LMFDB - Embedded newform 3680.2.i.b.1471.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1471,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3680.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.i (of order $2$, degree $1$, 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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.1
Character	$\chi$	$=$	3680.1471
Dual form			3680.2.i.b.1471.48

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.43521i q^{3} +1.00000i q^{5} +4.41643 q^{7} -8.80064 q^{9} +O(q^{10})$	$q-3.43521i q^{3} +1.00000i q^{5} +4.41643 q^{7} -8.80064 q^{9} +4.35346 q^{11} -0.975167 q^{13} +3.43521 q^{15} +2.69498i q^{17} +0.403177 q^{19} -15.1714i q^{21} +(3.21304 + 3.56039i) q^{23} -1.00000 q^{25} +19.9264i q^{27} +7.76355 q^{29} -9.87546i q^{31} -14.9550i q^{33} +4.41643i q^{35} +9.65028i q^{37} +3.34990i q^{39} -3.48144 q^{41} +3.29419 q^{43} -8.80064i q^{45} -6.43000i q^{47} +12.5049 q^{49} +9.25780 q^{51} -6.07996i q^{53} +4.35346i q^{55} -1.38500i q^{57} -3.24728i q^{59} +3.20587i q^{61} -38.8674 q^{63} -0.975167i q^{65} +3.73421 q^{67} +(12.2307 - 11.0375i) q^{69} -6.50005i q^{71} +3.36573 q^{73} +3.43521i q^{75} +19.2268 q^{77} -11.9608 q^{79} +42.0494 q^{81} +9.29653 q^{83} -2.69498 q^{85} -26.6694i q^{87} +14.9931i q^{89} -4.30676 q^{91} -33.9243 q^{93} +0.403177i q^{95} -1.72530i q^{97} -38.3133 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q + 8 * q^7 - 48 * q^9	$ 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) $ 48 * q + 8 * q^7 - 48 * q^9 + 16 * q^11 + 20 * q^23 - 48 * q^25 - 8 * q^29 + 8 * q^41 + 56 * q^49 - 24 * q^51 - 120 * q^63 - 32 * q^67 - 20 * q^69 + 32 * q^77 + 72 * q^81 + 64 * q^83 + 40 * q^91 - 32 * q^93 - 144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3680\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1381$	$3041$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	3.43521i		−	1.98332i	−0.128891	−	0.991659i	$-0.541142\pi$
$3$		−	3.43521i		−	1.98332i	0.128891	−	0.991659i	$-0.458858\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	4.41643			1.66925			0.834627	−	0.550816i	$-0.185683\pi$
$7$	4.41643			1.66925			0.834627	+	0.550816i	$0.185683\pi$
$8$		0			0
$9$	−8.80064			−2.93355
$10$		0			0
$11$	4.35346			1.31262			0.656309	−	0.754492i	$-0.272117\pi$
$11$	4.35346			1.31262			0.656309	+	0.754492i	$0.272117\pi$
$12$		0			0
$13$	−0.975167			−0.270463			−0.135231	−	0.990814i	$-0.543178\pi$
$13$	−0.975167			−0.270463			−0.135231	+	0.990814i	$0.543178\pi$
$14$		0			0
$15$	3.43521			0.886967
$16$		0			0
$17$			2.69498i			0.653627i	0.945089	+	0.326814i	$0.105975\pi$
$17$			2.69498i			0.653627i	−0.945089	+	0.326814i	$0.894025\pi$
$18$		0			0
$19$	0.403177			0.0924952			0.0462476	−	0.998930i	$-0.485274\pi$
$19$	0.403177			0.0924952			0.0462476	+	0.998930i	$0.485274\pi$
$20$		0			0
$21$		−	15.1714i		−	3.31066i
$22$		0			0
$23$	3.21304	+	3.56039i	0.669966	+	0.742392i
$23$	3.21304	+	3.56039i	0.669966	+	0.742392i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			19.9264i			3.83484i
$28$		0			0
$29$	7.76355			1.44166			0.720828	−	0.693114i	$-0.243762\pi$
$29$	7.76355			1.44166			0.720828	+	0.693114i	$0.243762\pi$
$30$		0			0
$31$		−	9.87546i		−	1.77369i	−0.462071	−	0.886843i	$-0.652894\pi$
$31$		−	9.87546i		−	1.77369i	0.462071	−	0.886843i	$-0.347106\pi$
$32$		0			0
$33$		−	14.9550i		−	2.60334i
$34$		0			0
$35$			4.41643i			0.746513i
$36$		0			0
$37$			9.65028i			1.58650i	0.608898	+	0.793248i	$0.291612\pi$
$37$			9.65028i			1.58650i	−0.608898	+	0.793248i	$0.708388\pi$
$38$		0			0
$39$			3.34990i			0.536413i
$40$		0			0
$41$	−3.48144			−0.543710			−0.271855	−	0.962338i	$-0.587637\pi$
$41$	−3.48144			−0.543710			−0.271855	+	0.962338i	$0.587637\pi$
$42$		0			0
$43$	3.29419			0.502359			0.251180	−	0.967940i	$-0.419182\pi$
$43$	3.29419			0.502359			0.251180	+	0.967940i	$0.419182\pi$
$44$		0			0
$45$		−	8.80064i		−	1.31192i
$46$		0			0
$47$		−	6.43000i		−	0.937912i	−0.883221	−	0.468956i	$-0.844630\pi$
$47$		−	6.43000i		−	0.937912i	0.883221	−	0.468956i	$-0.155370\pi$
$48$		0			0
$49$	12.5049			1.78641
$50$		0			0
$51$	9.25780			1.29635
$52$		0			0
$53$		−	6.07996i		−	0.835147i	−0.908643	−	0.417573i	$-0.862881\pi$
$53$		−	6.07996i		−	0.835147i	0.908643	−	0.417573i	$-0.137119\pi$
$54$		0			0
$55$			4.35346i			0.587021i
$56$		0			0
$57$		−	1.38500i		−	0.183447i
$58$		0			0
$59$		−	3.24728i		−	0.422760i	−0.977404	−	0.211380i	$-0.932204\pi$
$59$		−	3.24728i		−	0.422760i	0.977404	−	0.211380i	$-0.0677958\pi$
$60$		0			0
$61$			3.20587i			0.410470i	0.978713	+	0.205235i	$0.0657958\pi$
$61$			3.20587i			0.410470i	−0.978713	+	0.205235i	$0.934204\pi$
$62$		0			0
$63$	−38.8674			−4.89684
$64$		0			0
$65$		−	0.975167i		−	0.120955i
$66$		0			0
$67$	3.73421			0.456207			0.228103	−	0.973637i	$-0.426748\pi$
$67$	3.73421			0.456207			0.228103	+	0.973637i	$0.426748\pi$
$68$		0			0
$69$	12.2307	−	11.0375i	1.47240	−	1.32875i
$70$		0			0
$71$		−	6.50005i		−	0.771414i	−0.922621	−	0.385707i	$-0.873958\pi$
$71$		−	6.50005i		−	0.771414i	0.922621	−	0.385707i	$-0.126042\pi$
$72$		0			0
$73$	3.36573			0.393929			0.196965	−	0.980411i	$-0.436892\pi$
$73$	3.36573			0.393929			0.196965	+	0.980411i	$0.436892\pi$
$74$		0			0
$75$			3.43521i			0.396663i
$76$		0			0
$77$	19.2268			2.19109
$78$		0			0
$79$	−11.9608			−1.34570			−0.672850	−	0.739779i	$-0.734930\pi$
$79$	−11.9608			−1.34570			−0.672850	+	0.739779i	$0.734930\pi$
$80$		0			0
$81$	42.0494			4.67216
$82$		0			0
$83$	9.29653			1.02043			0.510213	−	0.860048i	$-0.329566\pi$
$83$	9.29653			1.02043			0.510213	+	0.860048i	$0.329566\pi$
$84$		0			0
$85$	−2.69498			−0.292311
$86$		0			0
$87$		−	26.6694i		−	2.85926i
$88$		0			0
$89$			14.9931i			1.58926i	0.607092	+	0.794632i	$0.292336\pi$
$89$			14.9931i			1.58926i	−0.607092	+	0.794632i	$0.707664\pi$
$90$		0			0
$91$	−4.30676			−0.451471
$92$		0			0
$93$	−33.9243			−3.51778
$94$		0			0
$95$			0.403177i			0.0413651i
$96$		0			0
$97$		−	1.72530i		−	0.175178i	−0.996157	−	0.0875891i	$-0.972084\pi$
$97$		−	1.72530i		−	0.175178i	0.996157	−	0.0875891i	$-0.0279162\pi$
$98$		0			0
$99$	−38.3133			−3.85063
$100$		0			0
$101$	8.21151			0.817076			0.408538	−	0.912741i	$-0.366039\pi$
$101$	8.21151			0.817076			0.408538	+	0.912741i	$0.366039\pi$
$102$		0			0
$103$	5.55289			0.547143			0.273571	−	0.961852i	$-0.411795\pi$
$103$	5.55289			0.547143			0.273571	+	0.961852i	$0.411795\pi$
$104$		0			0
$105$	15.1714			1.48057
$106$		0			0
$107$	10.4012			1.00552			0.502760	−	0.864426i	$-0.332318\pi$
$107$	10.4012			1.00552			0.502760	+	0.864426i	$0.332318\pi$
$108$		0			0
$109$		−	11.1960i		−	1.07238i	−0.844096	−	0.536191i	$-0.819863\pi$
$109$		−	11.1960i		−	1.07238i	0.844096	−	0.536191i	$-0.180137\pi$
$110$		0			0
$111$	33.1507			3.14653
$112$		0			0
$113$		−	16.6421i		−	1.56556i	−0.622301	−	0.782778i	$-0.713802\pi$
$113$		−	16.6421i		−	1.56556i	0.622301	−	0.782778i	$-0.286198\pi$
$114$		0			0
$115$	−3.56039	+	3.21304i	−0.332008	+	0.299618i
$116$		0			0
$117$	8.58210			0.793415
$118$		0			0
$119$			11.9022i			1.09107i
$120$		0			0
$121$	7.95264			0.722967
$122$		0			0
$123$			11.9595i			1.07835i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	8.06560i		−	0.715706i	−0.933778	−	0.357853i	$-0.883509\pi$
$127$		−	8.06560i		−	0.715706i	0.933778	−	0.357853i	$-0.116491\pi$
$128$		0			0
$129$		−	11.3162i		−	0.996338i
$130$		0			0
$131$			6.44214i			0.562852i	0.959583	+	0.281426i	$0.0908074\pi$
$131$			6.44214i			0.562852i	−0.959583	+	0.281426i	$0.909193\pi$
$132$		0			0
$133$	1.78060			0.154398
$134$		0			0
$135$	−19.9264			−1.71499
$136$		0			0
$137$		−	18.4771i		−	1.57861i	−0.614003	−	0.789304i	$-0.710442\pi$
$137$		−	18.4771i		−	1.57861i	0.614003	−	0.789304i	$-0.289558\pi$
$138$		0			0
$139$			1.64717i			0.139711i	0.997557	+	0.0698557i	$0.0222539\pi$
$139$			1.64717i			0.139711i	−0.997557	+	0.0698557i	$0.977746\pi$
$140$		0			0
$141$	−22.0884			−1.86018
$142$		0			0
$143$	−4.24535			−0.355014
$144$		0			0
$145$			7.76355i			0.644728i
$146$		0			0
$147$		−	42.9568i		−	3.54301i
$148$		0			0
$149$			14.2484i			1.16728i	0.812013	+	0.583639i	$0.198372\pi$
$149$			14.2484i			1.16728i	−0.812013	+	0.583639i	$0.801628\pi$
$150$		0			0
$151$			13.4199i			1.09210i	0.837753	+	0.546050i	$0.183869\pi$
$151$			13.4199i			1.09210i	−0.837753	+	0.546050i	$0.816131\pi$
$152$		0			0
$153$		−	23.7175i		−	1.91745i
$154$		0			0
$155$	9.87546			0.793216
$156$		0			0
$157$			0.175784i			0.0140291i	0.999975	+	0.00701455i	$0.00223282\pi$
$157$			0.175784i			0.0140291i	−0.999975	+	0.00701455i	$0.997767\pi$
$158$		0			0
$159$	−20.8859			−1.65636
$160$		0			0
$161$	14.1902	+	15.7242i	1.11834	+	1.23924i
$162$		0			0
$163$			8.42707i			0.660059i	0.943971	+	0.330030i	$0.107059\pi$
$163$			8.42707i			0.660059i	−0.943971	+	0.330030i	$0.892941\pi$
$164$		0			0
$165$	14.9550			1.16425
$166$		0			0
$167$		−	2.16700i		−	0.167688i	−0.996479	−	0.0838438i	$-0.973280\pi$
$167$		−	2.16700i		−	0.167688i	0.996479	−	0.0838438i	$-0.0267197\pi$
$168$		0			0
$169$	−12.0490			−0.926850
$170$		0			0
$171$	−3.54822			−0.271339
$172$		0			0
$173$	−2.30948			−0.175586			−0.0877931	−	0.996139i	$-0.527981\pi$
$173$	−2.30948			−0.175586			−0.0877931	+	0.996139i	$0.527981\pi$
$174$		0			0
$175$	−4.41643			−0.333851
$176$		0			0
$177$	−11.1551			−0.838468
$178$		0			0
$179$		−	0.503447i		−	0.0376294i	−0.999823	−	0.0188147i	$-0.994011\pi$
$179$		−	0.503447i		−	0.0376294i	0.999823	−	0.0188147i	$-0.00598926\pi$
$180$		0			0
$181$		−	6.13165i		−	0.455762i	−0.973689	−	0.227881i	$-0.926820\pi$
$181$		−	6.13165i		−	0.455762i	0.973689	−	0.227881i	$-0.0731797\pi$
$182$		0			0
$183$	11.0128			0.814092
$184$		0			0
$185$	−9.65028			−0.709503
$186$		0			0
$187$			11.7325i			0.857963i
$188$		0			0
$189$			88.0036i			6.40132i
$190$		0			0
$191$	−17.7194			−1.28213			−0.641065	−	0.767486i	$-0.721507\pi$
$191$	−17.7194			−1.28213			−0.641065	+	0.767486i	$0.721507\pi$
$192$		0			0
$193$	−19.6626			−1.41535			−0.707673	−	0.706540i	$-0.750255\pi$
$193$	−19.6626			−1.41535			−0.707673	+	0.706540i	$0.750255\pi$
$194$		0			0
$195$	−3.34990			−0.239891
$196$		0			0
$197$	−12.8758			−0.917363			−0.458682	−	0.888601i	$-0.651678\pi$
$197$	−12.8758			−0.917363			−0.458682	+	0.888601i	$0.651678\pi$
$198$		0			0
$199$	13.4014			0.950000			0.475000	−	0.879986i	$-0.342448\pi$
$199$	13.4014			0.950000			0.475000	+	0.879986i	$0.342448\pi$
$200$		0			0
$201$		−	12.8278i		−	0.904803i
$202$		0			0
$203$	34.2872			2.40649
$204$		0			0
$205$		−	3.48144i		−	0.243155i
$206$		0			0
$207$	−28.2768	−	31.3337i	−1.96538	−	2.17784i
$208$		0			0
$209$	1.75522			0.121411
$210$		0			0
$211$			19.4329i			1.33782i	0.743344	+	0.668909i	$0.233238\pi$
$211$			19.4329i			1.33782i	−0.743344	+	0.668909i	$0.766762\pi$
$212$		0			0
$213$	−22.3290			−1.52996
$214$		0			0
$215$			3.29419i			0.224662i
$216$		0			0
$217$		−	43.6143i		−	2.96073i
$218$		0			0
$219$		−	11.5620i		−	0.781287i
$220$		0			0
$221$		−	2.62805i		−	0.176782i
$222$		0			0
$223$		−	1.42967i		−	0.0957380i	−0.998854	−	0.0478690i	$-0.984757\pi$
$223$		−	1.42967i		−	0.0957380i	0.998854	−	0.0478690i	$-0.0152430\pi$
$224$		0			0
$225$	8.80064			0.586710
$226$		0			0
$227$	−22.8223			−1.51477			−0.757385	−	0.652969i	$-0.773523\pi$
$227$	−22.8223			−1.51477			−0.757385	+	0.652969i	$0.773523\pi$
$228$		0			0
$229$			6.48929i			0.428824i	0.976743	+	0.214412i	$0.0687836\pi$
$229$			6.48929i			0.428824i	−0.976743	+	0.214412i	$0.931216\pi$
$230$		0			0
$231$		−	66.0479i		−	4.34563i
$232$		0			0
$233$	−12.9406			−0.847764			−0.423882	−	0.905717i	$-0.639333\pi$
$233$	−12.9406			−0.847764			−0.423882	+	0.905717i	$0.639333\pi$
$234$		0			0
$235$	6.43000			0.419447
$236$		0			0
$237$			41.0879i			2.66895i
$238$		0			0
$239$		−	22.1545i		−	1.43305i	−0.697560	−	0.716527i	$-0.745731\pi$
$239$		−	22.1545i		−	1.43305i	0.697560	−	0.716527i	$-0.254269\pi$
$240$		0			0
$241$			2.37840i			0.153206i	0.997062	+	0.0766031i	$0.0244074\pi$
$241$			2.37840i			0.153206i	−0.997062	+	0.0766031i	$0.975593\pi$
$242$		0			0
$243$		−	84.6692i		−	5.43153i
$244$		0			0
$245$			12.5049i			0.798906i
$246$		0			0
$247$	−0.393165			−0.0250165
$248$		0			0
$249$		−	31.9355i		−	2.02383i
$250$		0			0
$251$	3.16505			0.199776			0.0998881	−	0.994999i	$-0.468152\pi$
$251$	3.16505			0.199776			0.0998881	+	0.994999i	$0.468152\pi$
$252$		0			0
$253$	13.9879	+	15.5000i	0.879409	+	0.974478i
$254$		0			0
$255$			9.25780i			0.579746i
$256$		0			0
$257$	24.4007			1.52208			0.761038	−	0.648707i	$-0.224690\pi$
$257$	24.4007			1.52208			0.761038	+	0.648707i	$0.224690\pi$
$258$		0			0
$259$			42.6198i			2.64827i
$260$		0			0
$261$	−68.3243			−4.22917
$262$		0			0
$263$	−15.1269			−0.932767			−0.466383	−	0.884583i	$-0.654443\pi$
$263$	−15.1269			−0.932767			−0.466383	+	0.884583i	$0.654443\pi$
$264$		0			0
$265$	6.07996			0.373489
$266$		0			0
$267$	51.5043			3.15201
$268$		0			0
$269$	−4.41297			−0.269064			−0.134532	−	0.990909i	$-0.542953\pi$
$269$	−4.41297			−0.269064			−0.134532	+	0.990909i	$0.542953\pi$
$270$		0			0
$271$			13.0324i			0.791661i	0.918324	+	0.395831i	$0.129543\pi$
$271$			13.0324i			0.791661i	−0.918324	+	0.395831i	$0.870457\pi$
$272$		0			0
$273$			14.7946i			0.895410i
$274$		0			0
$275$	−4.35346			−0.262524
$276$		0			0
$277$	7.25619			0.435982			0.217991	−	0.975951i	$-0.430050\pi$
$277$	7.25619			0.435982			0.217991	+	0.975951i	$0.430050\pi$
$278$		0			0
$279$			86.9104i			5.20319i
$280$		0			0
$281$			6.29647i			0.375616i	0.982206	+	0.187808i	$0.0601383\pi$
$281$			6.29647i			0.375616i	−0.982206	+	0.187808i	$0.939862\pi$
$282$		0			0
$283$	−30.9059			−1.83717			−0.918583	−	0.395229i	$-0.870665\pi$
$283$	−30.9059			−1.83717			−0.918583	+	0.395229i	$0.870665\pi$
$284$		0			0
$285$	1.38500			0.0820401
$286$		0			0
$287$	−15.3755			−0.907590
$288$		0			0
$289$	9.73711			0.572771
$290$		0			0
$291$	−5.92678			−0.347434
$292$		0			0
$293$		−	14.2313i		−	0.831399i	−0.909502	−	0.415700i	$-0.863537\pi$
$293$		−	14.2313i		−	0.831399i	0.909502	−	0.415700i	$-0.136463\pi$
$294$		0			0
$295$	3.24728			0.189064
$296$		0			0
$297$			86.7489i			5.03368i
$298$		0			0
$299$	−3.13325	−	3.47197i	−0.181201	−	0.200789i
$300$		0			0
$301$	14.5486			0.838565
$302$		0			0
$303$		−	28.2082i		−	1.62052i
$304$		0			0
$305$	−3.20587			−0.183568
$306$		0			0
$307$			9.55321i			0.545230i	0.962123	+	0.272615i	$0.0878886\pi$
$307$			9.55321i			0.545230i	−0.962123	+	0.272615i	$0.912111\pi$
$308$		0			0
$309$		−	19.0753i		−	1.08516i
$310$		0			0
$311$			5.51014i			0.312451i	0.987721	+	0.156226i	$0.0499327\pi$
$311$			5.51014i			0.312451i	−0.987721	+	0.156226i	$0.950067\pi$
$312$		0			0
$313$			30.0636i			1.69929i	0.527352	+	0.849647i	$0.323185\pi$
$313$			30.0636i			1.69929i	−0.527352	+	0.849647i	$0.676815\pi$
$314$		0			0
$315$		−	38.8674i		−	2.18993i
$316$		0			0
$317$	−21.0690			−1.18335			−0.591676	−	0.806176i	$-0.701534\pi$
$317$	−21.0690			−1.18335			−0.591676	+	0.806176i	$0.701534\pi$
$318$		0			0
$319$	33.7983			1.89234
$320$		0			0
$321$		−	35.7302i		−	1.99427i
$322$		0			0
$323$			1.08655i			0.0604574i
$324$		0			0
$325$	0.975167			0.0540925
$326$		0			0
$327$	−38.4606			−2.12688
$328$		0			0
$329$		−	28.3977i		−	1.56561i
$330$		0			0
$331$			0.660450i			0.0363016i	0.999835	+	0.0181508i	$0.00577790\pi$
$331$			0.660450i			0.0363016i	−0.999835	+	0.0181508i	$0.994222\pi$
$332$		0			0
$333$		−	84.9287i		−	4.65407i
$334$		0			0
$335$			3.73421i			0.204022i
$336$		0			0
$337$		−	24.4413i		−	1.33140i	−0.746218	−	0.665702i	$-0.768132\pi$
$337$		−	24.4413i		−	1.33140i	0.746218	−	0.665702i	$-0.231868\pi$
$338$		0			0
$339$	−57.1690			−3.10499
$340$		0			0
$341$		−	42.9925i		−	2.32817i
$342$		0			0
$343$	24.3118			1.31271
$344$		0			0
$345$	11.0375	+	12.2307i	0.594237	+	0.658477i
$346$		0			0
$347$		−	26.4969i		−	1.42243i	−0.702975	−	0.711215i	$-0.748145\pi$
$347$		−	26.4969i		−	1.42243i	0.702975	−	0.711215i	$-0.251855\pi$
$348$		0			0
$349$	0.0351292			0.00188042			0.000940210	−	1.00000i	$-0.499701\pi$
$349$	0.0351292			0.00188042			0.000940210		1.00000i	$0.499701\pi$
$350$		0			0
$351$		−	19.4316i		−	1.03718i
$352$		0			0
$353$	2.97512			0.158350			0.0791748	−	0.996861i	$-0.474771\pi$
$353$	2.97512			0.158350			0.0791748	+	0.996861i	$0.474771\pi$
$354$		0			0
$355$	6.50005			0.344987
$356$		0			0
$357$	40.8864			2.16394
$358$		0			0
$359$	−35.3941			−1.86803			−0.934015	−	0.357233i	$-0.883720\pi$
$359$	−35.3941			−1.86803			−0.934015	+	0.357233i	$0.883720\pi$
$360$		0			0
$361$	−18.8374			−0.991445
$362$		0			0
$363$		−	27.3190i		−	1.43387i
$364$		0			0
$365$			3.36573i			0.176170i
$366$		0			0
$367$	−8.55895			−0.446773			−0.223387	−	0.974730i	$-0.571711\pi$
$367$	−8.55895			−0.446773			−0.223387	+	0.974730i	$0.571711\pi$
$368$		0			0
$369$	30.6389			1.59500
$370$		0			0
$371$		−	26.8517i		−	1.39407i
$372$		0			0
$373$			26.5141i			1.37285i	0.727201	+	0.686424i	$0.240821\pi$
$373$			26.5141i			1.37285i	−0.727201	+	0.686424i	$0.759179\pi$
$374$		0			0
$375$	−3.43521			−0.177393
$376$		0			0
$377$	−7.57076			−0.389914
$378$		0			0
$379$	5.54459			0.284807			0.142403	−	0.989809i	$-0.454517\pi$
$379$	5.54459			0.284807			0.142403	+	0.989809i	$0.454517\pi$
$380$		0			0
$381$	−27.7070			−1.41947
$382$		0			0
$383$	19.8352			1.01353			0.506767	−	0.862083i	$-0.330841\pi$
$383$	19.8352			1.01353			0.506767	+	0.862083i	$0.330841\pi$
$384$		0			0
$385$			19.2268i			0.979887i
$386$		0			0
$387$	−28.9910			−1.47369
$388$		0			0
$389$		−	25.7503i		−	1.30559i	−0.757534	−	0.652796i	$-0.773596\pi$
$389$		−	25.7503i		−	1.30559i	0.757534	−	0.652796i	$-0.226404\pi$
$390$		0			0
$391$	−9.59516	+	8.65907i	−0.485248	+	0.437908i
$392$		0			0
$393$	22.1301			1.11631
$394$		0			0
$395$		−	11.9608i		−	0.601815i
$396$		0			0
$397$	−22.0020			−1.10425			−0.552124	−	0.833762i	$-0.686183\pi$
$397$	−22.0020			−1.10425			−0.552124	+	0.833762i	$0.686183\pi$
$398$		0			0
$399$		−	6.11674i		−	0.306220i
$400$		0			0
$401$		−	16.3819i		−	0.818075i	−0.912518	−	0.409037i	$-0.865865\pi$
$401$		−	16.3819i		−	0.818075i	0.912518	−	0.409037i	$-0.134135\pi$
$402$		0			0
$403$			9.63023i			0.479716i
$404$		0			0
$405$			42.0494i			2.08945i
$406$		0			0
$407$			42.0122i			2.08247i
$408$		0			0
$409$	17.5714			0.868850			0.434425	−	0.900708i	$-0.356952\pi$
$409$	17.5714			0.868850			0.434425	+	0.900708i	$0.356952\pi$
$410$		0			0
$411$	−63.4727			−3.13088
$412$		0			0
$413$		−	14.3414i		−	0.705694i
$414$		0			0
$415$			9.29653i			0.456349i
$416$		0			0
$417$	5.65838			0.277092
$418$		0			0
$419$	−28.5261			−1.39359			−0.696795	−	0.717270i	$-0.745391\pi$
$419$	−28.5261			−1.39359			−0.696795	+	0.717270i	$0.745391\pi$
$420$		0			0
$421$			4.90016i			0.238819i	0.992845	+	0.119410i	$0.0381001\pi$
$421$			4.90016i			0.238819i	−0.992845	+	0.119410i	$0.961900\pi$
$422$		0			0
$423$			56.5882i			2.75141i
$424$		0			0
$425$		−	2.69498i		−	0.130725i
$426$		0			0
$427$			14.1585i			0.685178i
$428$		0			0
$429$			14.5837i			0.704106i
$430$		0			0
$431$	24.6244			1.18612			0.593058	−	0.805160i	$-0.297920\pi$
$431$	24.6244			1.18612			0.593058	+	0.805160i	$0.297920\pi$
$432$		0			0
$433$			0.206569i			0.00992706i	0.999988	+	0.00496353i	$0.00157995\pi$
$433$			0.206569i			0.00992706i	−0.999988	+	0.00496353i	$0.998420\pi$
$434$		0			0
$435$	26.6694			1.27870
$436$		0			0
$437$	1.29543	+	1.43547i	0.0619686	+	0.0686677i
$438$		0			0
$439$		−	17.2265i		−	0.822178i	−0.911595	−	0.411089i	$-0.865149\pi$
$439$		−	17.2265i		−	0.822178i	0.911595	−	0.411089i	$-0.134851\pi$
$440$		0			0
$441$	−110.051			−5.24051
$442$		0			0
$443$			2.33228i			0.110810i	0.998464	+	0.0554051i	$0.0176450\pi$
$443$			2.33228i			0.110810i	−0.998464	+	0.0554051i	$0.982355\pi$
$444$		0			0
$445$	−14.9931			−0.710740
$446$		0			0
$447$	48.9463			2.31508
$448$		0			0
$449$	−37.6570			−1.77714			−0.888572	−	0.458736i	$-0.848302\pi$
$449$	−37.6570			−1.77714			−0.888572	+	0.458736i	$0.848302\pi$
$450$		0			0
$451$	−15.1563			−0.713684
$452$		0			0
$453$	46.1003			2.16598
$454$		0			0
$455$		−	4.30676i		−	0.201904i
$456$		0			0
$457$			11.8743i			0.555459i	0.960659	+	0.277729i	$0.0895818\pi$
$457$			11.8743i			0.555459i	−0.960659	+	0.277729i	$0.910418\pi$
$458$		0			0
$459$	−53.7012			−2.50656
$460$		0			0
$461$	33.9891			1.58303			0.791515	−	0.611149i	$-0.209293\pi$
$461$	33.9891			1.58303			0.791515	+	0.611149i	$0.209293\pi$
$462$		0			0
$463$			10.7012i			0.497329i	0.968590	+	0.248664i	$0.0799916\pi$
$463$			10.7012i			0.497329i	−0.968590	+	0.248664i	$0.920008\pi$
$464$		0			0
$465$		−	33.9243i		−	1.57320i
$466$		0			0
$467$	9.23975			0.427564			0.213782	−	0.976881i	$-0.431422\pi$
$467$	9.23975			0.427564			0.213782	+	0.976881i	$0.431422\pi$
$468$		0			0
$469$	16.4919			0.761525
$470$		0			0
$471$	0.603855			0.0278242
$472$		0			0
$473$	14.3411			0.659406
$474$		0			0
$475$	−0.403177			−0.0184990
$476$		0			0
$477$			53.5076i			2.44994i
$478$		0			0
$479$	−5.17157			−0.236295			−0.118148	−	0.992996i	$-0.537696\pi$
$479$	−5.17157			−0.236295			−0.118148	+	0.992996i	$0.537696\pi$
$480$		0			0
$481$		−	9.41064i		−	0.429088i
$482$		0			0
$483$	54.0159	−	48.7462i	2.45781	−	2.21803i
$484$		0			0
$485$	1.72530			0.0783421
$486$		0			0
$487$		−	10.5766i		−	0.479270i	−0.970863	−	0.239635i	$-0.922972\pi$
$487$		−	10.5766i		−	0.479270i	0.970863	−	0.239635i	$-0.0770278\pi$
$488$		0			0
$489$	28.9487			1.30911
$490$		0			0
$491$			31.4273i			1.41829i	0.705061	+	0.709147i	$0.250920\pi$
$491$			31.4273i			1.41829i	−0.705061	+	0.709147i	$0.749080\pi$
$492$		0			0
$493$			20.9226i			0.942306i
$494$		0			0
$495$		−	38.3133i		−	1.72205i
$496$		0			0
$497$		−	28.7070i		−	1.28769i
$498$		0			0
$499$			3.27175i			0.146464i	0.997315	+	0.0732318i	$0.0233313\pi$
$499$			3.27175i			0.146464i	−0.997315	+	0.0732318i	$0.976669\pi$
$500$		0			0
$501$	−7.44410			−0.332578
$502$		0			0
$503$	10.7291			0.478389			0.239194	−	0.970972i	$-0.423117\pi$
$503$	10.7291			0.478389			0.239194	+	0.970972i	$0.423117\pi$
$504$		0			0
$505$			8.21151i			0.365408i
$506$		0			0
$507$			41.3910i			1.83824i
$508$		0			0
$509$	31.1515			1.38077			0.690383	−	0.723444i	$-0.257442\pi$
$509$	31.1515			1.38077			0.690383	+	0.723444i	$0.257442\pi$
$510$		0			0
$511$	14.8645			0.657568
$512$		0			0
$513$			8.03388i			0.354704i
$514$		0			0
$515$			5.55289i			0.244690i
$516$		0			0
$517$		−	27.9928i		−	1.23112i
$518$		0			0
$519$			7.93353i			0.348243i
$520$		0			0
$521$			24.7239i			1.08318i	0.840644	+	0.541588i	$0.182177\pi$
$521$			24.7239i			1.08318i	−0.840644	+	0.541588i	$0.817823\pi$
$522$		0			0
$523$	5.64846			0.246990			0.123495	−	0.992345i	$-0.460590\pi$
$523$	5.64846			0.246990			0.123495	+	0.992345i	$0.460590\pi$
$524$		0			0
$525$			15.1714i			0.662132i
$526$		0			0
$527$	26.6141			1.15933
$528$		0			0
$529$	−2.35272	+	22.8794i	−0.102292	+	0.994754i
$530$		0			0
$531$			28.5782i			1.24019i
$532$		0			0
$533$	3.39499			0.147053
$534$		0			0
$535$			10.4012i			0.449683i
$536$		0			0
$537$	−1.72945			−0.0746311
$538$		0			0
$539$	54.4394			2.34487
$540$		0			0
$541$	11.4516			0.492342			0.246171	−	0.969226i	$-0.420828\pi$
$541$	11.4516			0.492342			0.246171	+	0.969226i	$0.420828\pi$
$542$		0			0
$543$	−21.0635			−0.903920
$544$		0			0
$545$	11.1960			0.479584
$546$		0			0
$547$			14.8045i			0.632994i	0.948594	+	0.316497i	$0.102507\pi$
$547$			14.8045i			0.632994i	−0.948594	+	0.316497i	$0.897493\pi$
$548$		0			0
$549$		−	28.2137i		−	1.20413i
$550$		0			0
$551$	3.13009			0.133346
$552$		0			0
$553$	−52.8242			−2.24631
$554$		0			0
$555$			33.1507i			1.40717i
$556$		0			0
$557$		−	0.0226284i		−	0.000958797i	−1.00000		0.000479398i	$-0.999847\pi$
$557$		−	0.0226284i		−	0.000958797i	1.00000		0.000479398i	$-0.000152597\pi$
$558$		0			0
$559$	−3.21239			−0.135869
$560$		0			0
$561$	40.3035			1.70161
$562$		0			0
$563$	42.3792			1.78607			0.893036	−	0.449986i	$-0.148571\pi$
$563$	42.3792			1.78607			0.893036	+	0.449986i	$0.148571\pi$
$564$		0			0
$565$	16.6421			0.700138
$566$		0			0
$567$	185.708			7.79902
$568$		0			0
$569$			19.7504i			0.827978i	0.910282	+	0.413989i	$0.135865\pi$
$569$			19.7504i			0.827978i	−0.910282	+	0.413989i	$0.864135\pi$
$570$		0			0
$571$	38.6401			1.61704			0.808518	−	0.588471i	$-0.200270\pi$
$571$	38.6401			1.61704			0.808518	+	0.588471i	$0.200270\pi$
$572$		0			0
$573$			60.8698i			2.54287i
$574$		0			0
$575$	−3.21304	−	3.56039i	−0.133993	−	0.148478i
$576$		0			0
$577$	31.0838			1.29404			0.647019	−	0.762474i	$-0.276016\pi$
$577$	31.0838			1.29404			0.647019	+	0.762474i	$0.276016\pi$
$578$		0			0
$579$			67.5452i			2.80708i
$580$		0			0
$581$	41.0575			1.70335
$582$		0			0
$583$		−	26.4689i		−	1.09623i
$584$		0			0
$585$			8.58210i			0.354826i
$586$		0			0
$587$		−	22.7359i		−	0.938411i	−0.883089	−	0.469206i	$-0.844540\pi$
$587$		−	22.7359i		−	0.938411i	0.883089	−	0.469206i	$-0.155460\pi$
$588$		0			0
$589$		−	3.98156i		−	0.164057i
$590$		0			0
$591$			44.2310i			1.81942i
$592$		0			0
$593$	25.5522			1.04930			0.524651	−	0.851317i	$-0.324196\pi$
$593$	25.5522			1.04930			0.524651	+	0.851317i	$0.324196\pi$
$594$		0			0
$595$	−11.9022			−0.487941
$596$		0			0
$597$		−	46.0366i		−	1.88415i
$598$		0			0
$599$			38.5027i			1.57318i	0.617478	+	0.786588i	$0.288154\pi$
$599$			38.5027i			1.57318i	−0.617478	+	0.786588i	$0.711846\pi$
$600$		0			0
$601$	14.6461			0.597428			0.298714	−	0.954343i	$-0.403442\pi$
$601$	14.6461			0.597428			0.298714	+	0.954343i	$0.403442\pi$
$602$		0			0
$603$	−32.8635			−1.33831
$604$		0			0
$605$			7.95264i			0.323321i
$606$		0			0
$607$		−	23.3560i		−	0.947990i	−0.880528	−	0.473995i	$-0.842812\pi$
$607$		−	23.3560i		−	0.947990i	0.880528	−	0.473995i	$-0.157188\pi$
$608$		0			0
$609$		−	117.784i		−	4.77283i
$610$		0			0
$611$			6.27033i			0.253670i
$612$		0			0
$613$		−	20.3525i		−	0.822029i	−0.911629	−	0.411014i	$-0.865175\pi$
$613$		−	20.3525i		−	0.822029i	0.911629	−	0.411014i	$-0.134825\pi$
$614$		0			0
$615$	−11.9595			−0.482253
$616$		0			0
$617$			19.2650i			0.775581i	0.921748	+	0.387790i	$0.126762\pi$
$617$			19.2650i			0.775581i	−0.921748	+	0.387790i	$0.873238\pi$
$618$		0			0
$619$	38.7084			1.55582			0.777911	−	0.628374i	$-0.216279\pi$
$619$	38.7084			1.55582			0.777911	+	0.628374i	$0.216279\pi$
$620$		0			0
$621$	−70.9458	+	64.0244i	−2.84696	+	2.56921i
$622$		0			0
$623$			66.2159i			2.65288i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	6.02953i		−	0.240796i
$628$		0			0
$629$	−26.0073			−1.03698
$630$		0			0
$631$	−27.0059			−1.07509			−0.537543	−	0.843236i	$-0.680647\pi$
$631$	−27.0059			−1.07509			−0.537543	+	0.843236i	$0.680647\pi$
$632$		0			0
$633$	66.7561			2.65332
$634$		0			0
$635$	8.06560			0.320073
$636$		0			0
$637$	−12.1943			−0.483157
$638$		0			0
$639$			57.2046i			2.26298i
$640$		0			0
$641$			44.1734i			1.74475i	0.488841	+	0.872373i	$0.337420\pi$
$641$			44.1734i			1.74475i	−0.488841	+	0.872373i	$0.662580\pi$
$642$		0			0
$643$	17.3784			0.685335			0.342668	−	0.939457i	$-0.388670\pi$
$643$	17.3784			0.685335			0.342668	+	0.939457i	$0.388670\pi$
$644$		0			0
$645$	11.3162			0.445576
$646$		0			0
$647$		−	36.8304i		−	1.44795i	−0.689826	−	0.723975i	$-0.742313\pi$
$647$		−	36.8304i		−	1.44795i	0.689826	−	0.723975i	$-0.257687\pi$
$648$		0			0
$649$		−	14.1369i		−	0.554923i
$650$		0			0
$651$	−149.824			−5.87207
$652$		0			0
$653$	−31.4727			−1.23162			−0.615811	−	0.787894i	$-0.711171\pi$
$653$	−31.4727			−1.23162			−0.615811	+	0.787894i	$0.711171\pi$
$654$		0			0
$655$	−6.44214			−0.251715
$656$		0			0
$657$	−29.6206			−1.15561
$658$		0			0
$659$	3.81732			0.148701			0.0743507	−	0.997232i	$-0.476312\pi$
$659$	3.81732			0.148701			0.0743507	+	0.997232i	$0.476312\pi$
$660$		0			0
$661$			4.47849i			0.174193i	0.996200	+	0.0870966i	$0.0277589\pi$
$661$			4.47849i			0.174193i	−0.996200	+	0.0870966i	$0.972241\pi$
$662$		0			0
$663$	−9.02790			−0.350615
$664$		0			0
$665$			1.78060i			0.0690489i
$666$		0			0
$667$	24.9446	+	27.6413i	0.965860	+	1.07027i
$668$		0			0
$669$	−4.91122			−0.189879
$670$		0			0
$671$			13.9566i			0.538790i
$672$		0			0
$673$	−13.8797			−0.535024			−0.267512	−	0.963554i	$-0.586202\pi$
$673$	−13.8797			−0.535024			−0.267512	+	0.963554i	$0.586202\pi$
$674$		0			0
$675$		−	19.9264i		−	0.766968i
$676$		0			0
$677$		−	11.5650i		−	0.444481i	−0.974992	−	0.222240i	$-0.928663\pi$
$677$		−	11.5650i		−	0.444481i	0.974992	−	0.222240i	$-0.0713370\pi$
$678$		0			0
$679$		−	7.61969i		−	0.292417i
$680$		0			0
$681$			78.3993i			3.00427i
$682$		0			0
$683$			6.82242i			0.261053i	0.991445	+	0.130526i	$0.0416667\pi$
$683$			6.82242i			0.261053i	−0.991445	+	0.130526i	$0.958333\pi$
$684$		0			0
$685$	18.4771			0.705975
$686$		0			0
$687$	22.2921			0.850495
$688$		0			0
$689$			5.92898i			0.225876i
$690$		0			0
$691$			45.2641i			1.72193i	0.508665	+	0.860964i	$0.330139\pi$
$691$			45.2641i			1.72193i	−0.508665	+	0.860964i	$0.669861\pi$
$692$		0			0
$693$	−169.208			−6.42768
$694$		0			0
$695$	−1.64717			−0.0624808
$696$		0			0
$697$		−	9.38240i		−	0.355384i
$698$		0			0
$699$			44.4535i			1.68139i
$700$		0			0
$701$		−	24.4524i		−	0.923553i	−0.886996	−	0.461777i	$-0.847212\pi$
$701$		−	24.4524i		−	0.923553i	0.886996	−	0.461777i	$-0.152788\pi$
$702$		0			0
$703$			3.89077i			0.146743i
$704$		0			0
$705$		−	22.0884i		−	0.831897i
$706$		0			0
$707$	36.2656			1.36391
$708$		0			0
$709$			16.0729i			0.603632i	0.953366	+	0.301816i	$0.0975927\pi$
$709$			16.0729i			0.603632i	−0.953366	+	0.301816i	$0.902407\pi$
$710$		0			0
$711$	105.263			3.94767
$712$		0			0
$713$	35.1605	−	31.7303i	1.31677	−	1.18831i
$714$		0			0
$715$		−	4.24535i		−	0.158767i
$716$		0			0
$717$	−76.1052			−2.84220
$718$		0			0
$719$		−	9.82467i		−	0.366398i	−0.983076	−	0.183199i	$-0.941355\pi$
$719$		−	9.82467i		−	0.366398i	0.983076	−	0.183199i	$-0.0586453\pi$
$720$		0			0
$721$	24.5240			0.913320
$722$		0			0
$723$	8.17029			0.303857
$724$		0			0
$725$	−7.76355			−0.288331
$726$		0			0
$727$	−18.7176			−0.694197			−0.347098	−	0.937829i	$-0.612833\pi$
$727$	−18.7176			−0.694197			−0.347098	+	0.937829i	$0.612833\pi$
$728$		0			0
$729$	−164.708			−6.10029
$730$		0			0
$731$			8.87776i			0.328356i
$732$		0			0
$733$		−	13.0602i		−	0.482390i	−0.970477	−	0.241195i	$-0.922461\pi$
$733$		−	13.0602i		−	0.482390i	0.970477	−	0.241195i	$-0.0775394\pi$
$734$		0			0
$735$	42.9568			1.58448
$736$		0			0
$737$	16.2568			0.598826
$738$		0			0
$739$		−	9.65579i		−	0.355194i	−0.984103	−	0.177597i	$-0.943168\pi$
$739$		−	9.65579i		−	0.355194i	0.984103	−	0.177597i	$-0.0568324\pi$
$740$		0			0
$741$			1.35060i			0.0496157i
$742$		0			0
$743$	1.29760			0.0476045			0.0238023	−	0.999717i	$-0.492423\pi$
$743$	1.29760			0.0476045			0.0238023	+	0.999717i	$0.492423\pi$
$744$		0			0
$745$	−14.2484			−0.522022
$746$		0			0
$747$	−81.8154			−2.99347
$748$		0			0
$749$	45.9361			1.67847
$750$		0			0
$751$	3.01674			0.110083			0.0550413	−	0.998484i	$-0.482471\pi$
$751$	3.01674			0.110083			0.0550413	+	0.998484i	$0.482471\pi$
$752$		0			0
$753$		−	10.8726i		−	0.396219i
$754$		0			0
$755$	−13.4199			−0.488402
$756$		0			0
$757$			47.7461i			1.73536i	0.497122	+	0.867680i	$0.334390\pi$
$757$			47.7461i			1.73536i	−0.497122	+	0.867680i	$0.665610\pi$
$758$		0			0
$759$	53.2458	−	48.0512i	1.93270	−	1.74415i
$760$		0			0
$761$	−10.0601			−0.364680			−0.182340	−	0.983236i	$-0.558367\pi$
$761$	−10.0601			−0.364680			−0.182340	+	0.983236i	$0.558367\pi$
$762$		0			0
$763$		−	49.4464i		−	1.79008i
$764$		0			0
$765$	23.7175			0.857509
$766$		0			0
$767$			3.16664i			0.114341i
$768$		0			0
$769$			14.8983i			0.537247i	0.963245	+	0.268624i	$0.0865688\pi$
$769$			14.8983i			0.537247i	−0.963245	+	0.268624i	$0.913431\pi$
$770$		0			0
$771$		−	83.8216i		−	3.01876i
$772$		0			0
$773$		−	42.2230i		−	1.51865i	−0.650709	−	0.759327i	$-0.725528\pi$
$773$		−	42.2230i		−	1.51865i	0.650709	−	0.759327i	$-0.274472\pi$
$774$		0			0
$775$			9.87546i			0.354737i
$776$		0			0
$777$	146.408			5.25235
$778$		0			0
$779$	−1.40364			−0.0502906
$780$		0			0
$781$		−	28.2977i		−	1.01257i
$782$		0			0
$783$			154.700i			5.52852i
$784$		0			0
$785$	−0.175784			−0.00627401
$786$		0			0
$787$	38.2495			1.36345			0.681724	−	0.731610i	$-0.261231\pi$
$787$	38.2495			1.36345			0.681724	+	0.731610i	$0.261231\pi$
$788$		0			0
$789$			51.9642i			1.84997i
$790$		0			0
$791$		−	73.4986i		−	2.61331i
$792$		0			0
$793$		−	3.12626i		−	0.111017i
$794$		0			0
$795$		−	20.8859i		−	0.740747i
$796$		0			0
$797$		−	28.2641i		−	1.00116i	−0.865689	−	0.500582i	$-0.833119\pi$
$797$		−	28.2641i		−	1.00116i	0.865689	−	0.500582i	$-0.166881\pi$
$798$		0			0
$799$	17.3287			0.613045
$800$		0			0
$801$		−	131.949i		−	4.66218i
$802$		0			0
$803$	14.6526			0.517079
$804$		0			0
$805$	−15.7242	+	14.1902i	−0.554205	+	0.500138i
$806$		0			0
$807$			15.1595i			0.533639i
$808$		0			0
$809$	−6.85815			−0.241120			−0.120560	−	0.992706i	$-0.538469\pi$
$809$	−6.85815			−0.241120			−0.120560	+	0.992706i	$0.538469\pi$
$810$		0			0
$811$		−	21.7032i		−	0.762101i	−0.924554	−	0.381051i	$-0.875562\pi$
$811$		−	21.7032i		−	0.762101i	0.924554	−	0.381051i	$-0.124438\pi$
$812$		0			0
$813$	44.7689			1.57012
$814$		0			0
$815$	−8.42707			−0.295187
$816$		0			0
$817$	1.32814			0.0464658
$818$		0			0
$819$	37.9022			1.32441
$820$		0			0
$821$	29.9682			1.04590			0.522948	−	0.852364i	$-0.324832\pi$
$821$	29.9682			1.04590			0.522948	+	0.852364i	$0.324832\pi$
$822$		0			0
$823$			10.9602i			0.382048i	0.981585	+	0.191024i	$0.0611808\pi$
$823$			10.9602i			0.382048i	−0.981585	+	0.191024i	$0.938819\pi$
$824$		0			0
$825$			14.9550i			0.520668i
$826$		0			0
$827$	−17.3224			−0.602358			−0.301179	−	0.953568i	$-0.597380\pi$
$827$	−17.3224			−0.602358			−0.301179	+	0.953568i	$0.597380\pi$
$828$		0			0
$829$	−32.0484			−1.11309			−0.556544	−	0.830818i	$-0.687873\pi$
$829$	−32.0484			−1.11309			−0.556544	+	0.830818i	$0.687873\pi$
$830$		0			0
$831$		−	24.9265i		−	0.864691i
$832$		0			0
$833$			33.7003i			1.16765i
$834$		0			0
$835$	2.16700			0.0749922
$836$		0			0
$837$	196.783			6.80180
$838$		0			0
$839$	−28.6827			−0.990237			−0.495119	−	0.868825i	$-0.664875\pi$
$839$	−28.6827			−0.990237			−0.495119	+	0.868825i	$0.664875\pi$
$840$		0			0
$841$	31.2728			1.07837
$842$		0			0
$843$	21.6297			0.744966
$844$		0			0
$845$		−	12.0490i		−	0.414500i
$846$		0			0
$847$	35.1223			1.20682
$848$		0			0
$849$			106.168i			3.64368i
$850$		0			0
$851$	−34.3588	+	31.0068i	−1.17780	+	1.06290i
$852$		0			0
$853$	−29.3744			−1.00576			−0.502880	−	0.864356i	$-0.667726\pi$
$853$	−29.3744			−1.00576			−0.502880	+	0.864356i	$0.667726\pi$
$854$		0			0
$855$		−	3.54822i		−	0.121347i
$856$		0			0
$857$	38.6386			1.31987			0.659935	−	0.751323i	$-0.270584\pi$
$857$	38.6386			1.31987			0.659935	+	0.751323i	$0.270584\pi$
$858$		0			0
$859$			19.6801i			0.671478i	0.941955	+	0.335739i	$0.108986\pi$
$859$			19.6801i			0.671478i	−0.941955	+	0.335739i	$0.891014\pi$
$860$		0			0
$861$			52.8182i			1.80004i
$862$		0			0
$863$			11.7309i			0.399326i	0.979865	+	0.199663i	$0.0639847\pi$
$863$			11.7309i			0.399326i	−0.979865	+	0.199663i	$0.936015\pi$
$864$		0			0
$865$		−	2.30948i		−	0.0785245i
$866$		0			0
$867$		−	33.4490i		−	1.13599i
$868$		0			0
$869$	−52.0711			−1.76639
$870$		0			0
$871$	−3.64148			−0.123387
$872$		0			0
$873$			15.1838i			0.513894i
$874$		0			0
$875$		−	4.41643i		−	0.149303i
$876$		0			0
$877$	−49.2188			−1.66200			−0.831001	−	0.556271i	$-0.812232\pi$
$877$	−49.2188			−1.66200			−0.831001	+	0.556271i	$0.812232\pi$
$878$		0			0
$879$	−48.8873			−1.64893
$880$		0			0
$881$		−	38.0948i		−	1.28345i	−0.766936	−	0.641724i	$-0.778220\pi$
$881$		−	38.0948i		−	1.28345i	0.766936	−	0.641724i	$-0.221780\pi$
$882$		0			0
$883$			6.05328i			0.203709i	0.994799	+	0.101855i	$0.0324776\pi$
$883$			6.05328i			0.203709i	−0.994799	+	0.101855i	$0.967522\pi$
$884$		0			0
$885$		−	11.1551i		−	0.374974i
$886$		0			0
$887$		−	9.52706i		−	0.319887i	−0.987126	−	0.159944i	$-0.948869\pi$
$887$		−	9.52706i		−	0.319887i	0.987126	−	0.159944i	$-0.0511312\pi$
$888$		0			0
$889$		−	35.6212i		−	1.19470i
$890$		0			0
$891$	183.061			6.13276
$892$		0			0
$893$		−	2.59243i		−	0.0867524i
$894$		0			0
$895$	0.503447			0.0168284
$896$		0			0
$897$	−11.9269	+	10.7634i	−0.398229	+	0.359378i
$898$		0			0
$899$		−	76.6687i		−	2.55704i
$900$		0			0
$901$	16.3853			0.545875
$902$		0			0
$903$		−	49.9773i		−	1.66314i
$904$		0			0
$905$	6.13165			0.203823
$906$		0			0
$907$	24.3222			0.807604			0.403802	−	0.914846i	$-0.367688\pi$
$907$	24.3222			0.807604			0.403802	+	0.914846i	$0.367688\pi$
$908$		0			0
$909$	−72.2666			−2.39693
$910$		0			0
$911$	−20.9773			−0.695010			−0.347505	−	0.937678i	$-0.612971\pi$
$911$	−20.9773			−0.695010			−0.347505	+	0.937678i	$0.612971\pi$
$912$		0			0
$913$	40.4721			1.33943
$914$		0			0
$915$			11.0128i			0.364073i
$916$		0			0
$917$			28.4512i			0.939543i
$918$		0			0
$919$	39.1085			1.29007			0.645036	−	0.764152i	$-0.276843\pi$
$919$	39.1085			1.29007			0.645036	+	0.764152i	$0.276843\pi$
$920$		0			0
$921$	32.8172			1.08137
$922$		0			0
$923$			6.33863i			0.208639i
$924$		0			0
$925$		−	9.65028i		−	0.317299i
$926$		0			0
$927$	−48.8690			−1.60507
$928$		0			0
$929$	−20.1356			−0.660628			−0.330314	−	0.943871i	$-0.607155\pi$
$929$	−20.1356			−0.660628			−0.330314	+	0.943871i	$0.607155\pi$
$930$		0			0
$931$	5.04167			0.165234
$932$		0			0
$933$	18.9285			0.619690
$934$		0			0
$935$	−11.7325			−0.383693
$936$		0			0
$937$		−	19.9806i		−	0.652736i	−0.945243	−	0.326368i	$-0.894175\pi$
$937$		−	19.9806i		−	0.652736i	0.945243	−	0.326368i	$-0.105825\pi$
$938$		0			0
$939$	103.275			3.37024
$940$		0			0
$941$			17.2366i			0.561897i	0.959723	+	0.280949i	$0.0906491\pi$
$941$			17.2366i			0.561897i	−0.959723	+	0.280949i	$0.909351\pi$
$942$		0			0
$943$	−11.1860	−	12.3953i	−0.364267	−	0.403646i
$944$		0			0
$945$	−88.0036			−2.86276
$946$		0			0
$947$		−	23.4184i		−	0.760996i	−0.924782	−	0.380498i	$-0.875753\pi$
$947$		−	23.4184i		−	0.760996i	0.924782	−	0.380498i	$-0.124247\pi$
$948$		0			0
$949$	−3.28215			−0.106543
$950$		0			0
$951$			72.3763i			2.34696i
$952$		0			0
$953$		−	41.1735i		−	1.33374i	−0.745173	−	0.666871i	$-0.767633\pi$
$953$		−	41.1735i		−	1.33374i	0.745173	−	0.666871i	$-0.232367\pi$
$954$		0			0
$955$		−	17.7194i		−	0.573386i
$956$		0			0
$957$		−	116.104i		−	3.75312i
$958$		0			0
$959$		−	81.6029i		−	2.63510i
$960$		0			0
$961$	−66.5248			−2.14596
$962$		0			0
$963$	−91.5372			−2.94974
$964$		0			0
$965$		−	19.6626i		−	0.632962i
$966$		0			0
$967$		−	32.7188i		−	1.05217i	−0.850434	−	0.526083i	$-0.823660\pi$
$967$		−	32.7188i		−	1.05217i	0.850434	−	0.526083i	$-0.176340\pi$
$968$		0			0
$969$	3.73253			0.119906
$970$		0			0
$971$	43.3663			1.39169			0.695846	−	0.718191i	$-0.255030\pi$
$971$	43.3663			1.39169			0.695846	+	0.718191i	$0.255030\pi$
$972$		0			0
$973$			7.27462i			0.233214i
$974$		0			0
$975$		−	3.34990i		−	0.107283i
$976$		0			0
$977$			8.34969i			0.267130i	0.991040	+	0.133565i	$0.0426425\pi$
$977$			8.34969i			0.267130i	−0.991040	+	0.133565i	$0.957357\pi$
$978$		0			0
$979$			65.2718i			2.08610i
$980$		0			0
$981$			98.5321i			3.14589i
$982$		0			0
$983$	−39.1176			−1.24766			−0.623829	−	0.781561i	$-0.714424\pi$
$983$	−39.1176			−1.24766			−0.623829	+	0.781561i	$0.714424\pi$
$984$		0			0
$985$		−	12.8758i		−	0.410257i
$986$		0			0
$987$	−97.5518			−3.10511
$988$		0			0
$989$	10.5844	+	11.7286i	0.336563	+	0.372948i
$990$		0			0
$991$		−	29.1297i		−	0.925337i	−0.886531	−	0.462668i	$-0.846892\pi$
$991$		−	29.1297i		−	0.925337i	0.886531	−	0.462668i	$-0.153108\pi$
$992$		0			0
$993$	2.26878			0.0719976
$994$		0			0
$995$			13.4014i			0.424853i
$996$		0			0
$997$	−8.56349			−0.271208			−0.135604	−	0.990763i	$-0.543298\pi$
$997$	−8.56349			−0.271208			−0.135604	+	0.990763i	$0.543298\pi$
$998$		0			0
$999$	−192.296			−6.08396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.i.b.1471.1	yes	48
4.3	odd	2		3680.2.i.a.1471.48	yes	48
23.22	odd	2		3680.2.i.a.1471.1	✓	48
92.91	even	2	inner	3680.2.i.b.1471.48	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.i.a.1471.1	✓	48	23.22	odd	2
3680.2.i.a.1471.48	yes	48	4.3	odd	2
3680.2.i.b.1471.1	yes	48	1.1	even	1	trivial
3680.2.i.b.1471.48	yes	48	92.91	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.43521i q^{3} +1.00000i q^{5} +4.41643 q^{7} -8.80064 q^{9} +O(q^{10})\)	\(q-3.43521i q^{3} +1.00000i q^{5} +4.41643 q^{7} -8.80064 q^{9} +4.35346 q^{11} -0.975167 q^{13} +3.43521 q^{15} +2.69498i q^{17} +0.403177 q^{19} -15.1714i q^{21} +(3.21304 + 3.56039i) q^{23} -1.00000 q^{25} +19.9264i q^{27} +7.76355 q^{29} -9.87546i q^{31} -14.9550i q^{33} +4.41643i q^{35} +9.65028i q^{37} +3.34990i q^{39} -3.48144 q^{41} +3.29419 q^{43} -8.80064i q^{45} -6.43000i q^{47} +12.5049 q^{49} +9.25780 q^{51} -6.07996i q^{53} +4.35346i q^{55} -1.38500i q^{57} -3.24728i q^{59} +3.20587i q^{61} -38.8674 q^{63} -0.975167i q^{65} +3.73421 q^{67} +(12.2307 - 11.0375i) q^{69} -6.50005i q^{71} +3.36573 q^{73} +3.43521i q^{75} +19.2268 q^{77} -11.9608 q^{79} +42.0494 q^{81} +9.29653 q^{83} -2.69498 q^{85} -26.6694i q^{87} +14.9931i q^{89} -4.30676 q^{91} -33.9243 q^{93} +0.403177i q^{95} -1.72530i q^{97} -38.3133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q + 8 * q^7 - 48 * q^9	\( 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) \) 48 * q + 8 * q^7 - 48 * q^9 + 16 * q^11 + 20 * q^23 - 48 * q^25 - 8 * q^29 + 8 * q^41 + 56 * q^49 - 24 * q^51 - 120 * q^63 - 32 * q^67 - 20 * q^69 + 32 * q^77 + 72 * q^81 + 64 * q^83 + 40 * q^91 - 32 * q^93 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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