LMFDB - Embedded newform 4028.2.c.a.3497.2

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			3497.2
Character	$\chi$	$=$	4028.3497
Dual form			4028.2.c.a.3497.81

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32000i q^{3} -0.481923i q^{5} +4.89204 q^{7} -8.02243 q^{9} +O(q^{10})$	$q-3.32000i q^{3} -0.481923i q^{5} +4.89204 q^{7} -8.02243 q^{9} -0.204345 q^{11} +3.56324 q^{13} -1.59998 q^{15} +4.82760 q^{17} +1.00000i q^{19} -16.2416i q^{21} -2.25872i q^{23} +4.76775 q^{25} +16.6745i q^{27} -6.69789 q^{29} +0.0173925i q^{31} +0.678426i q^{33} -2.35759i q^{35} +9.09621 q^{37} -11.8300i q^{39} -3.89009i q^{41} +0.556395 q^{43} +3.86619i q^{45} -0.386900 q^{47} +16.9321 q^{49} -16.0277i q^{51} +(-5.33366 - 4.95500i) q^{53} +0.0984784i q^{55} +3.32000 q^{57} +0.366646 q^{59} +14.3774i q^{61} -39.2461 q^{63} -1.71720i q^{65} -3.34426i q^{67} -7.49895 q^{69} -12.9229i q^{71} -1.21371i q^{73} -15.8290i q^{75} -0.999664 q^{77} -2.26735i q^{79} +31.2921 q^{81} -6.60842i q^{83} -2.32653i q^{85} +22.2370i q^{87} -1.49569 q^{89} +17.4315 q^{91} +0.0577430 q^{93} +0.481923 q^{95} +1.51509 q^{97} +1.63934 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	3.32000i		−	1.91681i	−0.285420	−	0.958403i	$-0.592133\pi$
$3$		−	3.32000i		−	1.91681i	0.285420	−	0.958403i	$-0.407867\pi$
$4$		0			0
$5$		−	0.481923i		−	0.215522i	−0.994177	−	0.107761i	$-0.965632\pi$
$5$		−	0.481923i		−	0.215522i	0.994177	−	0.107761i	$-0.0343682\pi$
$6$		0			0
$7$	4.89204			1.84902			0.924509	−	0.381160i	$-0.124475\pi$
$7$	4.89204			1.84902			0.924509	+	0.381160i	$0.124475\pi$
$8$		0			0
$9$	−8.02243			−2.67414
$10$		0			0
$11$	−0.204345			−0.0616123			−0.0308062	−	0.999525i	$-0.509807\pi$
$11$	−0.204345			−0.0616123			−0.0308062	+	0.999525i	$0.509807\pi$
$12$		0			0
$13$	3.56324			0.988264			0.494132	−	0.869387i	$-0.335486\pi$
$13$	3.56324			0.988264			0.494132	+	0.869387i	$0.335486\pi$
$14$		0			0
$15$	−1.59998			−0.413114
$16$		0			0
$17$	4.82760			1.17087			0.585433	−	0.810721i	$-0.300925\pi$
$17$	4.82760			1.17087			0.585433	+	0.810721i	$0.300925\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		0			0
$21$		−	16.2416i		−	3.54421i
$22$		0			0
$23$		−	2.25872i		−	0.470975i	−0.971877	−	0.235487i	$-0.924331\pi$
$23$		−	2.25872i		−	0.470975i	0.971877	−	0.235487i	$-0.0756687\pi$
$24$		0			0
$25$	4.76775			0.953550
$26$		0			0
$27$			16.6745i			3.20900i
$28$		0			0
$29$	−6.69789			−1.24377			−0.621884	−	0.783110i	$-0.713632\pi$
$29$	−6.69789			−1.24377			−0.621884	+	0.783110i	$0.713632\pi$
$30$		0			0
$31$			0.0173925i			0.00312378i	0.999999	+	0.00156189i	$0.000497165\pi$
$31$			0.0173925i			0.00312378i	−0.999999	+	0.00156189i	$0.999503\pi$
$32$		0			0
$33$			0.678426i			0.118099i
$34$		0			0
$35$		−	2.35759i		−	0.398505i
$36$		0			0
$37$	9.09621			1.49541			0.747704	−	0.664032i	$-0.231156\pi$
$37$	9.09621			1.49541			0.747704	+	0.664032i	$0.231156\pi$
$38$		0			0
$39$		−	11.8300i		−	1.89431i
$40$		0			0
$41$		−	3.89009i		−	0.607531i	−0.952747	−	0.303765i	$-0.901756\pi$
$41$		−	3.89009i		−	0.607531i	0.952747	−	0.303765i	$-0.0982439\pi$
$42$		0			0
$43$	0.556395			0.0848495			0.0424247	−	0.999100i	$-0.486492\pi$
$43$	0.556395			0.0848495			0.0424247	+	0.999100i	$0.486492\pi$
$44$		0			0
$45$			3.86619i			0.576337i
$46$		0			0
$47$	−0.386900			−0.0564352			−0.0282176	−	0.999602i	$-0.508983\pi$
$47$	−0.386900			−0.0564352			−0.0282176	+	0.999602i	$0.508983\pi$
$48$		0			0
$49$	16.9321			2.41887
$50$		0			0
$51$		−	16.0277i		−	2.24432i
$52$		0			0
$53$	−5.33366	−	4.95500i	−0.732635	−	0.680622i
$53$	−5.33366	−	4.95500i	−0.732635	−	0.680622i
$54$		0			0
$55$			0.0984784i			0.0132788i
$56$		0			0
$57$	3.32000			0.439745
$58$		0			0
$59$	0.366646			0.0477332			0.0238666	−	0.999715i	$-0.492402\pi$
$59$	0.366646			0.0477332			0.0238666	+	0.999715i	$0.492402\pi$
$60$		0			0
$61$			14.3774i			1.84083i	0.390941	+	0.920416i	$0.372150\pi$
$61$			14.3774i			1.84083i	−0.390941	+	0.920416i	$0.627850\pi$
$62$		0			0
$63$	−39.2461			−4.94454
$64$		0			0
$65$		−	1.71720i		−	0.212993i
$66$		0			0
$67$		−	3.34426i		−	0.408567i	−0.978912	−	0.204283i	$-0.934514\pi$
$67$		−	3.34426i		−	0.408567i	0.978912	−	0.204283i	$-0.0654864\pi$
$68$		0			0
$69$	−7.49895			−0.902767
$70$		0			0
$71$		−	12.9229i		−	1.53366i	−0.641849	−	0.766831i	$-0.721833\pi$
$71$		−	12.9229i		−	1.53366i	0.641849	−	0.766831i	$-0.278167\pi$
$72$		0			0
$73$		−	1.21371i		−	0.142054i	−0.997474	−	0.0710269i	$-0.977372\pi$
$73$		−	1.21371i		−	0.142054i	0.997474	−	0.0710269i	$-0.0226276\pi$
$74$		0			0
$75$		−	15.8290i		−	1.82777i
$76$		0			0
$77$	−0.999664			−0.113922
$78$		0			0
$79$		−	2.26735i		−	0.255096i	−0.991832	−	0.127548i	$-0.959289\pi$
$79$		−	2.26735i		−	0.255096i	0.991832	−	0.127548i	$-0.0407107\pi$
$80$		0			0
$81$	31.2921			3.47689
$82$		0			0
$83$		−	6.60842i		−	0.725368i	−0.931912	−	0.362684i	$-0.881860\pi$
$83$		−	6.60842i		−	0.725368i	0.931912	−	0.362684i	$-0.118140\pi$
$84$		0			0
$85$		−	2.32653i		−	0.252348i
$86$		0			0
$87$			22.2370i			2.38406i
$88$		0			0
$89$	−1.49569			−0.158542			−0.0792712	−	0.996853i	$-0.525259\pi$
$89$	−1.49569			−0.158542			−0.0792712	+	0.996853i	$0.525259\pi$
$90$		0			0
$91$	17.4315			1.82732
$92$		0			0
$93$	0.0577430			0.00598767
$94$		0			0
$95$	0.481923			0.0494442
$96$		0			0
$97$	1.51509			0.153834			0.0769171	−	0.997037i	$-0.475492\pi$
$97$	1.51509			0.153834			0.0769171	+	0.997037i	$0.475492\pi$
$98$		0			0
$99$	1.63934			0.164760
$100$		0			0
$101$			5.37424i			0.534757i	0.963592	+	0.267379i	$0.0861574\pi$
$101$			5.37424i			0.534757i	−0.963592	+	0.267379i	$0.913843\pi$
$102$		0			0
$103$		−	3.50727i		−	0.345582i	−0.984959	−	0.172791i	$-0.944722\pi$
$103$		−	3.50727i		−	0.345582i	0.984959	−	0.172791i	$-0.0552784\pi$
$104$		0			0
$105$	−7.82719			−0.763856
$106$		0			0
$107$	7.51940			0.726928			0.363464	−	0.931608i	$-0.381594\pi$
$107$	7.51940			0.726928			0.363464	+	0.931608i	$0.381594\pi$
$108$		0			0
$109$		−	7.24766i		−	0.694200i	−0.937828	−	0.347100i	$-0.887166\pi$
$109$		−	7.24766i		−	0.694200i	0.937828	−	0.347100i	$-0.112834\pi$
$110$		0			0
$111$		−	30.1994i		−	2.86640i
$112$		0			0
$113$	−14.8317			−1.39525			−0.697625	−	0.716463i	$-0.745760\pi$
$113$	−14.8317			−1.39525			−0.697625	+	0.716463i	$0.745760\pi$
$114$		0			0
$115$	−1.08853			−0.101506
$116$		0			0
$117$	−28.5858			−2.64276
$118$		0			0
$119$	23.6168			2.16495
$120$		0			0
$121$	−10.9582			−0.996204
$122$		0			0
$123$	−12.9151			−1.16452
$124$		0			0
$125$		−	4.70730i		−	0.421034i
$126$		0			0
$127$		−	12.5687i		−	1.11529i	−0.830078	−	0.557647i	$-0.811704\pi$
$127$		−	12.5687i		−	1.11529i	0.830078	−	0.557647i	$-0.188296\pi$
$128$		0			0
$129$		−	1.84723i		−	0.162640i
$130$		0			0
$131$	17.8953			1.56352			0.781759	−	0.623581i	$-0.214323\pi$
$131$	17.8953			1.56352			0.781759	+	0.623581i	$0.214323\pi$
$132$		0			0
$133$			4.89204i			0.424194i
$134$		0			0
$135$	8.03581			0.691612
$136$		0			0
$137$		−	8.40857i		−	0.718393i	−0.933262	−	0.359196i	$-0.883051\pi$
$137$		−	8.40857i		−	0.718393i	0.933262	−	0.359196i	$-0.116949\pi$
$138$		0			0
$139$		−	14.1730i		−	1.20214i	−0.799196	−	0.601071i	$-0.794741\pi$
$139$		−	14.1730i		−	1.20214i	0.799196	−	0.601071i	$-0.205259\pi$
$140$		0			0
$141$			1.28451i			0.108175i
$142$		0			0
$143$	−0.728129			−0.0608892
$144$		0			0
$145$			3.22786i			0.268060i
$146$		0			0
$147$		−	56.2146i		−	4.63650i
$148$		0			0
$149$	−0.0572233			−0.00468791			−0.00234396	−	0.999997i	$-0.500746\pi$
$149$	−0.0572233			−0.00468791			−0.00234396	+	0.999997i	$0.500746\pi$
$150$		0			0
$151$			14.1781i			1.15380i	0.816815	+	0.576899i	$0.195738\pi$
$151$			14.1781i			1.15380i	−0.816815	+	0.576899i	$0.804262\pi$
$152$		0			0
$153$	−38.7291			−3.13106
$154$		0			0
$155$	0.00838182			0.000673244
$156$		0			0
$157$			12.2977i			0.981463i	0.871311	+	0.490731i	$0.163270\pi$
$157$			12.2977i			0.981463i	−0.871311	+	0.490731i	$0.836730\pi$
$158$		0			0
$159$	−16.4506	+	17.7078i	−1.30462	+	1.40432i
$160$		0			0
$161$		−	11.0497i		−	0.870841i
$162$		0			0
$163$	−12.4489			−0.975071			−0.487535	−	0.873103i	$-0.662104\pi$
$163$	−12.4489			−0.975071			−0.487535	+	0.873103i	$0.662104\pi$
$164$		0			0
$165$	0.326949			0.0254529
$166$		0			0
$167$			7.48575i			0.579264i	0.957138	+	0.289632i	$0.0935330\pi$
$167$			7.48575i			0.579264i	−0.957138	+	0.289632i	$0.906467\pi$
$168$		0			0
$169$	−0.303353			−0.0233349
$170$		0			0
$171$		−	8.02243i		−	0.613490i
$172$		0			0
$173$			13.5501i			1.03020i	0.857132	+	0.515098i	$0.172244\pi$
$173$			13.5501i			1.03020i	−0.857132	+	0.515098i	$0.827756\pi$
$174$		0			0
$175$	23.3240			1.76313
$176$		0			0
$177$		−	1.21727i		−	0.0914953i
$178$		0			0
$179$			22.2290i			1.66147i	0.556664	+	0.830737i	$0.312081\pi$
$179$			22.2290i			1.66147i	−0.556664	+	0.830737i	$0.687919\pi$
$180$		0			0
$181$			20.2380i			1.50428i	0.659003	+	0.752140i	$0.270978\pi$
$181$			20.2380i			1.50428i	−0.659003	+	0.752140i	$0.729022\pi$
$182$		0			0
$183$	47.7329			3.52851
$184$		0			0
$185$		−	4.38367i		−	0.322294i
$186$		0			0
$187$	−0.986496			−0.0721397
$188$		0			0
$189$			81.5723i			5.93351i
$190$		0			0
$191$		−	7.83484i		−	0.566909i	−0.958986	−	0.283455i	$-0.908520\pi$
$191$		−	7.83484i		−	0.566909i	0.958986	−	0.283455i	$-0.0914805\pi$
$192$		0			0
$193$			23.2680i			1.67487i	0.546541	+	0.837433i	$0.315944\pi$
$193$			23.2680i			1.67487i	−0.546541	+	0.837433i	$0.684056\pi$
$194$		0			0
$195$	−5.70112			−0.408266
$196$		0			0
$197$	23.5454			1.67754			0.838771	−	0.544484i	$-0.183275\pi$
$197$	23.5454			1.67754			0.838771	+	0.544484i	$0.183275\pi$
$198$		0			0
$199$	2.80811			0.199062			0.0995308	−	0.995034i	$-0.468266\pi$
$199$	2.80811			0.199062			0.0995308	+	0.995034i	$0.468266\pi$
$200$		0			0
$201$	−11.1030			−0.783143
$202$		0			0
$203$	−32.7664			−2.29975
$204$		0			0
$205$	−1.87472			−0.130936
$206$		0			0
$207$			18.1204i			1.25945i
$208$		0			0
$209$		−	0.204345i		−	0.0141348i
$210$		0			0
$211$	−20.3697			−1.40231			−0.701153	−	0.713011i	$-0.747331\pi$
$211$	−20.3697			−1.40231			−0.701153	+	0.713011i	$0.747331\pi$
$212$		0			0
$213$	−42.9040			−2.93973
$214$		0			0
$215$		−	0.268139i		−	0.0182870i
$216$		0			0
$217$			0.0850846i			0.00577592i
$218$		0			0
$219$	−4.02951			−0.272289
$220$		0			0
$221$	17.2019			1.15712
$222$		0			0
$223$	−22.4219			−1.50148			−0.750742	−	0.660596i	$-0.770304\pi$
$223$	−22.4219			−1.50148			−0.750742	+	0.660596i	$0.770304\pi$
$224$		0			0
$225$	−38.2489			−2.54993
$226$		0			0
$227$	−26.0784			−1.73088			−0.865440	−	0.501012i	$-0.832961\pi$
$227$	−26.0784			−1.73088			−0.865440	+	0.501012i	$0.832961\pi$
$228$		0			0
$229$	3.95104			0.261092			0.130546	−	0.991442i	$-0.458327\pi$
$229$	3.95104			0.261092			0.130546	+	0.991442i	$0.458327\pi$
$230$		0			0
$231$			3.31889i			0.218367i
$232$		0			0
$233$		−	19.8178i		−	1.29830i	−0.760659	−	0.649152i	$-0.775124\pi$
$233$		−	19.8178i		−	1.29830i	0.760659	−	0.649152i	$-0.224876\pi$
$234$		0			0
$235$			0.186456i			0.0121630i
$236$		0			0
$237$	−7.52760			−0.488970
$238$		0			0
$239$			19.0307i			1.23100i	0.788138	+	0.615498i	$0.211045\pi$
$239$			19.0307i			1.23100i	−0.788138	+	0.615498i	$0.788955\pi$
$240$		0			0
$241$	15.1475			0.975733			0.487867	−	0.872918i	$-0.337775\pi$
$241$	15.1475			0.975733			0.487867	+	0.872918i	$0.337775\pi$
$242$		0			0
$243$		−	53.8663i		−	3.45552i
$244$		0			0
$245$		−	8.15995i		−	0.521320i
$246$		0			0
$247$			3.56324i			0.226723i
$248$		0			0
$249$	−21.9400			−1.39039
$250$		0			0
$251$			12.6089i			0.795867i	0.917414	+	0.397934i	$0.130273\pi$
$251$			12.6089i			0.795867i	−0.917414	+	0.397934i	$0.869727\pi$
$252$		0			0
$253$			0.461557i			0.0290179i
$254$		0			0
$255$	−7.72409			−0.483701
$256$		0			0
$257$			22.4751i			1.40196i	0.713182	+	0.700979i	$0.247253\pi$
$257$			22.4751i			1.40196i	−0.713182	+	0.700979i	$0.752747\pi$
$258$		0			0
$259$	44.4990			2.76504
$260$		0			0
$261$	53.7333			3.32601
$262$		0			0
$263$		−	2.58848i		−	0.159612i	−0.996810	−	0.0798061i	$-0.974570\pi$
$263$		−	2.58848i		−	0.159612i	0.996810	−	0.0798061i	$-0.0254301\pi$
$264$		0			0
$265$	−2.38793	+	2.57041i	−0.146689	+	0.157899i
$266$		0			0
$267$			4.96569i			0.303895i
$268$		0			0
$269$	−19.2163			−1.17164			−0.585821	−	0.810441i	$-0.699228\pi$
$269$	−19.2163			−1.17164			−0.585821	+	0.810441i	$0.699228\pi$
$270$		0			0
$271$	9.96655			0.605425			0.302712	−	0.953082i	$-0.402108\pi$
$271$	9.96655			0.605425			0.302712	+	0.953082i	$0.402108\pi$
$272$		0			0
$273$		−	57.8726i		−	3.50261i
$274$		0			0
$275$	−0.974266			−0.0587504
$276$		0			0
$277$		−	10.2927i		−	0.618426i	−0.950993	−	0.309213i	$-0.899934\pi$
$277$		−	10.2927i		−	0.618426i	0.950993	−	0.309213i	$-0.100066\pi$
$278$		0			0
$279$		−	0.139530i		−	0.00835342i
$280$		0			0
$281$	15.3246			0.914191			0.457095	−	0.889418i	$-0.348890\pi$
$281$	15.3246			0.914191			0.457095	+	0.889418i	$0.348890\pi$
$282$		0			0
$283$			28.4646i			1.69205i	0.533147	+	0.846023i	$0.321009\pi$
$283$			28.4646i			1.69205i	−0.533147	+	0.846023i	$0.678991\pi$
$284$		0			0
$285$		−	1.59998i		−	0.0947749i
$286$		0			0
$287$		−	19.0305i		−	1.12334i
$288$		0			0
$289$	6.30573			0.370925
$290$		0			0
$291$		−	5.03011i		−	0.294870i
$292$		0			0
$293$	−19.1097			−1.11640			−0.558201	−	0.829706i	$-0.688508\pi$
$293$	−19.1097			−1.11640			−0.558201	+	0.829706i	$0.688508\pi$
$294$		0			0
$295$		−	0.176695i		−	0.0102876i
$296$		0			0
$297$		−	3.40734i		−	0.197714i
$298$		0			0
$299$		−	8.04834i		−	0.465447i
$300$		0			0
$301$	2.72191			0.156888
$302$		0			0
$303$	17.8425			1.02503
$304$		0			0
$305$	6.92877			0.396740
$306$		0			0
$307$	−30.4086			−1.73551			−0.867756	−	0.496990i	$-0.834439\pi$
$307$	−30.4086			−1.73551			−0.867756	+	0.496990i	$0.834439\pi$
$308$		0			0
$309$	−11.6441			−0.662412
$310$		0			0
$311$	−5.86356			−0.332492			−0.166246	−	0.986084i	$-0.553165\pi$
$311$	−5.86356			−0.332492			−0.166246	+	0.986084i	$0.553165\pi$
$312$		0			0
$313$		−	16.2934i		−	0.920957i	−0.887671	−	0.460479i	$-0.847678\pi$
$313$		−	16.2934i		−	0.920957i	0.887671	−	0.460479i	$-0.152322\pi$
$314$		0			0
$315$			18.9136i			1.06566i
$316$		0			0
$317$	−17.6055			−0.988822			−0.494411	−	0.869228i	$-0.664616\pi$
$317$	−17.6055			−0.988822			−0.494411	+	0.869228i	$0.664616\pi$
$318$		0			0
$319$	1.36868			0.0766314
$320$		0			0
$321$		−	24.9644i		−	1.39338i
$322$		0			0
$323$			4.82760i			0.268615i
$324$		0			0
$325$	16.9886			0.942359
$326$		0			0
$327$	−24.0623			−1.33065
$328$		0			0
$329$	−1.89273			−0.104350
$330$		0			0
$331$	5.69033			0.312769			0.156384	−	0.987696i	$-0.450016\pi$
$331$	5.69033			0.312769			0.156384	+	0.987696i	$0.450016\pi$
$332$		0			0
$333$	−72.9737			−3.99893
$334$		0			0
$335$	−1.61168			−0.0880553
$336$		0			0
$337$		−	26.8673i		−	1.46355i	−0.681545	−	0.731776i	$-0.738692\pi$
$337$		−	26.8673i		−	1.46355i	0.681545	−	0.731776i	$-0.261308\pi$
$338$		0			0
$339$			49.2414i			2.67442i
$340$		0			0
$341$		−	0.00355406i		−	0.000192463i
$342$		0			0
$343$	48.5882			2.62351
$344$		0			0
$345$			3.61391i			0.194566i
$346$		0			0
$347$	13.2278			0.710105			0.355053	−	0.934846i	$-0.384463\pi$
$347$	13.2278			0.710105			0.355053	+	0.934846i	$0.384463\pi$
$348$		0			0
$349$			19.0175i			1.01798i	0.860772	+	0.508992i	$0.169982\pi$
$349$			19.0175i			1.01798i	−0.860772	+	0.508992i	$0.830018\pi$
$350$		0			0
$351$			59.4151i			3.17134i
$352$		0			0
$353$		−	12.2578i		−	0.652419i	−0.945298	−	0.326209i	$-0.894229\pi$
$353$		−	12.2578i		−	0.652419i	0.945298	−	0.326209i	$-0.105771\pi$
$354$		0			0
$355$	−6.22782			−0.330538
$356$		0			0
$357$		−	78.4080i		−	4.14979i
$358$		0			0
$359$		−	26.3599i		−	1.39122i	−0.718418	−	0.695612i	$-0.755133\pi$
$359$		−	26.3599i		−	1.39122i	0.718418	−	0.695612i	$-0.244867\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$			36.3814i			1.90953i
$364$		0			0
$365$	−0.584913			−0.0306158
$366$		0			0
$367$	−12.9561			−0.676302			−0.338151	−	0.941092i	$-0.609801\pi$
$367$	−12.9561			−0.676302			−0.338151	+	0.941092i	$0.609801\pi$
$368$		0			0
$369$			31.2080i			1.62462i
$370$		0			0
$371$	−26.0925	−	24.2401i	−1.35465	−	1.25848i
$372$		0			0
$373$		−	0.993397i		−	0.0514362i	−0.999669	−	0.0257181i	$-0.991813\pi$
$373$		−	0.993397i		−	0.0514362i	0.999669	−	0.0257181i	$-0.00818722\pi$
$374$		0			0
$375$	−15.6283			−0.807040
$376$		0			0
$377$	−23.8662			−1.22917
$378$		0			0
$379$		−	18.9877i		−	0.975331i	−0.873031	−	0.487665i	$-0.837849\pi$
$379$		−	18.9877i		−	0.975331i	0.873031	−	0.487665i	$-0.162151\pi$
$380$		0			0
$381$	−41.7282			−2.13780
$382$		0			0
$383$		−	32.2547i		−	1.64814i	−0.566491	−	0.824068i	$-0.691699\pi$
$383$		−	32.2547i		−	1.64814i	0.566491	−	0.824068i	$-0.308301\pi$
$384$		0			0
$385$			0.481761i			0.0245528i
$386$		0			0
$387$	−4.46364			−0.226900
$388$		0			0
$389$			26.8003i			1.35883i	0.733754	+	0.679416i	$0.237767\pi$
$389$			26.8003i			1.35883i	−0.733754	+	0.679416i	$0.762233\pi$
$390$		0			0
$391$		−	10.9042i		−	0.551448i
$392$		0			0
$393$		−	59.4124i		−	2.99696i
$394$		0			0
$395$	−1.09268			−0.0549789
$396$		0			0
$397$		−	17.2160i		−	0.864047i	−0.901862	−	0.432023i	$-0.857800\pi$
$397$		−	17.2160i		−	0.864047i	0.901862	−	0.432023i	$-0.142200\pi$
$398$		0			0
$399$	16.2416			0.813097
$400$		0			0
$401$			22.4871i			1.12295i	0.827493	+	0.561475i	$0.189766\pi$
$401$			22.4871i			1.12295i	−0.827493	+	0.561475i	$0.810234\pi$
$402$		0			0
$403$			0.0619734i			0.00308712i
$404$		0			0
$405$		−	15.0803i		−	0.749348i
$406$		0			0
$407$	−1.85876			−0.0921355
$408$		0			0
$409$	11.8108			0.584009			0.292004	−	0.956417i	$-0.405678\pi$
$409$	11.8108			0.584009			0.292004	+	0.956417i	$0.405678\pi$
$410$		0			0
$411$	−27.9165			−1.37702
$412$		0			0
$413$	1.79365			0.0882596
$414$		0			0
$415$	−3.18474			−0.156333
$416$		0			0
$417$	−47.0545			−2.30427
$418$		0			0
$419$		−	5.40591i		−	0.264096i	−0.991243	−	0.132048i	$-0.957845\pi$
$419$		−	5.40591i		−	0.264096i	0.991243	−	0.132048i	$-0.0421553\pi$
$420$		0			0
$421$			19.6155i			0.956002i	0.878359	+	0.478001i	$0.158638\pi$
$421$			19.6155i			0.956002i	−0.878359	+	0.478001i	$0.841362\pi$
$422$		0			0
$423$	3.10388			0.150916
$424$		0			0
$425$	23.0168			1.11648
$426$		0			0
$427$			70.3346i			3.40373i
$428$		0			0
$429$			2.41739i			0.116713i
$430$		0			0
$431$	32.7750			1.57871			0.789357	−	0.613934i	$-0.210414\pi$
$431$	32.7750			1.57871			0.789357	+	0.613934i	$0.210414\pi$
$432$		0			0
$433$	−13.7033			−0.658541			−0.329270	−	0.944236i	$-0.606803\pi$
$433$	−13.7033			−0.658541			−0.329270	+	0.944236i	$0.606803\pi$
$434$		0			0
$435$	10.7165			0.513818
$436$		0			0
$437$	2.25872			0.108049
$438$		0			0
$439$	11.9191			0.568866			0.284433	−	0.958696i	$-0.408195\pi$
$439$	11.9191			0.568866			0.284433	+	0.958696i	$0.408195\pi$
$440$		0			0
$441$	−135.836			−6.46840
$442$		0			0
$443$			20.6516i			0.981186i	0.871389	+	0.490593i	$0.163220\pi$
$443$			20.6516i			0.981186i	−0.871389	+	0.490593i	$0.836780\pi$
$444$		0			0
$445$			0.720805i			0.0341694i
$446$		0			0
$447$			0.189982i			0.00898582i
$448$		0			0
$449$	−20.6114			−0.972714			−0.486357	−	0.873760i	$-0.661674\pi$
$449$	−20.6114			−0.972714			−0.486357	+	0.873760i	$0.661674\pi$
$450$		0			0
$451$			0.794921i			0.0374314i
$452$		0			0
$453$	47.0714			2.21161
$454$		0			0
$455$		−	8.40063i		−	0.393828i
$456$		0			0
$457$			6.51590i			0.304801i	0.988319	+	0.152400i	$0.0487003\pi$
$457$			6.51590i			0.304801i	−0.988319	+	0.152400i	$0.951300\pi$
$458$		0			0
$459$			80.4977i			3.75731i
$460$		0			0
$461$	1.77610			0.0827213			0.0413606	−	0.999144i	$-0.486831\pi$
$461$	1.77610			0.0827213			0.0413606	+	0.999144i	$0.486831\pi$
$462$		0			0
$463$			12.9749i			0.602994i	0.953467	+	0.301497i	$0.0974864\pi$
$463$			12.9749i			0.602994i	−0.953467	+	0.301497i	$0.902514\pi$
$464$		0			0
$465$		−	0.0278277i		−	0.00129048i
$466$		0			0
$467$	0.0771129			0.00356836			0.00178418	−	0.999998i	$-0.499432\pi$
$467$	0.0771129			0.00356836			0.00178418	+	0.999998i	$0.499432\pi$
$468$		0			0
$469$		−	16.3603i		−	0.755448i
$470$		0			0
$471$	40.8284			1.88127
$472$		0			0
$473$	−0.113697			−0.00522777
$474$		0			0
$475$			4.76775i			0.218759i
$476$		0			0
$477$	42.7889	+	39.7512i	1.95917	+	1.82008i
$478$		0			0
$479$			31.8414i			1.45487i	0.686176	+	0.727436i	$0.259288\pi$
$479$			31.8414i			1.45487i	−0.686176	+	0.727436i	$0.740712\pi$
$480$		0			0
$481$	32.4119			1.47786
$482$		0			0
$483$	−36.6852			−1.66923
$484$		0			0
$485$		−	0.730157i		−	0.0331547i
$486$		0			0
$487$	−1.04935			−0.0475507			−0.0237754	−	0.999717i	$-0.507569\pi$
$487$	−1.04935			−0.0475507			−0.0237754	+	0.999717i	$0.507569\pi$
$488$		0			0
$489$			41.3303i			1.86902i
$490$		0			0
$491$		−	33.1403i		−	1.49560i	−0.663925	−	0.747800i	$-0.731110\pi$
$491$		−	33.1403i		−	1.49560i	0.663925	−	0.747800i	$-0.268890\pi$
$492$		0			0
$493$	−32.3347			−1.45628
$494$		0			0
$495$		−	0.790036i		−	0.0355095i
$496$		0			0
$497$		−	63.2192i		−	2.83577i
$498$		0			0
$499$			0.549409i			0.0245949i	0.999924	+	0.0122975i	$0.00391450\pi$
$499$			0.549409i			0.0245949i	−0.999924	+	0.0122975i	$0.996086\pi$
$500$		0			0
$501$	24.8527			1.11034
$502$		0			0
$503$		−	12.1401i		−	0.541299i	−0.962678	−	0.270649i	$-0.912762\pi$
$503$		−	12.1401i		−	0.541299i	0.962678	−	0.270649i	$-0.0872384\pi$
$504$		0			0
$505$	2.58997			0.115252
$506$		0			0
$507$			1.00713i			0.0447284i
$508$		0			0
$509$		−	31.1182i		−	1.37929i	−0.724148	−	0.689645i	$-0.757767\pi$
$509$		−	31.1182i		−	1.37929i	0.724148	−	0.689645i	$-0.242233\pi$
$510$		0			0
$511$		−	5.93751i		−	0.262660i
$512$		0			0
$513$	−16.6745			−0.736196
$514$		0			0
$515$	−1.69023			−0.0744805
$516$		0			0
$517$	0.0790611			0.00347710
$518$		0			0
$519$	44.9864			1.97468
$520$		0			0
$521$	−0.472069			−0.0206817			−0.0103409	−	0.999947i	$-0.503292\pi$
$521$	−0.472069			−0.0206817			−0.0103409	+	0.999947i	$0.503292\pi$
$522$		0			0
$523$	18.5962			0.813153			0.406577	−	0.913617i	$-0.366722\pi$
$523$	18.5962			0.813153			0.406577	+	0.913617i	$0.366722\pi$
$524$		0			0
$525$		−	77.4359i		−	3.37958i
$526$		0			0
$527$			0.0839638i			0.00365752i
$528$		0			0
$529$	17.8982			0.778183
$530$		0			0
$531$	−2.94139			−0.127645
$532$		0			0
$533$		−	13.8613i		−	0.600400i
$534$		0			0
$535$		−	3.62377i		−	0.156669i
$536$		0			0
$537$	73.8004			3.18472
$538$		0			0
$539$	−3.45998			−0.149032
$540$		0			0
$541$	−28.0106			−1.20427			−0.602134	−	0.798395i	$-0.705683\pi$
$541$	−28.0106			−1.20427			−0.602134	+	0.798395i	$0.705683\pi$
$542$		0			0
$543$	67.1903			2.88341
$544$		0			0
$545$	−3.49281			−0.149616
$546$		0			0
$547$	16.1720			0.691463			0.345731	−	0.938334i	$-0.387631\pi$
$547$	16.1720			0.691463			0.345731	+	0.938334i	$0.387631\pi$
$548$		0			0
$549$		−	115.341i		−	4.92264i
$550$		0			0
$551$		−	6.69789i		−	0.285340i
$552$		0			0
$553$		−	11.0919i		−	0.471678i
$554$		0			0
$555$	−14.5538			−0.617774
$556$		0			0
$557$			42.6243i			1.80605i	0.429589	+	0.903025i	$0.358658\pi$
$557$			42.6243i			1.80605i	−0.429589	+	0.903025i	$0.641342\pi$
$558$		0			0
$559$	1.98257			0.0838537
$560$		0			0
$561$			3.27517i			0.138278i
$562$		0			0
$563$			33.9603i			1.43126i	0.698482	+	0.715628i	$0.253859\pi$
$563$			33.9603i			1.43126i	−0.698482	+	0.715628i	$0.746141\pi$
$564$		0			0
$565$			7.14774i			0.300708i
$566$		0			0
$567$	153.082			6.42884
$568$		0			0
$569$			9.27746i			0.388931i	0.980909	+	0.194466i	$0.0622973\pi$
$569$			9.27746i			0.388931i	−0.980909	+	0.194466i	$0.937703\pi$
$570$		0			0
$571$			27.5645i			1.15354i	0.816907	+	0.576770i	$0.195687\pi$
$571$			27.5645i			1.15354i	−0.816907	+	0.576770i	$0.804313\pi$
$572$		0			0
$573$	−26.0117			−1.08665
$574$		0			0
$575$		−	10.7690i		−	0.449098i
$576$		0			0
$577$	−22.5801			−0.940020			−0.470010	−	0.882661i	$-0.655750\pi$
$577$	−22.5801			−0.940020			−0.470010	+	0.882661i	$0.655750\pi$
$578$		0			0
$579$	77.2498			3.21039
$580$		0			0
$581$		−	32.3287i		−	1.34122i
$582$		0			0
$583$	1.08991	+	1.01253i	0.0451393	+	0.0419347i
$584$		0			0
$585$			13.7761i			0.569573i
$586$		0			0
$587$	3.03165			0.125130			0.0625649	−	0.998041i	$-0.480072\pi$
$587$	3.03165			0.125130			0.0625649	+	0.998041i	$0.480072\pi$
$588$		0			0
$589$	−0.0173925			−0.000716644
$590$		0			0
$591$		−	78.1709i		−	3.21552i
$592$		0			0
$593$	−22.6411			−0.929760			−0.464880	−	0.885374i	$-0.653903\pi$
$593$	−22.6411			−0.929760			−0.464880	+	0.885374i	$0.653903\pi$
$594$		0			0
$595$		−	11.3815i		−	0.466595i
$596$		0			0
$597$		−	9.32293i		−	0.381562i
$598$		0			0
$599$	8.20625			0.335298			0.167649	−	0.985847i	$-0.446382\pi$
$599$	8.20625			0.335298			0.167649	+	0.985847i	$0.446382\pi$
$600$		0			0
$601$		−	10.8973i		−	0.444510i	−0.974989	−	0.222255i	$-0.928658\pi$
$601$		−	10.8973i		−	0.444510i	0.974989	−	0.222255i	$-0.0713418\pi$
$602$		0			0
$603$			26.8291i			1.09257i
$604$		0			0
$605$			5.28103i			0.214704i
$606$		0			0
$607$	44.0379			1.78744			0.893721	−	0.448623i	$-0.148086\pi$
$607$	44.0379			1.78744			0.893721	+	0.448623i	$0.148086\pi$
$608$		0			0
$609$			108.784i			4.40817i
$610$		0			0
$611$	−1.37862			−0.0557729
$612$		0			0
$613$			12.1406i			0.490356i	0.969478	+	0.245178i	$0.0788463\pi$
$613$			12.1406i			0.490356i	−0.969478	+	0.245178i	$0.921154\pi$
$614$		0			0
$615$			6.22409i			0.250980i
$616$		0			0
$617$		−	41.6830i		−	1.67809i	−0.544058	−	0.839047i	$-0.683113\pi$
$617$		−	41.6830i		−	1.67809i	0.544058	−	0.839047i	$-0.316887\pi$
$618$		0			0
$619$	18.9086			0.760001			0.380001	−	0.924986i	$-0.375924\pi$
$619$	18.9086			0.760001			0.380001	+	0.924986i	$0.375924\pi$
$620$		0			0
$621$	37.6629			1.51136
$622$		0			0
$623$	−7.31696			−0.293148
$624$		0			0
$625$	21.5702			0.862808
$626$		0			0
$627$	−0.678426			−0.0270937
$628$		0			0
$629$	43.9129			1.75092
$630$		0			0
$631$		−	25.3308i		−	1.00840i	−0.863587	−	0.504201i	$-0.831787\pi$
$631$		−	25.3308i		−	1.00840i	0.863587	−	0.504201i	$-0.168213\pi$
$632$		0			0
$633$			67.6274i			2.68795i
$634$		0			0
$635$	−6.05715			−0.240371
$636$		0			0
$637$	60.3330			2.39048
$638$		0			0
$639$			103.673i			4.10123i
$640$		0			0
$641$		−	14.7623i		−	0.583076i	−0.956559	−	0.291538i	$-0.905833\pi$
$641$		−	14.7623i		−	0.583076i	0.956559	−	0.291538i	$-0.0941671\pi$
$642$		0			0
$643$	13.3396			0.526063			0.263032	−	0.964787i	$-0.415278\pi$
$643$	13.3396			0.526063			0.263032	+	0.964787i	$0.415278\pi$
$644$		0			0
$645$	−0.890224			−0.0350525
$646$		0			0
$647$	20.5818			0.809153			0.404576	−	0.914504i	$-0.367419\pi$
$647$	20.5818			0.809153			0.404576	+	0.914504i	$0.367419\pi$
$648$		0			0
$649$	−0.0749222			−0.00294095
$650$		0			0
$651$	0.282481			0.0110713
$652$		0			0
$653$	4.01286			0.157035			0.0785177	−	0.996913i	$-0.474981\pi$
$653$	4.01286			0.157035			0.0785177	+	0.996913i	$0.474981\pi$
$654$		0			0
$655$		−	8.62414i		−	0.336973i
$656$		0			0
$657$			9.73688i			0.379872i
$658$		0			0
$659$		−	36.7960i		−	1.43337i	−0.697398	−	0.716684i	$-0.745659\pi$
$659$		−	36.7960i		−	1.43337i	0.697398	−	0.716684i	$-0.254341\pi$
$660$		0			0
$661$	−22.3647			−0.869886			−0.434943	−	0.900458i	$-0.643232\pi$
$661$	−22.3647			−0.869886			−0.434943	+	0.900458i	$0.643232\pi$
$662$		0			0
$663$		−	57.1103i		−	2.21798i
$664$		0			0
$665$	2.35759			0.0914233
$666$		0			0
$667$			15.1286i			0.585783i
$668$		0			0
$669$			74.4409i			2.87805i
$670$		0			0
$671$		−	2.93794i		−	0.113418i
$672$		0			0
$673$	−42.5273			−1.63931			−0.819654	−	0.572859i	$-0.805834\pi$
$673$	−42.5273			−1.63931			−0.819654	+	0.572859i	$0.805834\pi$
$674$		0			0
$675$			79.4997i			3.05995i
$676$		0			0
$677$		−	13.2041i		−	0.507473i	−0.967273	−	0.253737i	$-0.918340\pi$
$677$		−	13.2041i		−	0.507473i	0.967273	−	0.253737i	$-0.0816596\pi$
$678$		0			0
$679$	7.41189			0.284442
$680$		0			0
$681$			86.5802i			3.31776i
$682$		0			0
$683$	7.87903			0.301483			0.150741	−	0.988573i	$-0.451834\pi$
$683$	7.87903			0.301483			0.150741	+	0.988573i	$0.451834\pi$
$684$		0			0
$685$	−4.05228			−0.154830
$686$		0			0
$687$		−	13.1175i		−	0.500463i
$688$		0			0
$689$	−19.0051	−	17.6558i	−0.724036	−	0.672634i
$690$		0			0
$691$			30.1111i			1.14548i	0.819737	+	0.572740i	$0.194119\pi$
$691$			30.1111i			1.14548i	−0.819737	+	0.572740i	$0.805881\pi$
$692$		0			0
$693$	8.01973			0.304644
$694$		0			0
$695$	−6.83031			−0.259088
$696$		0			0
$697$		−	18.7798i		−	0.711336i
$698$		0			0
$699$	−65.7950			−2.48860
$700$		0			0
$701$			48.1945i			1.82028i	0.414301	+	0.910140i	$0.364026\pi$
$701$			48.1945i			1.82028i	−0.414301	+	0.910140i	$0.635974\pi$
$702$		0			0
$703$			9.09621i			0.343070i
$704$		0			0
$705$	0.619035			0.0233142
$706$		0			0
$707$			26.2910i			0.988776i
$708$		0			0
$709$			34.8028i			1.30705i	0.756907	+	0.653523i	$0.226710\pi$
$709$			34.8028i			1.30705i	−0.756907	+	0.653523i	$0.773290\pi$
$710$		0			0
$711$			18.1896i			0.682164i
$712$		0			0
$713$	0.0392846			0.00147122
$714$		0			0
$715$			0.350902i			0.0131230i
$716$		0			0
$717$	63.1822			2.35958
$718$		0			0
$719$			31.5187i			1.17545i	0.809061	+	0.587725i	$0.199976\pi$
$719$			31.5187i			1.17545i	−0.809061	+	0.587725i	$0.800024\pi$
$720$		0			0
$721$		−	17.1577i		−	0.638987i
$722$		0			0
$723$		−	50.2896i		−	1.87029i
$724$		0			0
$725$	−31.9339			−1.18599
$726$		0			0
$727$	41.0941			1.52410			0.762048	−	0.647521i	$-0.224194\pi$
$727$	41.0941			1.52410			0.762048	+	0.647521i	$0.224194\pi$
$728$		0			0
$729$	−84.9602			−3.14667
$730$		0			0
$731$	2.68605			0.0993473
$732$		0			0
$733$	−41.1422			−1.51962			−0.759810	−	0.650145i	$-0.774708\pi$
$733$	−41.1422			−1.51962			−0.759810	+	0.650145i	$0.774708\pi$
$734$		0			0
$735$	−27.0911			−0.999269
$736$		0			0
$737$			0.683383i			0.0251727i
$738$		0			0
$739$		−	14.7484i		−	0.542530i	−0.962505	−	0.271265i	$-0.912558\pi$
$739$		−	14.7484i		−	0.542530i	0.962505	−	0.271265i	$-0.0874419\pi$
$740$		0			0
$741$	11.8300			0.434584
$742$		0			0
$743$	14.4808			0.531249			0.265625	−	0.964077i	$-0.414422\pi$
$743$	14.4808			0.531249			0.265625	+	0.964077i	$0.414422\pi$
$744$		0			0
$745$			0.0275772i			0.00101035i
$746$		0			0
$747$			53.0155i			1.93974i
$748$		0			0
$749$	36.7852			1.34410
$750$		0			0
$751$	19.7494			0.720665			0.360332	−	0.932824i	$-0.382663\pi$
$751$	19.7494			0.720665			0.360332	+	0.932824i	$0.382663\pi$
$752$		0			0
$753$	41.8616			1.52552
$754$		0			0
$755$	6.83276			0.248669
$756$		0			0
$757$	30.7641			1.11814			0.559071	−	0.829120i	$-0.311158\pi$
$757$	30.7641			1.11814			0.559071	+	0.829120i	$0.311158\pi$
$758$		0			0
$759$	1.53237			0.0556216
$760$		0			0
$761$			12.4647i			0.451844i	0.974145	+	0.225922i	$0.0725395\pi$
$761$			12.4647i			0.451844i	−0.974145	+	0.225922i	$0.927461\pi$
$762$		0			0
$763$		−	35.4559i		−	1.28359i
$764$		0			0
$765$			18.6644i			0.674813i
$766$		0			0
$767$	1.30645			0.0471730
$768$		0			0
$769$		−	53.5103i		−	1.92963i	−0.262930	−	0.964815i	$-0.584689\pi$
$769$		−	53.5103i		−	1.92963i	0.262930	−	0.964815i	$-0.415311\pi$
$770$		0			0
$771$	74.6174			2.68728
$772$		0			0
$773$		−	3.21186i		−	0.115522i	−0.998330	−	0.0577612i	$-0.981604\pi$
$773$		−	3.21186i		−	0.115522i	0.998330	−	0.0577612i	$-0.0183962\pi$
$774$		0			0
$775$			0.0829229i			0.00297868i
$776$		0			0
$777$		−	147.737i		−	5.30004i
$778$		0			0
$779$	3.89009			0.139377
$780$		0			0
$781$			2.64072i			0.0944924i
$782$		0			0
$783$		−	111.684i		−	3.99125i
$784$		0			0
$785$	5.92654			0.211527
$786$		0			0
$787$		−	7.42057i		−	0.264515i	−0.991215	−	0.132257i	$-0.957777\pi$
$787$		−	7.42057i		−	0.264515i	0.991215	−	0.132257i	$-0.0422225\pi$
$788$		0			0
$789$	−8.59375			−0.305946
$790$		0			0
$791$	−72.5574			−2.57984
$792$		0			0
$793$			51.2299i			1.81923i
$794$		0			0
$795$	8.53378	+	7.92793i	0.302662	+	0.281175i
$796$		0			0
$797$			9.77745i			0.346335i	0.984892	+	0.173167i	$0.0554002\pi$
$797$			9.77745i			0.346335i	−0.984892	+	0.173167i	$0.944600\pi$
$798$		0			0
$799$	−1.86780			−0.0660780
$800$		0			0
$801$	11.9990			0.423965
$802$		0			0
$803$			0.248015i			0.00875226i
$804$		0			0
$805$	−5.32512			−0.187686
$806$		0			0
$807$			63.7983i			2.24581i
$808$		0			0
$809$			19.3143i			0.679054i	0.940596	+	0.339527i	$0.110267\pi$
$809$			19.3143i			0.679054i	−0.940596	+	0.339527i	$0.889733\pi$
$810$		0			0
$811$	−34.1921			−1.20065			−0.600323	−	0.799757i	$-0.704961\pi$
$811$	−34.1921			−1.20065			−0.600323	+	0.799757i	$0.704961\pi$
$812$		0			0
$813$		−	33.0890i		−	1.16048i
$814$		0			0
$815$			5.99939i			0.210150i
$816$		0			0
$817$			0.556395i			0.0194658i
$818$		0			0
$819$	−139.843			−4.88651
$820$		0			0
$821$		−	8.30612i		−	0.289886i	−0.989440	−	0.144943i	$-0.953700\pi$
$821$		−	8.30612i		−	0.289886i	0.989440	−	0.144943i	$-0.0462998\pi$
$822$		0			0
$823$	22.1661			0.772661			0.386330	−	0.922360i	$-0.373742\pi$
$823$	22.1661			0.772661			0.386330	+	0.922360i	$0.373742\pi$
$824$		0			0
$825$			3.23457i			0.112613i
$826$		0			0
$827$			20.6517i			0.718132i	0.933312	+	0.359066i	$0.116905\pi$
$827$			20.6517i			0.718132i	−0.933312	+	0.359066i	$0.883095\pi$
$828$		0			0
$829$		−	56.1562i		−	1.95039i	−0.221357	−	0.975193i	$-0.571049\pi$
$829$		−	56.1562i		−	1.95039i	0.221357	−	0.975193i	$-0.428951\pi$
$830$		0			0
$831$	−34.1717			−1.18540
$832$		0			0
$833$	81.7413			2.83217
$834$		0			0
$835$	3.60755			0.124844
$836$		0			0
$837$	−0.290010			−0.0100242
$838$		0			0
$839$	36.2972			1.25312			0.626559	−	0.779374i	$-0.284463\pi$
$839$	36.2972			1.25312			0.626559	+	0.779374i	$0.284463\pi$
$840$		0			0
$841$	15.8617			0.546956
$842$		0			0
$843$		−	50.8778i		−	1.75233i
$844$		0			0
$845$			0.146193i			0.00502919i
$846$		0			0
$847$	−53.6082			−1.84200
$848$		0			0
$849$	94.5026			3.24332
$850$		0			0
$851$		−	20.5458i		−	0.704299i
$852$		0			0
$853$		−	10.0429i		−	0.343864i	−0.985109	−	0.171932i	$-0.944999\pi$
$853$		−	10.0429i		−	0.343864i	0.985109	−	0.171932i	$-0.0550009\pi$
$854$		0			0
$855$	−3.86619			−0.132221
$856$		0			0
$857$	6.85328			0.234104			0.117052	−	0.993126i	$-0.462656\pi$
$857$	6.85328			0.234104			0.117052	+	0.993126i	$0.462656\pi$
$858$		0			0
$859$	22.0352			0.751832			0.375916	−	0.926654i	$-0.377328\pi$
$859$	22.0352			0.751832			0.375916	+	0.926654i	$0.377328\pi$
$860$		0			0
$861$	−63.1814			−2.15321
$862$		0			0
$863$	20.7636			0.706801			0.353401	−	0.935472i	$-0.385025\pi$
$863$	20.7636			0.706801			0.353401	+	0.935472i	$0.385025\pi$
$864$		0			0
$865$	6.53010			0.222030
$866$		0			0
$867$		−	20.9351i		−	0.710992i
$868$		0			0
$869$			0.463320i			0.0157171i
$870$		0			0
$871$		−	11.9164i		−	0.403772i
$872$		0			0
$873$	−12.1547			−0.411375
$874$		0			0
$875$		−	23.0283i		−	0.778499i
$876$		0			0
$877$	−54.3385			−1.83488			−0.917440	−	0.397873i	$-0.869748\pi$
$877$	−54.3385			−1.83488			−0.917440	+	0.397873i	$0.869748\pi$
$878$		0			0
$879$			63.4444i			2.13993i
$880$		0			0
$881$			35.1083i			1.18283i	0.806368	+	0.591414i	$0.201430\pi$
$881$			35.1083i			1.18283i	−0.806368	+	0.591414i	$0.798570\pi$
$882$		0			0
$883$		−	49.9167i		−	1.67983i	−0.542717	−	0.839915i	$-0.682605\pi$
$883$		−	49.9167i		−	1.67983i	0.542717	−	0.839915i	$-0.317395\pi$
$884$		0			0
$885$	−0.586628			−0.0197193
$886$		0			0
$887$			1.16066i			0.0389710i	0.999810	+	0.0194855i	$0.00620282\pi$
$887$			1.16066i			0.0389710i	−0.999810	+	0.0194855i	$0.993797\pi$
$888$		0			0
$889$		−	61.4868i		−	2.06220i
$890$		0			0
$891$	−6.39437			−0.214219
$892$		0			0
$893$		−	0.386900i		−	0.0129471i
$894$		0			0
$895$	10.7127			0.358085
$896$		0			0
$897$	−26.7205			−0.892172
$898$		0			0
$899$		−	0.116493i		−	0.00388525i
$900$		0			0
$901$	−25.7488	−	23.9208i	−0.857816	−	0.796917i
$902$		0			0
$903$		−	9.03675i		−	0.300724i
$904$		0			0
$905$	9.75316			0.324206
$906$		0			0
$907$	42.9080			1.42474			0.712369	−	0.701805i	$-0.247622\pi$
$907$	42.9080			1.42474			0.712369	+	0.701805i	$0.247622\pi$
$908$		0			0
$909$		−	43.1145i		−	1.43002i
$910$		0			0
$911$	−23.5389			−0.779878			−0.389939	−	0.920841i	$-0.627504\pi$
$911$	−23.5389			−0.779878			−0.389939	+	0.920841i	$0.627504\pi$
$912$		0			0
$913$			1.35040i			0.0446916i
$914$		0			0
$915$		−	23.0035i		−	0.760474i
$916$		0			0
$917$	87.5444			2.89097
$918$		0			0
$919$		−	7.62191i		−	0.251424i	−0.992067	−	0.125712i	$-0.959879\pi$
$919$		−	7.62191i		−	0.251424i	0.992067	−	0.125712i	$-0.0401215\pi$
$920$		0			0
$921$			100.957i			3.32664i
$922$		0			0
$923$		−	46.0472i		−	1.51566i
$924$		0			0
$925$	43.3685			1.42595
$926$		0			0
$927$			28.1368i			0.924134i
$928$		0			0
$929$	−32.9732			−1.08181			−0.540907	−	0.841082i	$-0.681919\pi$
$929$	−32.9732			−1.08181			−0.540907	+	0.841082i	$0.681919\pi$
$930$		0			0
$931$			16.9321i			0.554927i
$932$		0			0
$933$			19.4670i			0.637322i
$934$		0			0
$935$			0.475415i			0.0155477i
$936$		0			0
$937$	18.2362			0.595752			0.297876	−	0.954605i	$-0.403722\pi$
$937$	18.2362			0.595752			0.297876	+	0.954605i	$0.403722\pi$
$938$		0			0
$939$	−54.0942			−1.76530
$940$		0			0
$941$	46.5046			1.51601			0.758004	−	0.652250i	$-0.226175\pi$
$941$	46.5046			1.51601			0.758004	+	0.652250i	$0.226175\pi$
$942$		0			0
$943$	−8.78662			−0.286132
$944$		0			0
$945$	39.3115			1.27880
$946$		0			0
$947$	−60.4263			−1.96359			−0.981796	−	0.189940i	$-0.939171\pi$
$947$	−60.4263			−1.96359			−0.981796	+	0.189940i	$0.939171\pi$
$948$		0			0
$949$		−	4.32473i		−	0.140387i
$950$		0			0
$951$			58.4503i			1.89538i
$952$		0			0
$953$	−1.63449			−0.0529465			−0.0264732	−	0.999650i	$-0.508428\pi$
$953$	−1.63449			−0.0529465			−0.0264732	+	0.999650i	$0.508428\pi$
$954$		0			0
$955$	−3.77579			−0.122182
$956$		0			0
$957$		−	4.54402i		−	0.146887i
$958$		0			0
$959$		−	41.1351i		−	1.32832i
$960$		0			0
$961$	30.9997			0.999990
$962$		0			0
$963$	−60.3238			−1.94391
$964$		0			0
$965$	11.2134			0.360971
$966$		0			0
$967$	−35.2038			−1.13208			−0.566038	−	0.824379i	$-0.691525\pi$
$967$	−35.2038			−1.13208			−0.566038	+	0.824379i	$0.691525\pi$
$968$		0			0
$969$	16.0277			0.514882
$970$		0			0
$971$	31.2448			1.00269			0.501346	−	0.865247i	$-0.332838\pi$
$971$	31.2448			1.00269			0.501346	+	0.865247i	$0.332838\pi$
$972$		0			0
$973$		−	69.3351i		−	2.22278i
$974$		0			0
$975$		−	56.4023i		−	1.80632i
$976$		0			0
$977$		−	21.9862i		−	0.703400i	−0.936113	−	0.351700i	$-0.885604\pi$
$977$		−	21.9862i		−	0.703400i	0.936113	−	0.351700i	$-0.114396\pi$
$978$		0			0
$979$	0.305636			0.00976817
$980$		0			0
$981$			58.1438i			1.85639i
$982$		0			0
$983$	7.06087			0.225207			0.112603	−	0.993640i	$-0.464081\pi$
$983$	7.06087			0.225207			0.112603	+	0.993640i	$0.464081\pi$
$984$		0			0
$985$		−	11.3471i		−	0.361548i
$986$		0			0
$987$			6.28388i			0.200018i
$988$		0			0
$989$		−	1.25674i		−	0.0399620i
$990$		0			0
$991$	−7.74590			−0.246057			−0.123028	−	0.992403i	$-0.539261\pi$
$991$	−7.74590			−0.246057			−0.123028	+	0.992403i	$0.539261\pi$
$992$		0			0
$993$		−	18.8919i		−	0.599517i
$994$		0			0
$995$		−	1.35329i		−	0.0429022i
$996$		0			0
$997$	−1.52338			−0.0482459			−0.0241230	−	0.999709i	$-0.507679\pi$
$997$	−1.52338			−0.0482459			−0.0241230	+	0.999709i	$0.507679\pi$
$998$		0			0
$999$			151.675i			4.79877i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.c.a.3497.2	✓	82
53.52	even	2	inner	4028.2.c.a.3497.81	yes	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.c.a.3497.2	✓	82	1.1	even	1	trivial
4028.2.c.a.3497.81	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)

\(f(q)\)	\(=\)	\(q-3.32000i q^{3} -0.481923i q^{5} +4.89204 q^{7} -8.02243 q^{9} +O(q^{10})\)	\(q-3.32000i q^{3} -0.481923i q^{5} +4.89204 q^{7} -8.02243 q^{9} -0.204345 q^{11} +3.56324 q^{13} -1.59998 q^{15} +4.82760 q^{17} +1.00000i q^{19} -16.2416i q^{21} -2.25872i q^{23} +4.76775 q^{25} +16.6745i q^{27} -6.69789 q^{29} +0.0173925i q^{31} +0.678426i q^{33} -2.35759i q^{35} +9.09621 q^{37} -11.8300i q^{39} -3.89009i q^{41} +0.556395 q^{43} +3.86619i q^{45} -0.386900 q^{47} +16.9321 q^{49} -16.0277i q^{51} +(-5.33366 - 4.95500i) q^{53} +0.0984784i q^{55} +3.32000 q^{57} +0.366646 q^{59} +14.3774i q^{61} -39.2461 q^{63} -1.71720i q^{65} -3.34426i q^{67} -7.49895 q^{69} -12.9229i q^{71} -1.21371i q^{73} -15.8290i q^{75} -0.999664 q^{77} -2.26735i q^{79} +31.2921 q^{81} -6.60842i q^{83} -2.32653i q^{85} +22.2370i q^{87} -1.49569 q^{89} +17.4315 q^{91} +0.0577430 q^{93} +0.481923 q^{95} +1.51509 q^{97} +1.63934 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) 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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