LMFDB - Embedded newform 4028.2.c.a.3497.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(3497,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4028.3497");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.c (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:	$32.1637419342$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3497.1
Character	$\chi$	$=$	4028.3497
Dual form			4028.2.c.a.3497.82

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.43171i q^{3} +2.90667i q^{5} -0.616337 q^{7} -8.77663 q^{9} +O(q^{10})$	$q-3.43171i q^{3} +2.90667i q^{5} -0.616337 q^{7} -8.77663 q^{9} +3.74183 q^{11} +5.46385 q^{13} +9.97485 q^{15} -4.00726 q^{17} -1.00000i q^{19} +2.11509i q^{21} +0.915890i q^{23} -3.44873 q^{25} +19.8237i q^{27} +0.748258 q^{29} +10.8493i q^{31} -12.8409i q^{33} -1.79149i q^{35} -7.16590 q^{37} -18.7503i q^{39} +4.86217i q^{41} -10.0856 q^{43} -25.5108i q^{45} -12.4131 q^{47} -6.62013 q^{49} +13.7518i q^{51} +(-6.42733 - 3.41898i) q^{53} +10.8763i q^{55} -3.43171 q^{57} -6.32753 q^{59} -4.80845i q^{61} +5.40936 q^{63} +15.8816i q^{65} +7.35107i q^{67} +3.14307 q^{69} +6.89603i q^{71} +3.76387i q^{73} +11.8350i q^{75} -2.30623 q^{77} -8.97852i q^{79} +41.6994 q^{81} +7.05948i q^{83} -11.6478i q^{85} -2.56780i q^{87} +4.99838 q^{89} -3.36757 q^{91} +37.2316 q^{93} +2.90667 q^{95} +15.2003 q^{97} -32.8406 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) $ 82 * q - 8 * q^7 - 82 * q^9	$ 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) $ 82 * q - 8 * q^7 - 82 * q^9 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 58 * q^25 - 16 * q^29 - 12 * q^37 - 32 * q^43 + 8 * q^47 + 98 * q^49 + 6 * q^53 - 4 * q^57 + 4 * q^59 + 8 * q^63 + 28 * q^69 - 8 * q^77 + 154 * q^81 - 20 * q^89 + 48 * q^91 - 56 * q^93 - 44 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4028\mathbb{Z}\right)^\times$.

$n$	$2015$	$2281$	$2757$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	3.43171i		−	1.98130i	−0.136433	−	0.990649i	$-0.543564\pi$
$3$		−	3.43171i		−	1.98130i	0.136433	−	0.990649i	$-0.456436\pi$
$4$		0			0
$5$			2.90667i			1.29990i	0.759976	+	0.649951i	$0.225211\pi$
$5$			2.90667i			1.29990i	−0.759976	+	0.649951i	$0.774789\pi$
$6$		0			0
$7$	−0.616337			−0.232953			−0.116477	−	0.993193i	$-0.537160\pi$
$7$	−0.616337			−0.232953			−0.116477	+	0.993193i	$0.537160\pi$
$8$		0			0
$9$	−8.77663			−2.92554
$10$		0			0
$11$	3.74183			1.12820			0.564102	−	0.825705i	$-0.309223\pi$
$11$	3.74183			1.12820			0.564102	+	0.825705i	$0.309223\pi$
$12$		0			0
$13$	5.46385			1.51540			0.757699	−	0.652604i	$-0.226323\pi$
$13$	5.46385			1.51540			0.757699	+	0.652604i	$0.226323\pi$
$14$		0			0
$15$	9.97485			2.57549
$16$		0			0
$17$	−4.00726			−0.971904			−0.485952	−	0.873985i	$-0.661527\pi$
$17$	−4.00726			−0.971904			−0.485952	+	0.873985i	$0.661527\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$			2.11509i			0.461550i
$22$		0			0
$23$			0.915890i			0.190976i	0.995431	+	0.0954882i	$0.0304412\pi$
$23$			0.915890i			0.190976i	−0.995431	+	0.0954882i	$0.969559\pi$
$24$		0			0
$25$	−3.44873			−0.689746
$26$		0			0
$27$			19.8237i			3.81508i
$28$		0			0
$29$	0.748258			0.138948			0.0694740	−	0.997584i	$-0.477868\pi$
$29$	0.748258			0.138948			0.0694740	+	0.997584i	$0.477868\pi$
$30$		0			0
$31$			10.8493i			1.94859i	0.225275	+	0.974295i	$0.427672\pi$
$31$			10.8493i			1.94859i	−0.225275	+	0.974295i	$0.572328\pi$
$32$		0			0
$33$		−	12.8409i		−	2.23531i
$34$		0			0
$35$		−	1.79149i		−	0.302817i
$36$		0			0
$37$	−7.16590			−1.17807			−0.589034	−	0.808108i	$-0.700491\pi$
$37$	−7.16590			−1.17807			−0.589034	+	0.808108i	$0.700491\pi$
$38$		0			0
$39$		−	18.7503i		−	3.00246i
$40$		0			0
$41$			4.86217i			0.759343i	0.925121	+	0.379671i	$0.123963\pi$
$41$			4.86217i			0.759343i	−0.925121	+	0.379671i	$0.876037\pi$
$42$		0			0
$43$	−10.0856			−1.53804			−0.769020	−	0.639225i	$-0.779255\pi$
$43$	−10.0856			−1.53804			−0.769020	+	0.639225i	$0.779255\pi$
$44$		0			0
$45$		−	25.5108i		−	3.80292i
$46$		0			0
$47$	−12.4131			−1.81063			−0.905317	−	0.424736i	$-0.860367\pi$
$47$	−12.4131			−1.81063			−0.905317	+	0.424736i	$0.860367\pi$
$48$		0			0
$49$	−6.62013			−0.945733
$50$		0			0
$51$			13.7518i			1.92563i
$52$		0			0
$53$	−6.42733	−	3.41898i	−0.882861	−	0.469634i
$53$	−6.42733	−	3.41898i	−0.882861	−	0.469634i
$54$		0			0
$55$			10.8763i			1.46655i
$56$		0			0
$57$	−3.43171			−0.454541
$58$		0			0
$59$	−6.32753			−0.823774			−0.411887	−	0.911235i	$-0.635130\pi$
$59$	−6.32753			−0.823774			−0.411887	+	0.911235i	$0.635130\pi$
$60$		0			0
$61$		−	4.80845i		−	0.615659i	−0.951441	−	0.307830i	$-0.900397\pi$
$61$		−	4.80845i		−	0.615659i	0.951441	−	0.307830i	$-0.0996027\pi$
$62$		0			0
$63$	5.40936			0.681516
$64$		0			0
$65$			15.8816i			1.96987i
$66$		0			0
$67$			7.35107i			0.898076i	0.893513	+	0.449038i	$0.148233\pi$
$67$			7.35107i			0.898076i	−0.893513	+	0.449038i	$0.851767\pi$
$68$		0			0
$69$	3.14307			0.378381
$70$		0			0
$71$			6.89603i			0.818408i	0.912443	+	0.409204i	$0.134194\pi$
$71$			6.89603i			0.818408i	−0.912443	+	0.409204i	$0.865806\pi$
$72$		0			0
$73$			3.76387i			0.440527i	0.975440	+	0.220264i	$0.0706918\pi$
$73$			3.76387i			0.440527i	−0.975440	+	0.220264i	$0.929308\pi$
$74$		0			0
$75$			11.8350i			1.36659i
$76$		0			0
$77$	−2.30623			−0.262819
$78$		0			0
$79$		−	8.97852i		−	1.01016i	−0.863072	−	0.505081i	$-0.831463\pi$
$79$		−	8.97852i		−	1.01016i	0.863072	−	0.505081i	$-0.168537\pi$
$80$		0			0
$81$	41.6994			4.63326
$82$		0			0
$83$			7.05948i			0.774878i	0.921895	+	0.387439i	$0.126640\pi$
$83$			7.05948i			0.774878i	−0.921895	+	0.387439i	$0.873360\pi$
$84$		0			0
$85$		−	11.6478i		−	1.26338i
$86$		0			0
$87$		−	2.56780i		−	0.275297i
$88$		0			0
$89$	4.99838			0.529827			0.264914	−	0.964272i	$-0.414657\pi$
$89$	4.99838			0.529827			0.264914	+	0.964272i	$0.414657\pi$
$90$		0			0
$91$	−3.36757			−0.353017
$92$		0			0
$93$	37.2316			3.86074
$94$		0			0
$95$	2.90667			0.298218
$96$		0			0
$97$	15.2003			1.54335			0.771677	−	0.636014i	$-0.219418\pi$
$97$	15.2003			1.54335			0.771677	+	0.636014i	$0.219418\pi$
$98$		0			0
$99$	−32.8406			−3.30061
$100$		0			0
$101$		−	12.9260i		−	1.28618i	−0.765789	−	0.643092i	$-0.777651\pi$
$101$		−	12.9260i		−	1.28618i	0.765789	−	0.643092i	$-0.222349\pi$
$102$		0			0
$103$			3.26634i			0.321842i	0.986967	+	0.160921i	$0.0514464\pi$
$103$			3.26634i			0.321842i	−0.986967	+	0.160921i	$0.948554\pi$
$104$		0			0
$105$	−6.14787			−0.599970
$106$		0			0
$107$	10.6405			1.02865			0.514327	−	0.857594i	$-0.328042\pi$
$107$	10.6405			1.02865			0.514327	+	0.857594i	$0.328042\pi$
$108$		0			0
$109$			6.40178i			0.613180i	0.951842	+	0.306590i	$0.0991880\pi$
$109$			6.40178i			0.613180i	−0.951842	+	0.306590i	$0.900812\pi$
$110$		0			0
$111$			24.5913i			2.33410i
$112$		0			0
$113$	−0.227116			−0.0213653			−0.0106827	−	0.999943i	$-0.503400\pi$
$113$	−0.227116			−0.0213653			−0.0106827	+	0.999943i	$0.503400\pi$
$114$		0			0
$115$	−2.66219			−0.248251
$116$		0			0
$117$	−47.9542			−4.43337
$118$		0			0
$119$	2.46983			0.226409
$120$		0			0
$121$	3.00127			0.272843
$122$		0			0
$123$	16.6855			1.50448
$124$		0			0
$125$			4.50903i			0.403300i
$126$		0			0
$127$			16.8369i			1.49404i	0.664804	+	0.747018i	$0.268515\pi$
$127$			16.8369i			1.49404i	−0.664804	+	0.747018i	$0.731485\pi$
$128$		0			0
$129$			34.6109i			3.04732i
$130$		0			0
$131$	7.15709			0.625318			0.312659	−	0.949865i	$-0.398780\pi$
$131$	7.15709			0.625318			0.312659	+	0.949865i	$0.398780\pi$
$132$		0			0
$133$			0.616337i			0.0534432i
$134$		0			0
$135$	−57.6210			−4.95923
$136$		0			0
$137$			12.2193i			1.04396i	0.852957	+	0.521982i	$0.174807\pi$
$137$			12.2193i			1.04396i	−0.852957	+	0.521982i	$0.825193\pi$
$138$		0			0
$139$			0.128065i			0.0108623i	0.999985	+	0.00543116i	$0.00172880\pi$
$139$			0.128065i			0.0108623i	−0.999985	+	0.00543116i	$0.998271\pi$
$140$		0			0
$141$			42.5981i			3.58741i
$142$		0			0
$143$	20.4448			1.70968
$144$		0			0
$145$			2.17494i			0.180619i
$146$		0			0
$147$			22.7184i			1.87378i
$148$		0			0
$149$	−17.1833			−1.40771			−0.703855	−	0.710343i	$-0.748540\pi$
$149$	−17.1833			−1.40771			−0.703855	+	0.710343i	$0.748540\pi$
$150$		0			0
$151$		−	7.34847i		−	0.598010i	−0.954252	−	0.299005i	$-0.903345\pi$
$151$		−	7.34847i		−	0.598010i	0.954252	−	0.299005i	$-0.0966548\pi$
$152$		0			0
$153$	35.1703			2.84335
$154$		0			0
$155$	−31.5353			−2.53298
$156$		0			0
$157$		−	22.0507i		−	1.75984i	−0.475122	−	0.879920i	$-0.657596\pi$
$157$		−	22.0507i		−	1.75984i	0.475122	−	0.879920i	$-0.342404\pi$
$158$		0			0
$159$	−11.7330	+	22.0567i	−0.930484	+	1.74921i
$160$		0			0
$161$		−	0.564497i		−	0.0444886i
$162$		0			0
$163$	−8.94264			−0.700442			−0.350221	−	0.936667i	$-0.613893\pi$
$163$	−8.94264			−0.700442			−0.350221	+	0.936667i	$0.613893\pi$
$164$		0			0
$165$	37.3242			2.90568
$166$		0			0
$167$		−	6.90151i		−	0.534055i	−0.963689	−	0.267027i	$-0.913959\pi$
$167$		−	6.90151i		−	0.534055i	0.963689	−	0.267027i	$-0.0860414\pi$
$168$		0			0
$169$	16.8536			1.29643
$170$		0			0
$171$			8.77663i			0.671166i
$172$		0			0
$173$			10.7707i			0.818884i	0.912336	+	0.409442i	$0.134277\pi$
$173$			10.7707i			0.818884i	−0.912336	+	0.409442i	$0.865723\pi$
$174$		0			0
$175$	2.12558			0.160679
$176$		0			0
$177$			21.7143i			1.63214i
$178$		0			0
$179$			9.19901i			0.687566i	0.939049	+	0.343783i	$0.111708\pi$
$179$			9.19901i			0.687566i	−0.939049	+	0.343783i	$0.888292\pi$
$180$		0			0
$181$		−	14.3046i		−	1.06325i	−0.846978	−	0.531627i	$-0.821581\pi$
$181$		−	14.3046i		−	1.06325i	0.846978	−	0.531627i	$-0.178419\pi$
$182$		0			0
$183$	−16.5012			−1.21980
$184$		0			0
$185$		−	20.8289i		−	1.53137i
$186$		0			0
$187$	−14.9945			−1.09651
$188$		0			0
$189$		−	12.2181i		−	0.888735i
$190$		0			0
$191$			9.87367i			0.714433i	0.934022	+	0.357217i	$0.116274\pi$
$191$			9.87367i			0.714433i	−0.934022	+	0.357217i	$0.883726\pi$
$192$		0			0
$193$		−	6.33876i		−	0.456274i	−0.973629	−	0.228137i	$-0.926737\pi$
$193$		−	6.33876i		−	0.456274i	0.973629	−	0.228137i	$-0.0732634\pi$
$194$		0			0
$195$	54.5010			3.90290
$196$		0			0
$197$	−6.59303			−0.469734			−0.234867	−	0.972028i	$-0.575465\pi$
$197$	−6.59303			−0.469734			−0.234867	+	0.972028i	$0.575465\pi$
$198$		0			0
$199$	21.7169			1.53947			0.769735	−	0.638364i	$-0.220388\pi$
$199$	21.7169			1.53947			0.769735	+	0.638364i	$0.220388\pi$
$200$		0			0
$201$	25.2267			1.77936
$202$		0			0
$203$	−0.461179			−0.0323684
$204$		0			0
$205$	−14.1327			−0.987071
$206$		0			0
$207$		−	8.03843i		−	0.558710i
$208$		0			0
$209$		−	3.74183i		−	0.258828i
$210$		0			0
$211$	−12.4137			−0.854596			−0.427298	−	0.904111i	$-0.640535\pi$
$211$	−12.4137			−0.854596			−0.427298	+	0.904111i	$0.640535\pi$
$212$		0			0
$213$	23.6652			1.62151
$214$		0			0
$215$		−	29.3155i		−	1.99930i
$216$		0			0
$217$		−	6.68682i		−	0.453931i
$218$		0			0
$219$	12.9165			0.872816
$220$		0			0
$221$	−21.8951			−1.47282
$222$		0			0
$223$	15.1137			1.01209			0.506044	−	0.862508i	$-0.331107\pi$
$223$	15.1137			1.01209			0.506044	+	0.862508i	$0.331107\pi$
$224$		0			0
$225$	30.2682			2.01788
$226$		0			0
$227$	9.76004			0.647797			0.323898	−	0.946092i	$-0.395006\pi$
$227$	9.76004			0.647797			0.323898	+	0.946092i	$0.395006\pi$
$228$		0			0
$229$	5.21226			0.344436			0.172218	−	0.985059i	$-0.444907\pi$
$229$	5.21226			0.344436			0.172218	+	0.985059i	$0.444907\pi$
$230$		0			0
$231$			7.91430i			0.520723i
$232$		0			0
$233$			6.97567i			0.456992i	0.973545	+	0.228496i	$0.0733807\pi$
$233$			6.97567i			0.456992i	−0.973545	+	0.228496i	$0.926619\pi$
$234$		0			0
$235$		−	36.0807i		−	2.35365i
$236$		0			0
$237$	−30.8117			−2.00143
$238$		0			0
$239$			25.2006i			1.63009i	0.579399	+	0.815044i	$0.303287\pi$
$239$			25.2006i			1.63009i	−0.579399	+	0.815044i	$0.696713\pi$
$240$		0			0
$241$	12.1161			0.780469			0.390234	−	0.920716i	$-0.372394\pi$
$241$	12.1161			0.780469			0.390234	+	0.920716i	$0.372394\pi$
$242$		0			0
$243$		−	83.6290i		−	5.36480i
$244$		0			0
$245$		−	19.2425i		−	1.22936i
$246$		0			0
$247$		−	5.46385i		−	0.347656i
$248$		0			0
$249$	24.2261			1.53527
$250$		0			0
$251$			2.82463i			0.178289i	0.996019	+	0.0891445i	$0.0284133\pi$
$251$			2.82463i			0.178289i	−0.996019	+	0.0891445i	$0.971587\pi$
$252$		0			0
$253$			3.42710i			0.215460i
$254$		0			0
$255$	−39.9719			−2.50313
$256$		0			0
$257$			25.8046i			1.60964i	0.593516	+	0.804822i	$0.297739\pi$
$257$			25.8046i			1.60964i	−0.593516	+	0.804822i	$0.702261\pi$
$258$		0			0
$259$	4.41661			0.274435
$260$		0			0
$261$	−6.56718			−0.406498
$262$		0			0
$263$			17.1255i			1.05600i	0.849243	+	0.528002i	$0.177059\pi$
$263$			17.1255i			1.05600i	−0.849243	+	0.528002i	$0.822941\pi$
$264$		0			0
$265$	9.93786	−	18.6821i	0.610478	−	1.14763i
$266$		0			0
$267$		−	17.1530i		−	1.04975i
$268$		0			0
$269$	0.212698			0.0129684			0.00648420	−	0.999979i	$-0.497936\pi$
$269$	0.212698			0.0129684			0.00648420	+	0.999979i	$0.497936\pi$
$270$		0			0
$271$	−18.5535			−1.12704			−0.563522	−	0.826101i	$-0.690554\pi$
$271$	−18.5535			−1.12704			−0.563522	+	0.826101i	$0.690554\pi$
$272$		0			0
$273$			11.5565i			0.699433i
$274$		0			0
$275$	−12.9045			−0.778173
$276$		0			0
$277$			16.0462i			0.964124i	0.876137	+	0.482062i	$0.160112\pi$
$277$			16.0462i			0.964124i	−0.876137	+	0.482062i	$0.839888\pi$
$278$		0			0
$279$		−	95.2203i		−	5.70069i
$280$		0			0
$281$	21.9320			1.30836			0.654178	−	0.756341i	$-0.273015\pi$
$281$	21.9320			1.30836			0.654178	+	0.756341i	$0.273015\pi$
$282$		0			0
$283$			15.0152i			0.892560i	0.894893	+	0.446280i	$0.147252\pi$
$283$			15.0152i			0.892560i	−0.894893	+	0.446280i	$0.852748\pi$
$284$		0			0
$285$		−	9.97485i		−	0.590859i
$286$		0			0
$287$		−	2.99673i		−	0.176892i
$288$		0			0
$289$	−0.941829			−0.0554017
$290$		0			0
$291$		−	52.1630i		−	3.05785i
$292$		0			0
$293$	−32.5754			−1.90307			−0.951537	−	0.307534i	$-0.900496\pi$
$293$	−32.5754			−1.90307			−0.951537	+	0.307534i	$0.900496\pi$
$294$		0			0
$295$		−	18.3920i		−	1.07083i
$296$		0			0
$297$			74.1769i			4.30418i
$298$		0			0
$299$			5.00428i			0.289405i
$300$		0			0
$301$	6.21613			0.358292
$302$		0			0
$303$	−44.3583			−2.54832
$304$		0			0
$305$	13.9766			0.800297
$306$		0			0
$307$	2.70282			0.154258			0.0771289	−	0.997021i	$-0.475425\pi$
$307$	2.70282			0.154258			0.0771289	+	0.997021i	$0.475425\pi$
$308$		0			0
$309$	11.2091			0.637665
$310$		0			0
$311$	−21.3936			−1.21312			−0.606561	−	0.795037i	$-0.707451\pi$
$311$	−21.3936			−1.21312			−0.606561	+	0.795037i	$0.707451\pi$
$312$		0			0
$313$		−	1.09176i		−	0.0617098i	−0.999524	−	0.0308549i	$-0.990177\pi$
$313$		−	1.09176i		−	0.0617098i	0.999524	−	0.0308549i	$-0.00982297\pi$
$314$		0			0
$315$			15.7232i			0.885904i
$316$		0			0
$317$	0.873743			0.0490743			0.0245372	−	0.999699i	$-0.492189\pi$
$317$	0.873743			0.0490743			0.0245372	+	0.999699i	$0.492189\pi$
$318$		0			0
$319$	2.79985			0.156762
$320$		0			0
$321$		−	36.5150i		−	2.03807i
$322$		0			0
$323$			4.00726i			0.222970i
$324$		0			0
$325$	−18.8433			−1.04524
$326$		0			0
$327$	21.9691			1.21489
$328$		0			0
$329$	7.65064			0.421793
$330$		0			0
$331$	20.2977			1.11566			0.557831	−	0.829954i	$-0.311634\pi$
$331$	20.2977			1.11566			0.557831	+	0.829954i	$0.311634\pi$
$332$		0			0
$333$	62.8925			3.44649
$334$		0			0
$335$	−21.3671			−1.16741
$336$		0			0
$337$		−	11.9528i		−	0.651111i	−0.945523	−	0.325555i	$-0.894449\pi$
$337$		−	11.9528i		−	0.651111i	0.945523	−	0.325555i	$-0.105551\pi$
$338$		0			0
$339$			0.779397i			0.0423310i
$340$		0			0
$341$			40.5962i			2.19841i
$342$		0			0
$343$	8.39459			0.453265
$344$		0			0
$345$			9.13587i			0.491858i
$346$		0			0
$347$	−4.63739			−0.248948			−0.124474	−	0.992223i	$-0.539724\pi$
$347$	−4.63739			−0.248948			−0.124474	+	0.992223i	$0.539724\pi$
$348$		0			0
$349$			1.06456i			0.0569845i	0.999594	+	0.0284922i	$0.00907059\pi$
$349$			1.06456i			0.0569845i	−0.999594	+	0.0284922i	$0.990929\pi$
$350$		0			0
$351$			108.314i			5.78136i
$352$		0			0
$353$			27.1732i			1.44628i	0.690700	+	0.723142i	$0.257303\pi$
$353$			27.1732i			1.44628i	−0.690700	+	0.723142i	$0.742697\pi$
$354$		0			0
$355$	−20.0445			−1.06385
$356$		0			0
$357$		−	8.47572i		−	0.448583i
$358$		0			0
$359$		−	21.3032i		−	1.12434i	−0.827022	−	0.562170i	$-0.809967\pi$
$359$		−	21.3032i		−	1.12434i	0.827022	−	0.562170i	$-0.190033\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$		−	10.2995i		−	0.540583i
$364$		0			0
$365$	−10.9403			−0.572642
$366$		0			0
$367$	0.204036			0.0106506			0.00532530	−	0.999986i	$-0.498305\pi$
$367$	0.204036			0.0106506			0.00532530	+	0.999986i	$0.498305\pi$
$368$		0			0
$369$		−	42.6734i		−	2.22149i
$370$		0			0
$371$	3.96140	+	2.10725i	0.205666	+	0.109403i
$372$		0			0
$373$		−	6.10470i		−	0.316090i	−0.987432	−	0.158045i	$-0.949481\pi$
$373$		−	6.10470i		−	0.316090i	0.987432	−	0.158045i	$-0.0505190\pi$
$374$		0			0
$375$	15.4737			0.799058
$376$		0			0
$377$	4.08837			0.210562
$378$		0			0
$379$			0.808030i			0.0415057i	0.999785	+	0.0207529i	$0.00660631\pi$
$379$			0.808030i			0.0415057i	−0.999785	+	0.0207529i	$0.993394\pi$
$380$		0			0
$381$	57.7795			2.96013
$382$		0			0
$383$		−	4.36035i		−	0.222803i	−0.993775	−	0.111402i	$-0.964466\pi$
$383$		−	4.36035i		−	0.222803i	0.993775	−	0.111402i	$-0.0355340\pi$
$384$		0			0
$385$		−	6.70344i		−	0.341639i
$386$		0			0
$387$	88.5176			4.49960
$388$		0			0
$389$		−	31.3744i		−	1.59074i	−0.606122	−	0.795372i	$-0.707276\pi$
$389$		−	31.3744i		−	1.59074i	0.606122	−	0.795372i	$-0.292724\pi$
$390$		0			0
$391$		−	3.67021i		−	0.185611i
$392$		0			0
$393$		−	24.5611i		−	1.23894i
$394$		0			0
$395$	26.0976			1.31311
$396$		0			0
$397$		−	29.8365i		−	1.49745i	−0.662881	−	0.748725i	$-0.730667\pi$
$397$		−	29.8365i		−	1.49745i	0.662881	−	0.748725i	$-0.269333\pi$
$398$		0			0
$399$	2.11509			0.105887
$400$		0			0
$401$		−	11.5047i		−	0.574517i	−0.957853	−	0.287259i	$-0.907256\pi$
$401$		−	11.5047i		−	0.574517i	0.957853	−	0.287259i	$-0.0927439\pi$
$402$		0			0
$403$			59.2789i			2.95289i
$404$		0			0
$405$			121.206i			6.02279i
$406$		0			0
$407$	−26.8136			−1.32910
$408$		0			0
$409$	−26.8306			−1.32669			−0.663343	−	0.748316i	$-0.730863\pi$
$409$	−26.8306			−1.32669			−0.663343	+	0.748316i	$0.730863\pi$
$410$		0			0
$411$	41.9330			2.06840
$412$		0			0
$413$	3.89989			0.191901
$414$		0			0
$415$	−20.5196			−1.00727
$416$		0			0
$417$	0.439481			0.0215215
$418$		0			0
$419$		−	16.4825i		−	0.805223i	−0.915371	−	0.402611i	$-0.868103\pi$
$419$		−	16.4825i		−	0.805223i	0.915371	−	0.402611i	$-0.131897\pi$
$420$		0			0
$421$		−	35.0016i		−	1.70588i	−0.522013	−	0.852938i	$-0.674819\pi$
$421$		−	35.0016i		−	1.70588i	0.522013	−	0.852938i	$-0.325181\pi$
$422$		0			0
$423$	108.945			5.29709
$424$		0			0
$425$	13.8200			0.670367
$426$		0			0
$427$			2.96363i			0.143420i
$428$		0			0
$429$		−	70.1605i		−	3.38738i
$430$		0			0
$431$	−36.5719			−1.76161			−0.880803	−	0.473482i	$-0.842997\pi$
$431$	−36.5719			−1.76161			−0.880803	+	0.473482i	$0.842997\pi$
$432$		0			0
$433$	7.74275			0.372093			0.186046	−	0.982541i	$-0.440433\pi$
$433$	7.74275			0.372093			0.186046	+	0.982541i	$0.440433\pi$
$434$		0			0
$435$	7.46376			0.357860
$436$		0			0
$437$	0.915890			0.0438130
$438$		0			0
$439$	0.732074			0.0349400			0.0174700	−	0.999847i	$-0.494439\pi$
$439$	0.732074			0.0349400			0.0174700	+	0.999847i	$0.494439\pi$
$440$		0			0
$441$	58.1024			2.76678
$442$		0			0
$443$		−	1.38546i		−	0.0658253i	−0.999458	−	0.0329127i	$-0.989522\pi$
$443$		−	1.38546i		−	0.0658253i	0.999458	−	0.0329127i	$-0.0104783\pi$
$444$		0			0
$445$			14.5286i			0.688724i
$446$		0			0
$447$			58.9681i			2.78910i
$448$		0			0
$449$	12.6142			0.595302			0.297651	−	0.954675i	$-0.403797\pi$
$449$	12.6142			0.595302			0.297651	+	0.954675i	$0.403797\pi$
$450$		0			0
$451$			18.1934i			0.856693i
$452$		0			0
$453$	−25.2178			−1.18484
$454$		0			0
$455$		−	9.78842i		−	0.458888i
$456$		0			0
$457$			35.7924i			1.67430i	0.546974	+	0.837149i	$0.315780\pi$
$457$			35.7924i			1.67430i	−0.546974	+	0.837149i	$0.684220\pi$
$458$		0			0
$459$		−	79.4389i		−	3.70789i
$460$		0			0
$461$	21.5659			1.00442			0.502211	−	0.864745i	$-0.332520\pi$
$461$	21.5659			1.00442			0.502211	+	0.864745i	$0.332520\pi$
$462$		0			0
$463$			39.2869i			1.82581i	0.408167	+	0.912907i	$0.366168\pi$
$463$			39.2869i			1.82581i	−0.408167	+	0.912907i	$0.633832\pi$
$464$		0			0
$465$			108.220i			5.01858i
$466$		0			0
$467$	3.34533			0.154803			0.0774017	−	0.997000i	$-0.475338\pi$
$467$	3.34533			0.154803			0.0774017	+	0.997000i	$0.475338\pi$
$468$		0			0
$469$		−	4.53073i		−	0.209210i
$470$		0			0
$471$	−75.6717			−3.48677
$472$		0			0
$473$	−37.7386			−1.73522
$474$		0			0
$475$			3.44873i			0.158239i
$476$		0			0
$477$	56.4103	+	30.0072i	2.58285	+	1.37393i
$478$		0			0
$479$		−	17.5160i		−	0.800326i	−0.916444	−	0.400163i	$-0.868953\pi$
$479$		−	17.5160i		−	0.800326i	0.916444	−	0.400163i	$-0.131047\pi$
$480$		0			0
$481$	−39.1534			−1.78524
$482$		0			0
$483$	−1.93719			−0.0881452
$484$		0			0
$485$			44.1822i			2.00621i
$486$		0			0
$487$	14.2197			0.644354			0.322177	−	0.946679i	$-0.395585\pi$
$487$	14.2197			0.644354			0.322177	+	0.946679i	$0.395585\pi$
$488$		0			0
$489$			30.6885i			1.38778i
$490$		0			0
$491$			32.3959i			1.46201i	0.682373	+	0.731004i	$0.260948\pi$
$491$			32.3959i			1.46201i	−0.682373	+	0.731004i	$0.739052\pi$
$492$		0			0
$493$	−2.99847			−0.135044
$494$		0			0
$495$		−	95.4569i		−	4.29047i
$496$		0			0
$497$		−	4.25028i		−	0.190651i
$498$		0			0
$499$			3.45017i			0.154451i	0.997014	+	0.0772254i	$0.0246061\pi$
$499$			3.45017i			0.154451i	−0.997014	+	0.0772254i	$0.975394\pi$
$500$		0			0
$501$	−23.6840			−1.05812
$502$		0			0
$503$			10.0943i			0.450082i	0.974349	+	0.225041i	$0.0722516\pi$
$503$			10.0943i			0.450082i	−0.974349	+	0.225041i	$0.927748\pi$
$504$		0			0
$505$	37.5716			1.67191
$506$		0			0
$507$		−	57.8368i		−	2.56862i
$508$		0			0
$509$		−	19.9276i		−	0.883277i	−0.897193	−	0.441638i	$-0.854397\pi$
$509$		−	19.9276i		−	0.883277i	0.897193	−	0.441638i	$-0.145603\pi$
$510$		0			0
$511$		−	2.31981i		−	0.102622i
$512$		0			0
$513$	19.8237			0.875239
$514$		0			0
$515$	−9.49417			−0.418363
$516$		0			0
$517$	−46.4476			−2.04276
$518$		0			0
$519$	36.9620			1.62245
$520$		0			0
$521$	−10.8524			−0.475454			−0.237727	−	0.971332i	$-0.576402\pi$
$521$	−10.8524			−0.475454			−0.237727	+	0.971332i	$0.576402\pi$
$522$		0			0
$523$	−45.4217			−1.98615			−0.993075	−	0.117478i	$-0.962519\pi$
$523$	−45.4217			−1.98615			−0.993075	+	0.117478i	$0.962519\pi$
$524$		0			0
$525$		−	7.29437i		−	0.318352i
$526$		0			0
$527$		−	43.4760i		−	1.89384i
$528$		0			0
$529$	22.1611			0.963528
$530$		0			0
$531$	55.5344			2.40999
$532$		0			0
$533$			26.5661i			1.15071i
$534$		0			0
$535$			30.9283i			1.33715i
$536$		0			0
$537$	31.5683			1.36227
$538$		0			0
$539$	−24.7714			−1.06698
$540$		0			0
$541$	10.9835			0.472216			0.236108	−	0.971727i	$-0.424128\pi$
$541$	10.9835			0.472216			0.236108	+	0.971727i	$0.424128\pi$
$542$		0			0
$543$	−49.0893			−2.10662
$544$		0			0
$545$	−18.6079			−0.797073
$546$		0			0
$547$	−26.2395			−1.12192			−0.560960	−	0.827843i	$-0.689568\pi$
$547$	−26.2395			−1.12192			−0.560960	+	0.827843i	$0.689568\pi$
$548$		0			0
$549$			42.2020i			1.80114i
$550$		0			0
$551$		−	0.748258i		−	0.0318769i
$552$		0			0
$553$			5.53379i			0.235321i
$554$		0			0
$555$	−71.4788			−3.03411
$556$		0			0
$557$		−	4.97058i		−	0.210610i	−0.994440	−	0.105305i	$-0.966418\pi$
$557$		−	4.97058i		−	0.210610i	0.994440	−	0.105305i	$-0.0335819\pi$
$558$		0			0
$559$	−55.1062			−2.33074
$560$		0			0
$561$			51.4567i			2.17251i
$562$		0			0
$563$			18.7745i			0.791252i	0.918412	+	0.395626i	$0.129472\pi$
$563$			18.7745i			0.791252i	−0.918412	+	0.395626i	$0.870528\pi$
$564$		0			0
$565$		−	0.660152i		−	0.0277728i
$566$		0			0
$567$	−25.7009			−1.07933
$568$		0			0
$569$		−	37.3211i		−	1.56458i	−0.622913	−	0.782291i	$-0.714051\pi$
$569$		−	37.3211i		−	1.56458i	0.622913	−	0.782291i	$-0.285949\pi$
$570$		0			0
$571$			30.7261i			1.28585i	0.765931	+	0.642923i	$0.222278\pi$
$571$			30.7261i			1.28585i	−0.765931	+	0.642923i	$0.777722\pi$
$572$		0			0
$573$	33.8836			1.41551
$574$		0			0
$575$		−	3.15866i		−	0.131725i
$576$		0			0
$577$	−24.9189			−1.03739			−0.518693	−	0.854960i	$-0.673581\pi$
$577$	−24.9189			−1.03739			−0.518693	+	0.854960i	$0.673581\pi$
$578$		0			0
$579$	−21.7528			−0.904015
$580$		0			0
$581$		−	4.35102i		−	0.180511i
$582$		0			0
$583$	−24.0500	−	12.7932i	−0.996047	−	0.529842i
$584$		0			0
$585$		−	139.387i		−	5.76294i
$586$		0			0
$587$	25.7980			1.06480			0.532398	−	0.846494i	$-0.321291\pi$
$587$	25.7980			1.06480			0.532398	+	0.846494i	$0.321291\pi$
$588$		0			0
$589$	10.8493			0.447037
$590$		0			0
$591$			22.6254i			0.930683i
$592$		0			0
$593$	−43.8087			−1.79901			−0.899504	−	0.436913i	$-0.856072\pi$
$593$	−43.8087			−1.79901			−0.899504	+	0.436913i	$0.856072\pi$
$594$		0			0
$595$			7.17897i			0.294309i
$596$		0			0
$597$		−	74.5261i		−	3.05015i
$598$		0			0
$599$	18.2887			0.747254			0.373627	−	0.927579i	$-0.378114\pi$
$599$	18.2887			0.747254			0.373627	+	0.927579i	$0.378114\pi$
$600$		0			0
$601$		−	24.7213i		−	1.00840i	−0.863586	−	0.504202i	$-0.831787\pi$
$601$		−	24.7213i		−	1.00840i	0.863586	−	0.504202i	$-0.168213\pi$
$602$		0			0
$603$		−	64.5176i		−	2.62736i
$604$		0			0
$605$			8.72370i			0.354669i
$606$		0			0
$607$	20.8246			0.845243			0.422622	−	0.906306i	$-0.361110\pi$
$607$	20.8246			0.845243			0.422622	+	0.906306i	$0.361110\pi$
$608$		0			0
$609$			1.58263i			0.0641315i
$610$		0			0
$611$	−67.8232			−2.74383
$612$		0			0
$613$		−	28.3881i		−	1.14658i	−0.819351	−	0.573292i	$-0.805666\pi$
$613$		−	28.3881i		−	1.14658i	0.819351	−	0.573292i	$-0.194334\pi$
$614$		0			0
$615$			48.4994i			1.95568i
$616$		0			0
$617$			39.1701i			1.57693i	0.615080	+	0.788464i	$0.289124\pi$
$617$			39.1701i			1.57693i	−0.615080	+	0.788464i	$0.710876\pi$
$618$		0			0
$619$	36.2921			1.45870			0.729351	−	0.684140i	$-0.239822\pi$
$619$	36.2921			1.45870			0.729351	+	0.684140i	$0.239822\pi$
$620$		0			0
$621$	−18.1564			−0.728589
$622$		0			0
$623$	−3.08069			−0.123425
$624$		0			0
$625$	−30.3499			−1.21400
$626$		0			0
$627$	−12.8409			−0.512815
$628$		0			0
$629$	28.7157			1.14497
$630$		0			0
$631$		−	25.3653i		−	1.00978i	−0.863184	−	0.504889i	$-0.831534\pi$
$631$		−	25.3653i		−	1.00978i	0.863184	−	0.504889i	$-0.168466\pi$
$632$		0			0
$633$			42.6003i			1.69321i
$634$		0			0
$635$	−48.9394			−1.94210
$636$		0			0
$637$	−36.1714			−1.43316
$638$		0			0
$639$		−	60.5239i		−	2.39429i
$640$		0			0
$641$			28.8252i			1.13853i	0.822155	+	0.569263i	$0.192771\pi$
$641$			28.8252i			1.13853i	−0.822155	+	0.569263i	$0.807229\pi$
$642$		0			0
$643$	20.4352			0.805887			0.402943	−	0.915225i	$-0.367987\pi$
$643$	20.4352			0.805887			0.402943	+	0.915225i	$0.367987\pi$
$644$		0			0
$645$	−100.602			−3.96121
$646$		0			0
$647$	−8.19221			−0.322069			−0.161035	−	0.986949i	$-0.551483\pi$
$647$	−8.19221			−0.322069			−0.161035	+	0.986949i	$0.551483\pi$
$648$		0			0
$649$	−23.6765			−0.929385
$650$		0			0
$651$	−22.9472			−0.899373
$652$		0			0
$653$	2.67007			0.104488			0.0522439	−	0.998634i	$-0.483363\pi$
$653$	2.67007			0.104488			0.0522439	+	0.998634i	$0.483363\pi$
$654$		0			0
$655$			20.8033i			0.812853i
$656$		0			0
$657$		−	33.0341i		−	1.28878i
$658$		0			0
$659$		−	10.5159i		−	0.409641i	−0.978800	−	0.204820i	$-0.934339\pi$
$659$		−	10.5159i		−	0.409641i	0.978800	−	0.204820i	$-0.0656610\pi$
$660$		0			0
$661$	15.1600			0.589654			0.294827	−	0.955551i	$-0.404738\pi$
$661$	15.1600			0.589654			0.294827	+	0.955551i	$0.404738\pi$
$662$		0			0
$663$			75.1376i			2.91810i
$664$		0			0
$665$	−1.79149			−0.0694709
$666$		0			0
$667$			0.685322i			0.0265358i
$668$		0			0
$669$		−	51.8658i		−	2.00525i
$670$		0			0
$671$		−	17.9924i		−	0.694589i
$672$		0			0
$673$	34.7809			1.34071			0.670353	−	0.742042i	$-0.266142\pi$
$673$	34.7809			1.34071			0.670353	+	0.742042i	$0.266142\pi$
$674$		0			0
$675$		−	68.3666i		−	2.63143i
$676$		0			0
$677$			17.7247i			0.681214i	0.940206	+	0.340607i	$0.110633\pi$
$677$			17.7247i			0.681214i	−0.940206	+	0.340607i	$0.889367\pi$
$678$		0			0
$679$	−9.36849			−0.359530
$680$		0			0
$681$		−	33.4936i		−	1.28348i
$682$		0			0
$683$	0.0888533			0.00339988			0.00169994	−	0.999999i	$-0.499459\pi$
$683$	0.0888533			0.00339988			0.00169994	+	0.999999i	$0.499459\pi$
$684$		0			0
$685$	−35.5174			−1.35705
$686$		0			0
$687$		−	17.8870i		−	0.682431i
$688$		0			0
$689$	−35.1179	−	18.6808i	−1.33789	−	0.711682i
$690$		0			0
$691$		−	17.3000i		−	0.658125i	−0.944308	−	0.329062i	$-0.893267\pi$
$691$		−	17.3000i		−	0.658125i	0.944308	−	0.329062i	$-0.106733\pi$
$692$		0			0
$693$	20.2409			0.768888
$694$		0			0
$695$	−0.372242			−0.0141200
$696$		0			0
$697$		−	19.4840i		−	0.738009i
$698$		0			0
$699$	23.9385			0.905437
$700$		0			0
$701$		−	22.3320i		−	0.843469i	−0.906719	−	0.421735i	$-0.861421\pi$
$701$		−	22.3320i		−	0.843469i	0.906719	−	0.421735i	$-0.138579\pi$
$702$		0			0
$703$			7.16590i			0.270267i
$704$		0			0
$705$	−123.819			−4.66328
$706$		0			0
$707$			7.96677i			0.299621i
$708$		0			0
$709$		−	9.49731i		−	0.356679i	−0.983969	−	0.178339i	$-0.942927\pi$
$709$		−	9.49731i		−	0.356679i	0.983969	−	0.178339i	$-0.0570725\pi$
$710$		0			0
$711$			78.8012i			2.95527i
$712$		0			0
$713$	−9.93676			−0.372135
$714$		0			0
$715$			59.4262i			2.22241i
$716$		0			0
$717$	86.4810			3.22969
$718$		0			0
$719$			24.6475i			0.919197i	0.888127	+	0.459598i	$0.152007\pi$
$719$			24.6475i			0.919197i	−0.888127	+	0.459598i	$0.847993\pi$
$720$		0			0
$721$		−	2.01316i		−	0.0749742i
$722$		0			0
$723$		−	41.5791i		−	1.54634i
$724$		0			0
$725$	−2.58054			−0.0958388
$726$		0			0
$727$	31.9442			1.18474			0.592372	−	0.805665i	$-0.298192\pi$
$727$	31.9442			1.18474			0.592372	+	0.805665i	$0.298192\pi$
$728$		0			0
$729$	−161.892			−5.99601
$730$		0			0
$731$	40.4157			1.49483
$732$		0			0
$733$	−47.6603			−1.76037			−0.880187	−	0.474627i	$-0.842583\pi$
$733$	−47.6603			−1.76037			−0.880187	+	0.474627i	$0.842583\pi$
$734$		0			0
$735$	−66.0348			−2.43573
$736$		0			0
$737$			27.5064i			1.01321i
$738$		0			0
$739$		−	43.5631i		−	1.60249i	−0.598334	−	0.801247i	$-0.704170\pi$
$739$		−	43.5631i		−	1.60249i	0.598334	−	0.801247i	$-0.295830\pi$
$740$		0			0
$741$	−18.7503			−0.688811
$742$		0			0
$743$	−45.7604			−1.67879			−0.839394	−	0.543524i	$-0.817090\pi$
$743$	−45.7604			−1.67879			−0.839394	+	0.543524i	$0.817090\pi$
$744$		0			0
$745$		−	49.9462i		−	1.82989i
$746$		0			0
$747$		−	61.9584i		−	2.26694i
$748$		0			0
$749$	−6.55811			−0.239628
$750$		0			0
$751$	27.1345			0.990153			0.495076	−	0.868849i	$-0.335140\pi$
$751$	27.1345			0.990153			0.495076	+	0.868849i	$0.335140\pi$
$752$		0			0
$753$	9.69330			0.353244
$754$		0			0
$755$	21.3596			0.777355
$756$		0			0
$757$	26.9147			0.978231			0.489116	−	0.872219i	$-0.337320\pi$
$757$	26.9147			0.978231			0.489116	+	0.872219i	$0.337320\pi$
$758$		0			0
$759$	11.7608			0.426891
$760$		0			0
$761$		−	45.0945i		−	1.63467i	−0.576161	−	0.817337i	$-0.695450\pi$
$761$		−	45.0945i		−	1.63467i	0.576161	−	0.817337i	$-0.304550\pi$
$762$		0			0
$763$		−	3.94565i		−	0.142842i
$764$		0			0
$765$			102.228i			3.69608i
$766$		0			0
$767$	−34.5727			−1.24835
$768$		0			0
$769$		−	42.3652i		−	1.52773i	−0.645377	−	0.763864i	$-0.723300\pi$
$769$		−	42.3652i		−	1.52773i	0.645377	−	0.763864i	$-0.276700\pi$
$770$		0			0
$771$	88.5537			3.18919
$772$		0			0
$773$			33.1017i			1.19058i	0.803509	+	0.595292i	$0.202964\pi$
$773$			33.1017i			1.19058i	−0.803509	+	0.595292i	$0.797036\pi$
$774$		0			0
$775$		−	37.4163i		−	1.34403i
$776$		0			0
$777$		−	15.1565i		−	0.543737i
$778$		0			0
$779$	4.86217			0.174205
$780$		0			0
$781$			25.8037i			0.923331i
$782$		0			0
$783$			14.8333i			0.530097i
$784$		0			0
$785$	64.0942			2.28762
$786$		0			0
$787$		−	31.3530i		−	1.11761i	−0.829298	−	0.558807i	$-0.811259\pi$
$787$		−	31.3530i		−	1.11761i	0.829298	−	0.558807i	$-0.188741\pi$
$788$		0			0
$789$	58.7698			2.09226
$790$		0			0
$791$	0.139980			0.00497712
$792$		0			0
$793$		−	26.2727i		−	0.932969i
$794$		0			0
$795$	−64.1116	−	34.1038i	−2.27380	−	1.20954i
$796$		0			0
$797$			16.7609i			0.593700i	0.954924	+	0.296850i	$0.0959362\pi$
$797$			16.7609i			0.593700i	−0.954924	+	0.296850i	$0.904064\pi$
$798$		0			0
$799$	49.7425			1.75976
$800$		0			0
$801$	−43.8689			−1.55003
$802$		0			0
$803$			14.0837i			0.497004i
$804$		0			0
$805$	1.64081			0.0578308
$806$		0			0
$807$		−	0.729917i		−	0.0256943i
$808$		0			0
$809$			15.9648i			0.561293i	0.959811	+	0.280647i	$0.0905489\pi$
$809$			15.9648i			0.561293i	−0.959811	+	0.280647i	$0.909451\pi$
$810$		0			0
$811$	−7.72895			−0.271400			−0.135700	−	0.990750i	$-0.543328\pi$
$811$	−7.72895			−0.271400			−0.135700	+	0.990750i	$0.543328\pi$
$812$		0			0
$813$			63.6702i			2.23301i
$814$		0			0
$815$		−	25.9933i		−	0.910506i
$816$		0			0
$817$			10.0856i			0.352851i
$818$		0			0
$819$	29.5559			1.03277
$820$		0			0
$821$		−	13.6065i		−	0.474870i	−0.971403	−	0.237435i	$-0.923693\pi$
$821$		−	13.6065i		−	0.474870i	0.971403	−	0.237435i	$-0.0763067\pi$
$822$		0			0
$823$	−19.2420			−0.670732			−0.335366	−	0.942088i	$-0.608860\pi$
$823$	−19.2420			−0.670732			−0.335366	+	0.942088i	$0.608860\pi$
$824$		0			0
$825$			44.2847i			1.54179i
$826$		0			0
$827$			17.8668i			0.621290i	0.950526	+	0.310645i	$0.100545\pi$
$827$			17.8668i			0.621290i	−0.950526	+	0.310645i	$0.899455\pi$
$828$		0			0
$829$		−	37.8022i		−	1.31292i	−0.754359	−	0.656462i	$-0.772052\pi$
$829$		−	37.8022i		−	1.31292i	0.754359	−	0.656462i	$-0.227948\pi$
$830$		0			0
$831$	55.0660			1.91022
$832$		0			0
$833$	26.5286			0.919162
$834$		0			0
$835$	20.0604			0.694219
$836$		0			0
$837$	−215.073			−7.43402
$838$		0			0
$839$	−27.4087			−0.946252			−0.473126	−	0.880995i	$-0.656875\pi$
$839$	−27.4087			−0.946252			−0.473126	+	0.880995i	$0.656875\pi$
$840$		0			0
$841$	−28.4401			−0.980693
$842$		0			0
$843$		−	75.2644i		−	2.59224i
$844$		0			0
$845$			48.9879i			1.68524i
$846$		0			0
$847$	−1.84979			−0.0635597
$848$		0			0
$849$	51.5278			1.76843
$850$		0			0
$851$		−	6.56318i		−	0.224983i
$852$		0			0
$853$			36.1136i			1.23651i	0.785979	+	0.618254i	$0.212160\pi$
$853$			36.1136i			1.23651i	−0.785979	+	0.618254i	$0.787840\pi$
$854$		0			0
$855$	−25.5108			−0.872450
$856$		0			0
$857$	47.1216			1.60964			0.804821	−	0.593518i	$-0.202261\pi$
$857$	47.1216			1.60964			0.804821	+	0.593518i	$0.202261\pi$
$858$		0			0
$859$	−20.6627			−0.705003			−0.352501	−	0.935811i	$-0.614669\pi$
$859$	−20.6627			−0.705003			−0.352501	+	0.935811i	$0.614669\pi$
$860$		0			0
$861$	−10.2839			−0.350475
$862$		0			0
$863$	29.2760			0.996565			0.498283	−	0.867015i	$-0.333964\pi$
$863$	29.2760			0.996565			0.498283	+	0.867015i	$0.333964\pi$
$864$		0			0
$865$	−31.3070			−1.06447
$866$		0			0
$867$			3.23208i			0.109767i
$868$		0			0
$869$		−	33.5961i		−	1.13967i
$870$		0			0
$871$			40.1651i			1.36094i
$872$		0			0
$873$	−133.407			−4.51515
$874$		0			0
$875$		−	2.77908i		−	0.0939502i
$876$		0			0
$877$	−30.9810			−1.04616			−0.523078	−	0.852285i	$-0.675216\pi$
$877$	−30.9810			−1.04616			−0.523078	+	0.852285i	$0.675216\pi$
$878$		0			0
$879$			111.789i			3.77056i
$880$		0			0
$881$			52.4238i			1.76620i	0.469181	+	0.883102i	$0.344549\pi$
$881$			52.4238i			1.76620i	−0.469181	+	0.883102i	$0.655451\pi$
$882$		0			0
$883$			20.3379i			0.684425i	0.939623	+	0.342213i	$0.111176\pi$
$883$			20.3379i			0.684425i	−0.939623	+	0.342213i	$0.888824\pi$
$884$		0			0
$885$	−63.1162			−2.12163
$886$		0			0
$887$		−	27.1731i		−	0.912383i	−0.889882	−	0.456191i	$-0.849213\pi$
$887$		−	27.1731i		−	0.912383i	0.889882	−	0.456191i	$-0.150787\pi$
$888$		0			0
$889$		−	10.3772i		−	0.348041i
$890$		0			0
$891$	156.032			5.22726
$892$		0			0
$893$			12.4131i			0.415388i
$894$		0			0
$895$	−26.7385			−0.893769
$896$		0			0
$897$	17.1733			0.573398
$898$		0			0
$899$			8.11807i			0.270753i
$900$		0			0
$901$	25.7560	+	13.7008i	0.858057	+	0.456439i
$902$		0			0
$903$		−	21.3319i		−	0.709883i
$904$		0			0
$905$	41.5788			1.38213
$906$		0			0
$907$	−16.9024			−0.561236			−0.280618	−	0.959819i	$-0.590539\pi$
$907$	−16.9024			−0.561236			−0.280618	+	0.959819i	$0.590539\pi$
$908$		0			0
$909$			113.447i			3.76279i
$910$		0			0
$911$	−15.5540			−0.515326			−0.257663	−	0.966235i	$-0.582952\pi$
$911$	−15.5540			−0.515326			−0.257663	+	0.966235i	$0.582952\pi$
$912$		0			0
$913$			26.4153i			0.874220i
$914$		0			0
$915$		−	47.9636i		−	1.58563i
$916$		0			0
$917$	−4.41118			−0.145670
$918$		0			0
$919$			0.288503i			0.00951682i	0.999989	+	0.00475841i	$0.00151465\pi$
$919$			0.288503i			0.00951682i	−0.999989	+	0.00475841i	$0.998485\pi$
$920$		0			0
$921$		−	9.27528i		−	0.305631i
$922$		0			0
$923$			37.6788i			1.24021i
$924$		0			0
$925$	24.7133			0.812567
$926$		0			0
$927$		−	28.6674i		−	0.941563i
$928$		0			0
$929$	38.0100			1.24707			0.623534	−	0.781796i	$-0.285696\pi$
$929$	38.0100			1.24707			0.623534	+	0.781796i	$0.285696\pi$
$930$		0			0
$931$			6.62013i			0.216966i
$932$		0			0
$933$			73.4167i			2.40356i
$934$		0			0
$935$		−	43.5840i		−	1.42535i
$936$		0			0
$937$	−37.5545			−1.22685			−0.613426	−	0.789752i	$-0.710209\pi$
$937$	−37.5545			−1.22685			−0.613426	+	0.789752i	$0.710209\pi$
$938$		0			0
$939$	−3.74659			−0.122265
$940$		0			0
$941$	−43.5858			−1.42086			−0.710428	−	0.703769i	$-0.751499\pi$
$941$	−43.5858			−1.42086			−0.710428	+	0.703769i	$0.751499\pi$
$942$		0			0
$943$	−4.45321			−0.145017
$944$		0			0
$945$	35.5140			1.15527
$946$		0			0
$947$	40.0596			1.30176			0.650881	−	0.759180i	$-0.274400\pi$
$947$	40.0596			1.30176			0.650881	+	0.759180i	$0.274400\pi$
$948$		0			0
$949$			20.5652i			0.667575i
$950$		0			0
$951$		−	2.99843i		−	0.0972309i
$952$		0			0
$953$	20.3833			0.660280			0.330140	−	0.943932i	$-0.392904\pi$
$953$	20.3833			0.660280			0.330140	+	0.943932i	$0.392904\pi$
$954$		0			0
$955$	−28.6995			−0.928694
$956$		0			0
$957$		−	9.60828i		−	0.310592i
$958$		0			0
$959$		−	7.53119i		−	0.243195i
$960$		0			0
$961$	−86.7072			−2.79701
$962$		0			0
$963$	−93.3875			−3.00937
$964$		0			0
$965$	18.4247			0.593111
$966$		0			0
$967$	−3.53265			−0.113602			−0.0568012	−	0.998386i	$-0.518090\pi$
$967$	−3.53265			−0.113602			−0.0568012	+	0.998386i	$0.518090\pi$
$968$		0			0
$969$	13.7518			0.441770
$970$		0			0
$971$	34.7658			1.11569			0.557844	−	0.829946i	$-0.311629\pi$
$971$	34.7658			1.11569			0.557844	+	0.829946i	$0.311629\pi$
$972$		0			0
$973$		−	0.0789311i		−	0.00253041i
$974$		0			0
$975$			64.6648i			2.07093i
$976$		0			0
$977$			23.6861i			0.757784i	0.925441	+	0.378892i	$0.123695\pi$
$977$			23.6861i			0.757784i	−0.925441	+	0.378892i	$0.876305\pi$
$978$		0			0
$979$	18.7031			0.597753
$980$		0			0
$981$		−	56.1861i		−	1.79388i
$982$		0			0
$983$	11.7945			0.376188			0.188094	−	0.982151i	$-0.439769\pi$
$983$	11.7945			0.376188			0.188094	+	0.982151i	$0.439769\pi$
$984$		0			0
$985$		−	19.1638i		−	0.610608i
$986$		0			0
$987$		−	26.2548i		−	0.835699i
$988$		0			0
$989$		−	9.23730i		−	0.293729i
$990$		0			0
$991$	42.3562			1.34549			0.672745	−	0.739874i	$-0.265115\pi$
$991$	42.3562			1.34549			0.672745	+	0.739874i	$0.265115\pi$
$992$		0			0
$993$		−	69.6558i		−	2.21046i
$994$		0			0
$995$			63.1238i			2.00116i
$996$		0			0
$997$	19.4379			0.615605			0.307803	−	0.951450i	$-0.400406\pi$
$997$	19.4379			0.615605			0.307803	+	0.951450i	$0.400406\pi$
$998$		0			0
$999$		−	142.055i		−	4.49442i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.c.a.3497.1	✓	82
53.52	even	2	inner	4028.2.c.a.3497.82	yes	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.c.a.3497.1	✓	82	1.1	even	1	trivial
4028.2.c.a.3497.82	yes	82	53.52	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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.43171i q^{3} +2.90667i q^{5} -0.616337 q^{7} -8.77663 q^{9} +O(q^{10})\)	\(q-3.43171i q^{3} +2.90667i q^{5} -0.616337 q^{7} -8.77663 q^{9} +3.74183 q^{11} +5.46385 q^{13} +9.97485 q^{15} -4.00726 q^{17} -1.00000i q^{19} +2.11509i q^{21} +0.915890i q^{23} -3.44873 q^{25} +19.8237i q^{27} +0.748258 q^{29} +10.8493i q^{31} -12.8409i q^{33} -1.79149i q^{35} -7.16590 q^{37} -18.7503i q^{39} +4.86217i q^{41} -10.0856 q^{43} -25.5108i q^{45} -12.4131 q^{47} -6.62013 q^{49} +13.7518i q^{51} +(-6.42733 - 3.41898i) q^{53} +10.8763i q^{55} -3.43171 q^{57} -6.32753 q^{59} -4.80845i q^{61} +5.40936 q^{63} +15.8816i q^{65} +7.35107i q^{67} +3.14307 q^{69} +6.89603i q^{71} +3.76387i q^{73} +11.8350i q^{75} -2.30623 q^{77} -8.97852i q^{79} +41.6994 q^{81} +7.05948i q^{83} -11.6478i q^{85} -2.56780i q^{87} +4.99838 q^{89} -3.36757 q^{91} +37.2316 q^{93} +2.90667 q^{95} +15.2003 q^{97} -32.8406 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) 82 * q - 8 * q^7 - 82 * q^9	\( 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) 82 * q - 8 * q^7 - 82 * q^9 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 58 * q^25 - 16 * q^29 - 12 * q^37 - 32 * q^43 + 8 * q^47 + 98 * q^49 + 6 * q^53 - 4 * q^57 + 4 * q^59 + 8 * q^63 + 28 * q^69 - 8 * q^77 + 154 * q^81 - 20 * q^89 + 48 * q^91 - 56 * q^93 - 44 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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