$	−5.18843	−0.273835	−0.136917	+	0.990582i	$0.543719\pi$
$360$	0	0
$361$	8.82095	0.464260
$362$	0	0
$363$	5.01826i	0.263391i
$364$	0	0
$365$	10.5694i	0.553226i
$366$	0	0
$367$	−20.9256	−1.09231	−0.546153	−	0.837685i	$-0.683908\pi$
$367$	−20.9256	−1.09231	−0.546153	+	0.837685i	$0.683908\pi$
$368$	0	0
$369$	8.41299	0.437963
$370$	0	0
$371$	13.5814i	0.705114i
$372$	0	0
$373$	− 5.93536i	− 0.307321i	−0.988124	−	0.153661i	$-0.950894\pi$
$373$	− 5.93536i	− 0.307321i	0.988124	−	0.153661i	$-0.0491063\pi$
$374$	0	0
$375$	−10.9420	−0.565040
$376$	0	0
$377$	3.25931	0.167863
$378$	0	0
$379$	− 6.04243i	− 0.310379i	−0.987885	−	0.155189i	$-0.950401\pi$
$379$	− 6.04243i	− 0.310379i	0.987885	−	0.155189i	$-0.0495988\pi$
$380$	0	0
$381$	2.51328i	0.128759i
$382$	0	0
$383$	3.53306	0.180531	0.0902653	−	0.995918i	$-0.471228\pi$
$383$	3.53306	0.180531	0.0902653	+	0.995918i	$0.471228\pi$
$384$	0	0
$385$	−5.15533	−0.262740
$386$	0	0
$387$	− 10.0577i	− 0.511259i
$388$	0	0
$389$	− 2.75123i	− 0.139493i	−0.997565	−	0.0697466i	$-0.977781\pi$
$389$	− 2.75123i	− 0.139493i	0.997565	−	0.0697466i	$-0.0222191\pi$
$390$	0	0
$391$	14.7892	0.747922
$392$	0	0
$393$	3.71477	0.187385
$394$	0	0
$395$	− 1.24078i	− 0.0624304i
$396$	0	0
$397$	− 27.5791i	− 1.38416i	−0.721822	−	0.692079i	$-0.756695\pi$
$397$	− 27.5791i	− 1.38416i	0.721822	−	0.692079i	$-0.243305\pi$
$398$	0	0
$399$	−3.24424	−0.162415
$400$	0	0
$401$	−1.51409	−0.0756099	−0.0378050	−	0.999285i	$-0.512037\pi$
$401$	−1.51409	−0.0756099	−0.0378050	+	0.999285i	$0.512037\pi$
$402$	0	0
$403$	3.05157i	0.152009i
$404$	0	0
$405$	− 0.985638i	− 0.0489767i
$406$	0	0
$407$	25.7080	1.27430
$408$	0	0
$409$	3.50489	0.173306	0.0866529	−	0.996239i	$-0.472383\pi$
$409$	3.50489	0.173306	0.0866529	+	0.996239i	$0.472383\pi$
$410$	0	0
$411$	− 13.6275i	− 0.672196i
$412$	0	0
$413$	9.71935i	0.478258i
$414$	0	0
$415$	−6.98159	−0.342713
$416$	0	0
$417$	−16.6320	−0.814471
$418$	0	0
$419$	0.0423241i	0.00206767i	0.999999	+	0.00103383i	$0.000329080\pi$
$419$	0.0423241i	0.00206767i	−0.999999	+	0.00103383i	$0.999671\pi$
$420$	0	0
$421$	25.8137i	1.25808i	0.777372	+	0.629041i	$0.216552\pi$
$421$	25.8137i	1.25808i	−0.777372	+	0.629041i	$0.783448\pi$
$422$	0	0
$423$	15.4134	0.749423
$424$	0	0
$425$	−21.5716	−1.04638
$426$	0	0
$427$	2.06814i	0.100084i
$428$	0	0
$429$	2.17140i	0.104836i
$430$	0	0
$431$	33.0398	1.59147	0.795735	−	0.605645i	$-0.207085\pi$
$431$	33.0398	1.59147	0.795735	+	0.605645i	$0.207085\pi$
$432$	0	0
$433$	−10.7807	−0.518086	−0.259043	−	0.965866i	$-0.583407\pi$
$433$	−10.7807	−0.518086	−0.259043	+	0.965866i	$0.583407\pi$
$434$	0	0
$435$	8.00129i	0.383632i
$436$	0	0
$437$	− 7.28853i	− 0.348657i
$438$	0	0
$439$	−23.5740	−1.12513	−0.562564	−	0.826754i	$-0.690185\pi$
$439$	−23.5740	−1.12513	−0.562564	+	0.826754i	$0.690185\pi$
$440$	0	0
$441$	1.96601	0.0936193
$442$	0	0
$443$	35.7764i	1.69979i	0.526955	+	0.849893i	$0.323334\pi$
$443$	35.7764i	1.69979i	−0.526955	+	0.849893i	$0.676666\pi$
$444$	0	0
$445$	6.42014i	0.304344i
$446$	0	0
$447$	1.95290	0.0923688
$448$	0	0
$449$	−12.9320	−0.610300	−0.305150	−	0.952304i	$-0.598707\pi$
$449$	−12.9320	−0.610300	−0.305150	+	0.952304i	$0.598707\pi$
$450$	0	0
$451$	17.0822i	0.804367i
$452$	0	0
$453$	− 4.54613i	− 0.213596i
$454$	0	0
$455$	−0.690848	−0.0323875
$456$	0	0
$457$	30.5150	1.42743	0.713715	−	0.700436i	$-0.247011\pi$
$457$	30.5150	1.42743	0.713715	+	0.700436i	$0.247011\pi$
$458$	0	0
$459$	32.6908i	1.52587i
$460$	0	0
$461$	− 19.6155i	− 0.913585i	−0.889573	−	0.456793i	$-0.848998\pi$
$461$	− 19.6155i	− 0.913585i	0.889573	−	0.456793i	$-0.151002\pi$
$462$	0	0
$463$	15.3254	0.712231	0.356116	−	0.934442i	$-0.384101\pi$
$463$	15.3254	0.712231	0.356116	+	0.934442i	$0.384101\pi$
$464$	0	0
$465$	−7.49131	−0.347401
$466$	0	0
$467$	− 30.2233i	− 1.39857i	−0.714845	−	0.699283i	$-0.753503\pi$
$467$	− 30.2233i	− 1.39857i	0.714845	−	0.699283i	$-0.246497\pi$
$468$	0	0
$469$	− 9.06175i	− 0.418433i
$470$	0	0
$471$	−5.00706	−0.230713
$472$	0	0
$473$	20.4216	0.938984
$474$	0	0
$475$	10.6311i	0.487788i
$476$	0	0
$477$	− 26.7012i	− 1.22256i
$478$	0	0
$479$	−18.2121	−0.832131	−0.416066	−	0.909335i	$-0.636591\pi$
$479$	−18.2121	−0.832131	−0.416066	+	0.909335i	$0.636591\pi$
$480$	0	0
$481$	3.44505	0.157081
$482$	0	0
$483$	− 2.32298i	− 0.105699i
$484$	0	0
$485$	− 9.00831i	− 0.409046i
$486$	0	0
$487$	−33.1641	−1.50281	−0.751404	−	0.659843i	$-0.770623\pi$
$487$	−33.1641	−1.50281	−0.751404	+	0.659843i	$0.770623\pi$
$488$	0	0
$489$	11.9627	0.540971
$490$	0	0
$491$	− 38.1547i	− 1.72190i	−0.508690	−	0.860950i	$-0.669870\pi$
$491$	− 38.1547i	− 1.72190i	0.508690	−	0.860950i	$-0.330130\pi$
$492$	0	0
$493$	39.4440i	1.77647i
$494$	0	0
$495$	10.1354	0.455553
$496$	0	0
$497$	−16.4535	−0.738039
$498$	0	0
$499$	1.44819i	0.0648298i	0.999474	+	0.0324149i	$0.0103198\pi$
$499$	1.44819i	0.0648298i	−0.999474	+	0.0324149i	$0.989680\pi$
$500$	0	0
$501$	11.7641i	0.525582i
$502$	0	0
$503$	−14.4905	−0.646098	−0.323049	−	0.946382i	$-0.604708\pi$
$503$	−14.4905	−0.646098	−0.323049	+	0.946382i	$0.604708\pi$
$504$	0	0
$505$	−7.99642	−0.355836
$506$	0	0
$507$	− 12.9281i	− 0.574158i
$508$	0	0
$509$	− 29.1421i	− 1.29170i	−0.763464	−	0.645850i	$-0.776503\pi$
$509$	− 29.1421i	− 1.29170i	0.763464	−	0.645850i	$-0.223497\pi$
$510$	0	0
$511$	8.18408	0.362042
$512$	0	0
$513$	16.1109	0.711314
$514$	0	0
$515$	12.1175i	0.533960i
$516$	0	0
$517$	31.2961i	1.37640i
$518$	0	0
$519$	−26.0323	−1.14269
$520$	0	0
$521$	−1.25214	−0.0548573	−0.0274287	−	0.999624i	$-0.508732\pi$
$521$	−1.25214	−0.0548573	−0.0274287	+	0.999624i	$0.508732\pi$
$522$	0	0
$523$	− 34.3401i	− 1.50159i	−0.660536	−	0.750795i	$-0.729671\pi$
$523$	− 34.3401i	− 1.50159i	0.660536	−	0.750795i	$-0.270329\pi$
$524$	0	0
$525$	3.38831i	0.147878i
$526$	0	0
$527$	−36.9299	−1.60869
$528$	0	0
$529$	−17.7812	−0.773095
$530$	0	0
$531$	− 19.1083i	− 0.829229i
$532$	0	0
$533$	2.28912i	0.0991529i
$534$	0	0
$535$	2.77058	0.119783
$536$	0	0
$537$	10.4247	0.449861
$538$	0	0
$539$	3.99188i	0.171942i
$540$	0	0
$541$	37.9611i	1.63207i	0.578000	+	0.816037i	$0.303833\pi$
$541$	37.9611i	1.63207i	−0.578000	+	0.816037i	$0.696167\pi$
$542$	0	0
$543$	−26.6949	−1.14559
$544$	0	0
$545$	10.9818	0.470409
$546$	0	0
$547$	− 9.82521i	− 0.420096i	−0.977691	−	0.210048i	$-0.932638\pi$
$547$	− 9.82521i	− 0.420096i	0.977691	−	0.210048i	$-0.0673620\pi$
$548$	0	0
$549$	− 4.06598i	− 0.173532i
$550$	0	0
$551$	19.4391	0.828132
$552$	0	0
$553$	−0.960761	−0.0408557
$554$	0	0
$555$	8.45726i	0.358991i
$556$	0	0
$557$	32.5759i	1.38028i	0.723674	+	0.690142i	$0.242452\pi$
$557$	32.5759i	1.38028i	−0.723674	+	0.690142i	$0.757548\pi$
$558$	0	0
$559$	2.73662	0.115747
$560$	0	0
$561$	−26.2782	−1.10946
$562$	0	0
$563$	− 40.1157i	− 1.69067i	−0.534234	−	0.845337i	$-0.679400\pi$
$563$	− 40.1157i	− 1.69067i	0.534234	−	0.845337i	$-0.320600\pi$
$564$	0	0
$565$	16.8140i	0.707369i
$566$	0	0
$567$	−0.763200	−0.0320514
$568$	0	0
$569$	−0.461975	−0.0193670	−0.00968351	−	0.999953i	$-0.503082\pi$
$569$	−0.461975	−0.0193670	−0.00968351	+	0.999953i	$0.503082\pi$
$570$	0	0
$571$	41.6492i	1.74296i	0.490428	+	0.871482i	$0.336841\pi$
$571$	41.6492i	1.74296i	−0.490428	+	0.871482i	$0.663159\pi$
$572$	0	0
$573$	− 8.32514i	− 0.347788i
$574$	0	0
$575$	−7.61220	−0.317451
$576$	0	0
$577$	−12.1029	−0.503851	−0.251925	−	0.967747i	$-0.581064\pi$
$577$	−12.1029	−0.503851	−0.251925	+	0.967747i	$0.581064\pi$
$578$	0	0
$579$	− 25.7495i	− 1.07011i
$580$	0	0
$581$	5.40599i	0.224278i
$582$	0	0
$583$	54.2155	2.24538
$584$	0	0
$585$	1.35821	0.0561551
$586$	0	0
$587$	14.2165i	0.586779i	0.955993	+	0.293389i	$0.0947833\pi$
$587$	14.2165i	0.586779i	−0.955993	+	0.293389i	$0.905217\pi$
$588$	0	0
$589$	18.2001i	0.749921i
$590$	0	0
$591$	−21.1980	−0.871971
$592$	0	0
$593$	−12.7044	−0.521705	−0.260853	−	0.965379i	$-0.584004\pi$
$593$	−12.7044	−0.521705	−0.260853	+	0.965379i	$0.584004\pi$
$594$	0	0
$595$	− 8.36061i	− 0.342752i
$596$	0	0
$597$	− 20.5460i	− 0.840894i
$598$	0	0
$599$	14.0608	0.574510	0.287255	−	0.957854i	$-0.407257\pi$
$599$	14.0608	0.574510	0.287255	+	0.957854i	$0.407257\pi$
$600$	0	0
$601$	−26.0142	−1.06114	−0.530570	−	0.847641i	$-0.678022\pi$
$601$	−26.0142	−1.06114	−0.530570	+	0.847641i	$0.678022\pi$
$602$	0	0
$603$	17.8155i	0.725502i
$604$	0	0
$605$	6.37344i	0.259117i
$606$	0	0
$607$	28.0201	1.13730	0.568650	−	0.822579i	$-0.307466\pi$
$607$	28.0201	1.13730	0.568650	+	0.822579i	$0.307466\pi$
$608$	0	0
$609$	6.19556	0.251057
$610$	0	0
$611$	4.19388i	0.169666i
$612$	0	0
$613$	− 18.1315i	− 0.732323i	−0.930551	−	0.366161i	$-0.880672\pi$
$613$	− 18.1315i	− 0.732323i	0.930551	−	0.366161i	$-0.119328\pi$
$614$	0	0
$615$	−5.61957	−0.226603
$616$	0	0
$617$	−24.2692	−0.977042	−0.488521	−	0.872552i	$-0.662463\pi$
$617$	−24.2692	−0.977042	−0.488521	+	0.872552i	$0.662463\pi$
$618$	0	0
$619$	34.4384i	1.38420i	0.721803	+	0.692099i	$0.243314\pi$
$619$	34.4384i	1.38420i	−0.721803	+	0.692099i	$0.756686\pi$
$620$	0	0
$621$	11.5359i	0.462921i
$622$	0	0
$623$	4.97125	0.199169
$624$	0	0
$625$	2.76392	0.110557
$626$	0	0
$627$	12.9506i	0.517197i
$628$	0	0
$629$	41.6918i	1.66236i
$630$	0	0
$631$	45.3042	1.80353	0.901766	−	0.432224i	$-0.142271\pi$
$631$	45.3042	1.80353	0.901766	+	0.432224i	$0.142271\pi$
$632$	0	0
$633$	−0.235545	−0.00936206
$634$	0	0
$635$	3.19199i	0.126670i
$636$	0	0
$637$	0.534938i	0.0211950i
$638$	0	0
$639$	32.3476	1.27965
$640$	0	0
$641$	46.2702	1.82756	0.913782	−	0.406204i	$-0.133148\pi$
$641$	46.2702	1.82756	0.913782	+	0.406204i	$0.133148\pi$
$642$	0	0
$643$	− 20.9674i	− 0.826872i	−0.910533	−	0.413436i	$-0.864329\pi$
$643$	− 20.9674i	− 0.826872i	0.910533	−	0.413436i	$-0.135671\pi$
$644$	0	0
$645$	6.71815i	0.264527i
$646$	0	0
$647$	44.7211	1.75817	0.879083	−	0.476668i	$-0.158156\pi$
$647$	44.7211	1.75817	0.879083	+	0.476668i	$0.158156\pi$
$648$	0	0
$649$	38.7984	1.52297
$650$	0	0
$651$	5.80068i	0.227346i
$652$	0	0
$653$	2.23312i	0.0873886i	0.999045	+	0.0436943i	$0.0139128\pi$
$653$	2.23312i	0.0873886i	−0.999045	+	0.0436943i	$0.986087\pi$
$654$	0	0
$655$	4.71793	0.184345
$656$	0	0
$657$	−16.0899	−0.627729
$658$	0	0
$659$	− 45.3644i	− 1.76715i	−0.468294	−	0.883573i	$-0.655131\pi$
$659$	− 45.3644i	− 1.76715i	0.468294	−	0.883573i	$-0.344869\pi$
$660$	0	0
$661$	36.0951i	1.40394i	0.712208	+	0.701969i	$0.247695\pi$
$661$	36.0951i	1.40394i	−0.712208	+	0.701969i	$0.752305\pi$
$662$	0	0
$663$	−3.52145	−0.136762
$664$	0	0
$665$	−4.12034	−0.159780
$666$	0	0
$667$	13.9190i	0.538946i
$668$	0	0
$669$	22.7659i	0.880181i
$670$	0	0
$671$	8.25576	0.318710
$672$	0	0
$673$	9.23324	0.355915	0.177958	−	0.984038i	$-0.443051\pi$
$673$	9.23324	0.355915	0.177958	+	0.984038i	$0.443051\pi$
$674$	0	0
$675$	− 16.8264i	− 0.647647i
$676$	0	0
$677$	42.4578i	1.63179i	0.578202	+	0.815893i	$0.303754\pi$
$677$	42.4578i	1.63179i	−0.578202	+	0.815893i	$0.696246\pi$
$678$	0	0
$679$	−6.97532	−0.267688
$680$	0	0
$681$	17.4052	0.666969
$682$	0	0
$683$	29.5156i	1.12938i	0.825302	+	0.564692i	$0.191005\pi$
$683$	29.5156i	1.12938i	−0.825302	+	0.564692i	$0.808995\pi$
$684$	0	0
$685$	− 17.3076i	− 0.661289i
$686$	0	0
$687$	15.3177	0.584407
$688$	0	0
$689$	7.26523	0.276783
$690$	0	0
$691$	− 21.1911i	− 0.806147i	−0.915168	−	0.403073i	$-0.867942\pi$
$691$	− 21.1911i	− 0.806147i	0.915168	−	0.403073i	$-0.132058\pi$
$692$	0	0
$693$	− 7.84806i	− 0.298123i
$694$	0	0
$695$	−21.1234	−0.801256
$696$	0	0
$697$	−27.7028	−1.04932
$698$	0	0
$699$	12.0564i	0.456017i
$700$	0	0
$701$	− 10.3625i	− 0.391386i	−0.980665	−	0.195693i	$-0.937304\pi$
$701$	− 10.3625i	− 0.391386i	0.980665	−	0.195693i	$-0.0626956\pi$
$702$	0	0
$703$	20.5468	0.774939
$704$	0	0
$705$	−10.2956	−0.387754
$706$	0	0
$707$	6.19179i	0.232866i
$708$	0	0
$709$	14.9196i	0.560318i	0.959954	+	0.280159i	$0.0903871\pi$
$709$	14.9196i	0.560318i	−0.959954	+	0.280159i	$0.909613\pi$
$710$	0	0
$711$	1.88886	0.0708379
$712$	0	0
$713$	−13.0318	−0.488046
$714$	0	0
$715$	2.75778i	0.103135i
$716$	0	0
$717$	− 8.69863i	− 0.324856i
$718$	0	0
$719$	−19.3570	−0.721896	−0.360948	−	0.932586i	$-0.617547\pi$
$719$	−19.3570	−0.721896	−0.360948	+	0.932586i	$0.617547\pi$
$720$	0	0
$721$	9.38281	0.349434
$722$	0	0
$723$	14.5583i	0.541428i
$724$	0	0
$725$	− 20.3023i	− 0.754010i
$726$	0	0
$727$	11.1685	0.414216	0.207108	−	0.978318i	$-0.433595\pi$
$727$	11.1685	0.414216	0.207108	+	0.978318i	$0.433595\pi$
$728$	0	0
$729$	−13.9040	−0.514963
$730$	0	0
$731$	33.1185i	1.22493i
$732$	0	0
$733$	27.4759i	1.01484i	0.861698	+	0.507422i	$0.169401\pi$
$733$	27.4759i	1.01484i	−0.861698	+	0.507422i	$0.830599\pi$
$734$	0	0
$735$	−1.31322	−0.0484389
$736$	0	0
$737$	−36.1734	−1.33246
$738$	0	0
$739$	47.1619i	1.73488i	0.497542	+	0.867440i	$0.334236\pi$
$739$	47.1619i	1.73488i	−0.497542	+	0.867440i	$0.665764\pi$
$740$	0	0
$741$	1.73547i	0.0637539i
$742$	0	0
$743$	−47.0796	−1.72718	−0.863591	−	0.504193i	$-0.831790\pi$
$743$	−47.0796	−1.72718	−0.863591	+	0.504193i	$0.831790\pi$
$744$	0	0
$745$	2.48027	0.0908701
$746$	0	0
$747$	− 10.6282i	− 0.388866i
$748$	0	0
$749$	− 2.14532i	− 0.0783882i
$750$	0	0
$751$	44.5767	1.62663	0.813313	−	0.581827i	$-0.197662\pi$
$751$	44.5767	1.62663	0.813313	+	0.581827i	$0.197662\pi$
$752$	0	0
$753$	0.352047	0.0128293
$754$	0	0
$755$	− 5.77381i	− 0.210130i
$756$	0	0
$757$	28.9656i	1.05277i	0.850246	+	0.526386i	$0.176453\pi$
$757$	28.9656i	1.05277i	−0.850246	+	0.526386i	$0.823547\pi$
$758$	0	0
$759$	−9.27304	−0.336590
$760$	0	0
$761$	9.89164	0.358572	0.179286	−	0.983797i	$-0.442621\pi$
$761$	9.89164	0.358572	0.179286	+	0.983797i	$0.442621\pi$
$762$	0	0
$763$	− 8.50343i	− 0.307845i
$764$	0	0
$765$	16.4370i	0.594281i
$766$	0	0
$767$	5.19925	0.187734
$768$	0	0
$769$	−22.1642	−0.799260	−0.399630	−	0.916677i	$-0.630861\pi$
$769$	−22.1642	−0.799260	−0.399630	+	0.916677i	$0.630861\pi$
$770$	0	0
$771$	− 22.1372i	− 0.797252i
$772$	0	0
$773$	− 40.6860i	− 1.46337i	−0.681641	−	0.731686i	$-0.738734\pi$
$773$	− 40.6860i	− 1.46337i	0.681641	−	0.731686i	$-0.261266\pi$
$774$	0	0
$775$	19.0083	0.682799
$776$	0	0
$777$	6.54863	0.234931
$778$	0	0
$779$	13.6527i	0.489159i
$780$	0	0
$781$	65.6803i	2.35022i
$782$	0	0
$783$	−30.7672	−1.09953
$784$	0	0
$785$	−6.35921	−0.226970
$786$	0	0
$787$	50.6622i	1.80591i	0.429734	+	0.902955i	$0.358607\pi$
$787$	50.6622i	1.80591i	−0.429734	+	0.902955i	$0.641393\pi$
$788$	0	0
$789$	16.3083i	0.580592i
$790$	0	0
$791$	13.0194	0.462917
$792$	0	0
$793$	1.10633	0.0392868
$794$	0	0
$795$	17.8355i	0.632558i
$796$	0	0
$797$	− 4.92249i	− 0.174364i	−0.996192	−	0.0871818i	$-0.972214\pi$
$797$	− 4.92249i	− 0.174364i	0.996192	−	0.0871818i	$-0.0277861\pi$
$798$	0	0
$799$	−50.7541	−1.79555
$800$	0	0
$801$	−9.77351	−0.345330
$802$	0	0
$803$	− 32.6698i	− 1.15289i
$804$	0	0
$805$	− 2.95029i	− 0.103984i
$806$	0	0
$807$	7.09924	0.249905
$808$	0	0
$809$	−22.1155	−0.777538	−0.388769	−	0.921335i	$-0.627100\pi$
$809$	−22.1155	−0.777538	−0.388769	+	0.921335i	$0.627100\pi$
$810$	0	0
$811$	− 46.3318i	− 1.62693i	−0.581615	−	0.813464i	$-0.697579\pi$
$811$	− 46.3318i	− 1.62693i	0.581615	−	0.813464i	$-0.302421\pi$
$812$	0	0
$813$	3.18511i	0.111707i
$814$	0	0
$815$	15.1932	0.532194
$816$	0	0
$817$	16.3217	0.571024
$818$	0	0
$819$	− 1.05169i	− 0.0367491i
$820$	0	0
$821$	− 41.9810i	− 1.46515i	−0.680689	−	0.732573i	$-0.738319\pi$
$821$	− 41.9810i	− 1.46515i	0.680689	−	0.732573i	$-0.261681\pi$
$822$	0	0
$823$	33.7030	1.17481	0.587406	−	0.809292i	$-0.300149\pi$
$823$	33.7030	1.17481	0.587406	+	0.809292i	$0.300149\pi$
$824$	0	0
$825$	13.5257	0.470905
$826$	0	0
$827$	− 7.51764i	− 0.261414i	−0.991421	−	0.130707i	$-0.958275\pi$
$827$	− 7.51764i	− 0.261414i	0.991421	−	0.130707i	$-0.0417247\pi$
$828$	0	0
$829$	− 12.0080i	− 0.417055i	−0.978016	−	0.208528i	$-0.933133\pi$
$829$	− 12.0080i	− 0.417055i	0.978016	−	0.208528i	$-0.0668671\pi$
$830$	0	0
$831$	−2.10625	−0.0730649
$832$	0	0
$833$	−6.47379	−0.224304
$834$	0	0
$835$	14.9410i	0.517054i
$836$	0	0
$837$	− 28.8062i	− 0.995688i
$838$	0	0
$839$	−0.308970	−0.0106668	−0.00533341	−	0.999986i	$-0.501698\pi$
$839$	−0.308970	−0.0106668	−0.00533341	+	0.999986i	$0.501698\pi$
$840$	0	0
$841$	−8.12306	−0.280105
$842$	0	0
$843$	10.3641i	0.356959i
$844$	0	0
$845$	− 16.4193i	− 0.564843i
$846$	0	0
$847$	4.93509	0.169572
$848$	0	0
$849$	15.0948	0.518053
$850$	0	0
$851$	14.7122i	0.504328i
$852$	0	0
$853$	48.4197i	1.65786i	0.559354	+	0.828929i	$0.311049\pi$
$853$	48.4197i	1.65786i	−0.559354	+	0.828929i	$0.688951\pi$
$854$	0	0
$855$	8.10061	0.277035
$856$	0	0
$857$	−8.09941	−0.276671	−0.138335	−	0.990385i	$-0.544175\pi$
$857$	−8.09941	−0.276671	−0.138335	+	0.990385i	$0.544175\pi$
$858$	0	0
$859$	− 40.2757i	− 1.37419i	−0.726568	−	0.687094i	$-0.758886\pi$
$859$	− 40.2757i	− 1.37419i	0.726568	−	0.687094i	$-0.241114\pi$
$860$	0	0
$861$	4.35135i	0.148294i
$862$	0	0
$863$	−2.93426	−0.0998835	−0.0499417	−	0.998752i	$-0.515904\pi$
$863$	−2.93426	−0.0998835	−0.0499417	+	0.998752i	$0.515904\pi$
$864$	0	0
$865$	−33.0623	−1.12415
$866$	0	0
$867$	− 25.3299i	− 0.860247i
$868$	0	0
$869$	3.83524i	0.130102i
$870$	0	0
$871$	−4.84748	−0.164250
$872$	0	0
$873$	13.7135	0.464132
$874$	0	0
$875$	10.7606i	0.363774i
$876$	0	0
$877$	54.3937i	1.83674i	0.395719	+	0.918372i	$0.370495\pi$
$877$	54.3937i	1.83674i	−0.395719	+	0.918372i	$0.629505\pi$
$878$	0	0
$879$	10.4088	0.351079
$880$	0	0
$881$	27.0359	0.910862	0.455431	−	0.890271i	$-0.349485\pi$
$881$	27.0359	0.910862	0.455431	+	0.890271i	$0.349485\pi$
$882$	0	0
$883$	− 13.3882i	− 0.450549i	−0.974295	−	0.225274i	$-0.927672\pi$
$883$	− 13.3882i	− 0.450549i	0.974295	−	0.225274i	$-0.0723279\pi$
$884$	0	0
$885$	12.7637i	0.429046i
$886$	0	0
$887$	−12.6712	−0.425457	−0.212729	−	0.977111i	$-0.568235\pi$
$887$	−12.6712	−0.425457	−0.212729	+	0.977111i	$0.568235\pi$
$888$	0	0
$889$	2.47162	0.0828955
$890$	0	0
$891$	3.04660i	0.102065i
$892$	0	0
$893$	25.0130i	0.837029i
$894$	0	0
$895$	13.2399	0.442562
$896$	0	0
$897$	−1.24265	−0.0414908
$898$	0	0
$899$	− 34.7569i	− 1.15921i
$900$	0	0
$901$	87.9235i	2.92916i
$902$	0	0
$903$	5.20200	0.173112
$904$	0	0
$905$	−33.9038	−1.12700
$906$	0	0
$907$	22.0547i	0.732313i	0.930553	+	0.366157i	$0.119327\pi$
$907$	22.0547i	0.732313i	−0.930553	+	0.366157i	$0.880673\pi$
$908$	0	0
$909$	− 12.1731i	− 0.403756i
$910$	0	0
$911$	−43.3934	−1.43769	−0.718844	−	0.695171i	$-0.755328\pi$
$911$	−43.3934	−1.43769	−0.718844	+	0.695171i	$0.755328\pi$
$912$	0	0
$913$	21.5800	0.714195
$914$	0	0
$915$	2.71593i	0.0897857i
$916$	0	0
$917$	− 3.65319i	− 0.120639i
$918$	0	0
$919$	−52.4028	−1.72861	−0.864304	−	0.502970i	$-0.832241\pi$
$919$	−52.4028	−1.72861	−0.864304	+	0.502970i	$0.832241\pi$
$920$	0	0
$921$	19.9437	0.657168
$922$	0	0
$923$	8.80159i	0.289708i
$924$	0	0
$925$	− 21.4593i	− 0.705577i
$926$	0	0
$927$	−18.4467	−0.605868
$928$	0	0
$929$	−8.43753	−0.276826	−0.138413	−	0.990375i	$-0.544200\pi$
$929$	−8.43753	−0.276826	−0.138413	+	0.990375i	$0.544200\pi$
$930$	0	0
$931$	3.19046i	0.104563i
$932$	0	0
$933$	10.0498i	0.329016i
$934$	0	0
$935$	−33.3745	−1.09146
$936$	0	0
$937$	35.8332	1.17062	0.585310	−	0.810810i	$-0.300973\pi$
$937$	35.8332	1.17062	0.585310	+	0.810810i	$0.300973\pi$
$938$	0	0
$939$	0.582673i	0.0190148i
$940$	0	0
$941$	21.7906i	0.710352i	0.934800	+	0.355176i	$0.115579\pi$
$941$	21.7906i	0.710352i	−0.934800	+	0.355176i	$0.884421\pi$
$942$	0	0
$943$	−9.77578	−0.318343
$944$	0	0
$945$	6.52147	0.212143
$946$	0	0
$947$	− 11.1570i	− 0.362555i	−0.983432	−	0.181277i	$-0.941977\pi$
$947$	− 11.1570i	− 0.362555i	0.983432	−	0.181277i	$-0.0580232\pi$
$948$	0	0
$949$	− 4.37797i	− 0.142115i
$950$	0	0
$951$	20.0820	0.651203
$952$	0	0
$953$	49.6642	1.60878	0.804391	−	0.594101i	$-0.202492\pi$
$953$	49.6642	1.60878	0.804391	+	0.594101i	$0.202492\pi$
$954$	0	0
$955$	− 10.5733i	− 0.342145i
$956$	0	0
$957$	− 24.7319i	− 0.799470i
$958$	0	0
$959$	−13.4016	−0.432761
$960$	0	0
$961$	1.54163	0.0497299
$962$	0	0
$963$	4.21771i	0.135914i
$964$	0	0
$965$	− 32.7030i	− 1.05275i
$966$	0	0
$967$	−15.8683	−0.510291	−0.255145	−	0.966903i	$-0.582123\pi$
$967$	−15.8683	−0.510291	−0.255145	+	0.966903i	$0.582123\pi$
$968$	0	0
$969$	−21.0025	−0.674698
$970$	0	0
$971$	1.01819i	0.0326752i	0.999867	+	0.0163376i	$0.00520065\pi$
$971$	1.01819i	0.0326752i	−0.999867	+	0.0163376i	$0.994799\pi$
$972$	0	0
$973$	16.3563i	0.524358i
$974$	0	0
$975$	1.81253	0.0580476
$976$	0	0
$977$	34.8975	1.11647	0.558235	−	0.829683i	$-0.311479\pi$
$977$	34.8975	1.11647	0.558235	+	0.829683i	$0.311479\pi$
$978$	0	0
$979$	− 19.8446i	− 0.634237i
$980$	0	0
$981$	16.7178i	0.533758i
$982$	0	0
$983$	−56.4029	−1.79897	−0.899487	−	0.436947i	$-0.856060\pi$
$983$	−56.4029	−1.79897	−0.899487	+	0.436947i	$0.856060\pi$
$984$	0	0
$985$	−26.9225	−0.857823
$986$	0	0
$987$	7.97207i	0.253754i
$988$	0	0
$989$	11.6869i	0.371620i
$990$	0	0
$991$	−24.2515	−0.770374	−0.385187	−	0.922839i	$-0.625863\pi$
$991$	−24.2515	−0.770374	−0.385187	+	0.922839i	$0.625863\pi$
$992$	0	0
$993$	−23.2286	−0.737138
$994$	0	0
$995$	− 26.0945i	− 0.827250i
$996$	0	0
$997$	34.7021i	1.09902i	0.835486	+	0.549512i	$0.185187\pi$
$997$	34.7021i	1.09902i	−0.835486	+	0.549512i	$0.814813\pi$
$998$	0	0
$999$	−32.5205	−1.02890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.i.1793.5		12
4.3	odd	2		3584.2.b.k.1793.8		12
8.3	odd	2		3584.2.b.k.1793.5		12
8.5	even	2	inner	3584.2.b.i.1793.8		12
16.3	odd	4		3584.2.a.k.1.2	yes	6
16.5	even	4		3584.2.a.l.1.2	yes	6
16.11	odd	4		3584.2.a.e.1.5	✓	6
16.13	even	4		3584.2.a.f.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.e.1.5	✓	6	16.11	odd	4
3584.2.a.f.1.5	yes	6	16.13	even	4
3584.2.a.k.1.2	yes	6	16.3	odd	4
3584.2.a.l.1.2	yes	6	16.5	even	4
3584.2.b.i.1793.5		12	1.1	even	1	trivial
3584.2.b.i.1793.8		12	8.5	even	2	inner
3584.2.b.k.1793.5		12	8.3	odd	2
3584.2.b.k.1793.8		12	4.3	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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.01685i q^{3} -1.29145i q^{5} -1.00000 q^{7} +1.96601 q^{9} +O(q^{10})\)	\(q-1.01685i q^{3} -1.29145i q^{5} -1.00000 q^{7} +1.96601 q^{9} +3.99188i q^{11} +0.534938i q^{13} -1.31322 q^{15} -6.47379 q^{17} +3.19046i q^{19} +1.01685i q^{21} -2.28447 q^{23} +3.33215 q^{25} -5.04971i q^{27} -6.09287i q^{29} +5.70453 q^{31} +4.05916 q^{33} +1.29145i q^{35} -6.44008i q^{37} +0.543954 q^{39} +4.27923 q^{41} -5.11578i q^{43} -2.53901i q^{45} +7.83993 q^{47} +1.00000 q^{49} +6.58291i q^{51} -13.5814i q^{53} +5.15533 q^{55} +3.24424 q^{57} -9.71935i q^{59} -2.06814i q^{61} -1.96601 q^{63} +0.690848 q^{65} +9.06175i q^{67} +2.32298i q^{69} +16.4535 q^{71} -8.18408 q^{73} -3.38831i q^{75} -3.99188i q^{77} +0.960761 q^{79} +0.763200 q^{81} -5.40599i q^{83} +8.36061i q^{85} -6.19556 q^{87} -4.97125 q^{89} -0.534938i q^{91} -5.80068i q^{93} +4.12034 q^{95} +6.97532 q^{97} +7.84806i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 20 * q^9	\( 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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