LMFDB - Embedded newform 4028.2.a.d.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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, 0]))

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

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

chi := DirichletCharacter("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.a (trivial)

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:	yes
Analytic conductor:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.01911$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.01911 q^{3} +3.39604 q^{5} -2.78127 q^{7} -1.96142 q^{9} +O(q^{10})$	$q-1.01911 q^{3} +3.39604 q^{5} -2.78127 q^{7} -1.96142 q^{9} +1.42621 q^{11} -5.82753 q^{13} -3.46093 q^{15} +0.730044 q^{17} -1.00000 q^{19} +2.83442 q^{21} +2.62707 q^{23} +6.53310 q^{25} +5.05622 q^{27} +9.98145 q^{29} +8.88856 q^{31} -1.45346 q^{33} -9.44531 q^{35} -1.61419 q^{37} +5.93889 q^{39} +0.419992 q^{41} -9.18535 q^{43} -6.66106 q^{45} -7.19973 q^{47} +0.735466 q^{49} -0.743994 q^{51} +1.00000 q^{53} +4.84346 q^{55} +1.01911 q^{57} -11.5341 q^{59} +0.616251 q^{61} +5.45523 q^{63} -19.7905 q^{65} -12.1943 q^{67} -2.67727 q^{69} -8.09796 q^{71} +1.60322 q^{73} -6.65794 q^{75} -3.96667 q^{77} -3.28668 q^{79} +0.731414 q^{81} -13.0549 q^{83} +2.47926 q^{85} -10.1722 q^{87} +1.73855 q^{89} +16.2079 q^{91} -9.05841 q^{93} -3.39604 q^{95} -12.9515 q^{97} -2.79739 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.01911	−0.588383	−0.294191	−	0.955747i	$-0.595050\pi$
$3$	−1.01911	−0.588383	−0.294191	+	0.955747i	$0.595050\pi$
$4$	0	0
$5$	3.39604	1.51876	0.759378	−	0.650650i	$-0.225503\pi$
$5$	3.39604	1.51876	0.759378	+	0.650650i	$0.225503\pi$
$6$	0	0
$7$	−2.78127	−1.05122	−0.525611	−	0.850725i	$-0.676163\pi$
$7$	−2.78127	−1.05122	−0.525611	+	0.850725i	$0.676163\pi$
$8$	0	0
$9$	−1.96142	−0.653806
$10$	0	0
$11$	1.42621	0.430018	0.215009	−	0.976612i	$-0.431022\pi$
$11$	1.42621	0.430018	0.215009	+	0.976612i	$0.431022\pi$
$12$	0	0
$13$	−5.82753	−1.61627	−0.808133	−	0.589000i	$-0.799522\pi$
$13$	−5.82753	−1.61627	−0.808133	+	0.589000i	$0.799522\pi$
$14$	0	0
$15$	−3.46093	−0.893610
$16$	0	0
$17$	0.730044	0.177062	0.0885308	−	0.996073i	$-0.471783\pi$
$17$	0.730044	0.177062	0.0885308	+	0.996073i	$0.471783\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.83442	0.618520
$22$	0	0
$23$	2.62707	0.547782	0.273891	−	0.961761i	$-0.411689\pi$
$23$	2.62707	0.547782	0.273891	+	0.961761i	$0.411689\pi$
$24$	0	0
$25$	6.53310	1.30662
$26$	0	0
$27$	5.05622	0.973071
$28$	0	0
$29$	9.98145	1.85351	0.926754	−	0.375669i	$-0.122587\pi$
$29$	9.98145	1.85351	0.926754	+	0.375669i	$0.122587\pi$
$30$	0	0
$31$	8.88856	1.59643	0.798217	−	0.602371i	$-0.205777\pi$
$31$	8.88856	1.59643	0.798217	+	0.602371i	$0.205777\pi$
$32$	0	0
$33$	−1.45346	−0.253015
$34$	0	0
$35$	−9.44531	−1.59655
$36$	0	0
$37$	−1.61419	−0.265370	−0.132685	−	0.991158i	$-0.542360\pi$
$37$	−1.61419	−0.265370	−0.132685	+	0.991158i	$0.542360\pi$
$38$	0	0
$39$	5.93889	0.950983
$40$	0	0
$41$	0.419992	0.0655917	0.0327959	−	0.999462i	$-0.489559\pi$
$41$	0.419992	0.0655917	0.0327959	+	0.999462i	$0.489559\pi$
$42$	0	0
$43$	−9.18535	−1.40075	−0.700376	−	0.713774i	$-0.746984\pi$
$43$	−9.18535	−1.40075	−0.700376	+	0.713774i	$0.746984\pi$
$44$	0	0
$45$	−6.66106	−0.992972
$46$	0	0
$47$	−7.19973	−1.05019	−0.525094	−	0.851044i	$-0.675970\pi$
$47$	−7.19973	−1.05019	−0.525094	+	0.851044i	$0.675970\pi$
$48$	0	0
$49$	0.735466	0.105067
$50$	0	0
$51$	−0.743994	−0.104180
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	4.84346	0.653092
$56$	0	0
$57$	1.01911	0.134984
$58$	0	0
$59$	−11.5341	−1.50161	−0.750806	−	0.660522i	$-0.770335\pi$
$59$	−11.5341	−1.50161	−0.750806	+	0.660522i	$0.770335\pi$
$60$	0	0
$61$	0.616251	0.0789029	0.0394514	−	0.999221i	$-0.487439\pi$
$61$	0.616251	0.0789029	0.0394514	+	0.999221i	$0.487439\pi$
$62$	0	0
$63$	5.45523	0.687295
$64$	0	0
$65$	−19.7905	−2.45471
$66$	0	0
$67$	−12.1943	−1.48978	−0.744888	−	0.667189i	$-0.767497\pi$
$67$	−12.1943	−1.48978	−0.744888	+	0.667189i	$0.767497\pi$
$68$	0	0
$69$	−2.67727	−0.322306
$70$	0	0
$71$	−8.09796	−0.961051	−0.480525	−	0.876981i	$-0.659554\pi$
$71$	−8.09796	−0.961051	−0.480525	+	0.876981i	$0.659554\pi$
$72$	0	0
$73$	1.60322	0.187643	0.0938216	−	0.995589i	$-0.470092\pi$
$73$	1.60322	0.187643	0.0938216	+	0.995589i	$0.470092\pi$
$74$	0	0
$75$	−6.65794	−0.768792
$76$	0	0
$77$	−3.96667	−0.452044
$78$	0	0
$79$	−3.28668	−0.369781	−0.184890	−	0.982759i	$-0.559193\pi$
$79$	−3.28668	−0.369781	−0.184890	+	0.982759i	$0.559193\pi$
$80$	0	0
$81$	0.731414	0.0812683
$82$	0	0
$83$	−13.0549	−1.43296	−0.716482	−	0.697606i	$-0.754249\pi$
$83$	−13.0549	−1.43296	−0.716482	+	0.697606i	$0.754249\pi$
$84$	0	0
$85$	2.47926	0.268913
$86$	0	0
$87$	−10.1722	−1.09057
$88$	0	0
$89$	1.73855	0.184286	0.0921428	−	0.995746i	$-0.470628\pi$
$89$	1.73855	0.184286	0.0921428	+	0.995746i	$0.470628\pi$
$90$	0	0
$91$	16.2079	1.69905
$92$	0	0
$93$	−9.05841	−0.939313
$94$	0	0
$95$	−3.39604	−0.348427
$96$	0	0
$97$	−12.9515	−1.31503	−0.657514	−	0.753442i	$-0.728392\pi$
$97$	−12.9515	−1.31503	−0.657514	+	0.753442i	$0.728392\pi$
$98$	0	0
$99$	−2.79739	−0.281148
$100$	0	0
$101$	0.977436	0.0972585	0.0486292	−	0.998817i	$-0.484515\pi$
$101$	0.977436	0.0972585	0.0486292	+	0.998817i	$0.484515\pi$
$102$	0	0
$103$	−0.0403488	−0.00397568	−0.00198784	−	0.999998i	$-0.500633\pi$
$103$	−0.0403488	−0.00397568	−0.00198784	+	0.999998i	$0.500633\pi$
$104$	0	0
$105$	9.62580	0.939382
$106$	0	0
$107$	−12.4248	−1.20115	−0.600576	−	0.799568i	$-0.705062\pi$
$107$	−12.4248	−1.20115	−0.600576	+	0.799568i	$0.705062\pi$
$108$	0	0
$109$	19.3523	1.85362	0.926809	−	0.375534i	$-0.122541\pi$
$109$	19.3523	1.85362	0.926809	+	0.375534i	$0.122541\pi$
$110$	0	0
$111$	1.64503	0.156139
$112$	0	0
$113$	−14.3330	−1.34834	−0.674168	−	0.738578i	$-0.735498\pi$
$113$	−14.3330	−1.34834	−0.674168	+	0.738578i	$0.735498\pi$
$114$	0	0
$115$	8.92164	0.831948
$116$	0	0
$117$	11.4302	1.05672
$118$	0	0
$119$	−2.03045	−0.186131
$120$	0	0
$121$	−8.96593	−0.815085
$122$	0	0
$123$	−0.428017	−0.0385930
$124$	0	0
$125$	5.20647	0.465681
$126$	0	0
$127$	−18.8326	−1.67112	−0.835562	−	0.549395i	$-0.814858\pi$
$127$	−18.8326	−1.67112	−0.835562	+	0.549395i	$0.814858\pi$
$128$	0	0
$129$	9.36086	0.824178
$130$	0	0
$131$	−6.91591	−0.604246	−0.302123	−	0.953269i	$-0.597695\pi$
$131$	−6.91591	−0.604246	−0.302123	+	0.953269i	$0.597695\pi$
$132$	0	0
$133$	2.78127	0.241167
$134$	0	0
$135$	17.1711	1.47786
$136$	0	0
$137$	11.4324	0.976733	0.488366	−	0.872639i	$-0.337593\pi$
$137$	11.4324	0.976733	0.488366	+	0.872639i	$0.337593\pi$
$138$	0	0
$139$	−4.18394	−0.354877	−0.177439	−	0.984132i	$-0.556781\pi$
$139$	−4.18394	−0.354877	−0.177439	+	0.984132i	$0.556781\pi$
$140$	0	0
$141$	7.33730	0.617912
$142$	0	0
$143$	−8.31127	−0.695023
$144$	0	0
$145$	33.8974	2.81503
$146$	0	0
$147$	−0.749520	−0.0618193
$148$	0	0
$149$	−12.9467	−1.06063	−0.530315	−	0.847800i	$-0.677927\pi$
$149$	−12.9467	−1.06063	−0.530315	+	0.847800i	$0.677927\pi$
$150$	0	0
$151$	20.5029	1.66850	0.834251	−	0.551384i	$-0.185900\pi$
$151$	20.5029	1.66850	0.834251	+	0.551384i	$0.185900\pi$
$152$	0	0
$153$	−1.43192	−0.115764
$154$	0	0
$155$	30.1859	2.42459
$156$	0	0
$157$	−19.3543	−1.54464	−0.772320	−	0.635233i	$-0.780904\pi$
$157$	−19.3543	−1.54464	−0.772320	+	0.635233i	$0.780904\pi$
$158$	0	0
$159$	−1.01911	−0.0808206
$160$	0	0
$161$	−7.30660	−0.575840
$162$	0	0
$163$	16.8234	1.31771	0.658855	−	0.752270i	$-0.271041\pi$
$163$	16.8234	1.31771	0.658855	+	0.752270i	$0.271041\pi$
$164$	0	0
$165$	−4.93601	−0.384268
$166$	0	0
$167$	−5.34045	−0.413256	−0.206628	−	0.978420i	$-0.566249\pi$
$167$	−5.34045	−0.413256	−0.206628	+	0.978420i	$0.566249\pi$
$168$	0	0
$169$	20.9601	1.61232
$170$	0	0
$171$	1.96142	0.149993
$172$	0	0
$173$	21.5842	1.64102	0.820509	−	0.571634i	$-0.193690\pi$
$173$	21.5842	1.64102	0.820509	+	0.571634i	$0.193690\pi$
$174$	0	0
$175$	−18.1703	−1.37355
$176$	0	0
$177$	11.7545	0.883523
$178$	0	0
$179$	−21.7623	−1.62659	−0.813297	−	0.581849i	$-0.802329\pi$
$179$	−21.7623	−1.62659	−0.813297	+	0.581849i	$0.802329\pi$
$180$	0	0
$181$	−13.4511	−0.999816	−0.499908	−	0.866079i	$-0.666633\pi$
$181$	−13.4511	−0.999816	−0.499908	+	0.866079i	$0.666633\pi$
$182$	0	0
$183$	−0.628027	−0.0464251
$184$	0	0
$185$	−5.48184	−0.403033
$186$	0	0
$187$	1.04119	0.0761396
$188$	0	0
$189$	−14.0627	−1.02291
$190$	0	0
$191$	−14.5835	−1.05523	−0.527614	−	0.849484i	$-0.676913\pi$
$191$	−14.5835	−1.05523	−0.527614	+	0.849484i	$0.676913\pi$
$192$	0	0
$193$	15.0472	1.08312	0.541562	−	0.840661i	$-0.317833\pi$
$193$	15.0472	1.08312	0.541562	+	0.840661i	$0.317833\pi$
$194$	0	0
$195$	20.1687	1.44431
$196$	0	0
$197$	−8.55539	−0.609547	−0.304773	−	0.952425i	$-0.598581\pi$
$197$	−8.55539	−0.609547	−0.304773	+	0.952425i	$0.598581\pi$
$198$	0	0
$199$	−3.84835	−0.272802	−0.136401	−	0.990654i	$-0.543554\pi$
$199$	−3.84835	−0.272802	−0.136401	+	0.990654i	$0.543554\pi$
$200$	0	0
$201$	12.4274	0.876558
$202$	0	0
$203$	−27.7611	−1.94845
$204$	0	0
$205$	1.42631	0.0996178
$206$	0	0
$207$	−5.15279	−0.358143
$208$	0	0
$209$	−1.42621	−0.0986528
$210$	0	0
$211$	−12.3323	−0.848990	−0.424495	−	0.905430i	$-0.639548\pi$
$211$	−12.3323	−0.848990	−0.424495	+	0.905430i	$0.639548\pi$
$212$	0	0
$213$	8.25270	0.565465
$214$	0	0
$215$	−31.1938	−2.12740
$216$	0	0
$217$	−24.7215	−1.67820
$218$	0	0
$219$	−1.63386	−0.110406
$220$	0	0
$221$	−4.25435	−0.286179
$222$	0	0
$223$	−16.3464	−1.09464	−0.547318	−	0.836925i	$-0.684351\pi$
$223$	−16.3464	−1.09464	−0.547318	+	0.836925i	$0.684351\pi$
$224$	0	0
$225$	−12.8141	−0.854276
$226$	0	0
$227$	13.2943	0.882374	0.441187	−	0.897415i	$-0.354558\pi$
$227$	13.2943	0.882374	0.441187	+	0.897415i	$0.354558\pi$
$228$	0	0
$229$	21.9953	1.45349	0.726747	−	0.686905i	$-0.241031\pi$
$229$	21.9953	1.45349	0.726747	+	0.686905i	$0.241031\pi$
$230$	0	0
$231$	4.04246	0.265975
$232$	0	0
$233$	8.43894	0.552853	0.276427	−	0.961035i	$-0.410850\pi$
$233$	8.43894	0.552853	0.276427	+	0.961035i	$0.410850\pi$
$234$	0	0
$235$	−24.4506	−1.59498
$236$	0	0
$237$	3.34949	0.217573
$238$	0	0
$239$	−26.2390	−1.69726	−0.848631	−	0.528985i	$-0.822573\pi$
$239$	−26.2390	−1.69726	−0.848631	+	0.528985i	$0.822573\pi$
$240$	0	0
$241$	2.68226	0.172780	0.0863898	−	0.996261i	$-0.472467\pi$
$241$	2.68226	0.172780	0.0863898	+	0.996261i	$0.472467\pi$
$242$	0	0
$243$	−15.9141	−1.02089
$244$	0	0
$245$	2.49767	0.159571
$246$	0	0
$247$	5.82753	0.370797
$248$	0	0
$249$	13.3044	0.843131
$250$	0	0
$251$	1.61739	0.102089	0.0510443	−	0.998696i	$-0.483745\pi$
$251$	1.61739	0.102089	0.0510443	+	0.998696i	$0.483745\pi$
$252$	0	0
$253$	3.74675	0.235556
$254$	0	0
$255$	−2.52663	−0.158224
$256$	0	0
$257$	−8.49474	−0.529887	−0.264944	−	0.964264i	$-0.585353\pi$
$257$	−8.49474	−0.529887	−0.264944	+	0.964264i	$0.585353\pi$
$258$	0	0
$259$	4.48949	0.278963
$260$	0	0
$261$	−19.5778	−1.21183
$262$	0	0
$263$	14.0296	0.865102	0.432551	−	0.901609i	$-0.357614\pi$
$263$	14.0296	0.865102	0.432551	+	0.901609i	$0.357614\pi$
$264$	0	0
$265$	3.39604	0.208617
$266$	0	0
$267$	−1.77177	−0.108430
$268$	0	0
$269$	20.2241	1.23309	0.616544	−	0.787321i	$-0.288532\pi$
$269$	20.2241	1.23309	0.616544	+	0.787321i	$0.288532\pi$
$270$	0	0
$271$	4.56823	0.277500	0.138750	−	0.990327i	$-0.455692\pi$
$271$	4.56823	0.277500	0.138750	+	0.990327i	$0.455692\pi$
$272$	0	0
$273$	−16.5177	−0.999694
$274$	0	0
$275$	9.31755	0.561870
$276$	0	0
$277$	−20.9886	−1.26108	−0.630542	−	0.776155i	$-0.717167\pi$
$277$	−20.9886	−1.26108	−0.630542	+	0.776155i	$0.717167\pi$
$278$	0	0
$279$	−17.4342	−1.04376
$280$	0	0
$281$	−7.48708	−0.446642	−0.223321	−	0.974745i	$-0.571690\pi$
$281$	−7.48708	−0.446642	−0.223321	+	0.974745i	$0.571690\pi$
$282$	0	0
$283$	25.8026	1.53381	0.766903	−	0.641763i	$-0.221797\pi$
$283$	25.8026	1.53381	0.766903	+	0.641763i	$0.221797\pi$
$284$	0	0
$285$	3.46093	0.205008
$286$	0	0
$287$	−1.16811	−0.0689514
$288$	0	0
$289$	−16.4670	−0.968649
$290$	0	0
$291$	13.1990	0.773739
$292$	0	0
$293$	21.4452	1.25284	0.626420	−	0.779486i	$-0.284519\pi$
$293$	21.4452	1.25284	0.626420	+	0.779486i	$0.284519\pi$
$294$	0	0
$295$	−39.1703	−2.28058
$296$	0	0
$297$	7.21122	0.418438
$298$	0	0
$299$	−15.3093	−0.885362
$300$	0	0
$301$	25.5469	1.47250
$302$	0	0
$303$	−0.996113	−0.0572252
$304$	0	0
$305$	2.09282	0.119834
$306$	0	0
$307$	17.7094	1.01073	0.505366	−	0.862905i	$-0.331358\pi$
$307$	17.7094	1.01073	0.505366	+	0.862905i	$0.331358\pi$
$308$	0	0
$309$	0.0411198	0.00233922
$310$	0	0
$311$	0.565422	0.0320622	0.0160311	−	0.999871i	$-0.494897\pi$
$311$	0.565422	0.0320622	0.0160311	+	0.999871i	$0.494897\pi$
$312$	0	0
$313$	−11.9488	−0.675389	−0.337694	−	0.941256i	$-0.609647\pi$
$313$	−11.9488	−0.675389	−0.337694	+	0.941256i	$0.609647\pi$
$314$	0	0
$315$	18.5262	1.04383
$316$	0	0
$317$	21.0263	1.18095	0.590477	−	0.807054i	$-0.298940\pi$
$317$	21.0263	1.18095	0.590477	+	0.807054i	$0.298940\pi$
$318$	0	0
$319$	14.2356	0.797041
$320$	0	0
$321$	12.6622	0.706737
$322$	0	0
$323$	−0.730044	−0.0406207
$324$	0	0
$325$	−38.0718	−2.11185
$326$	0	0
$327$	−19.7221	−1.09064
$328$	0	0
$329$	20.0244	1.10398
$330$	0	0
$331$	9.00189	0.494788	0.247394	−	0.968915i	$-0.420426\pi$
$331$	9.00189	0.494788	0.247394	+	0.968915i	$0.420426\pi$
$332$	0	0
$333$	3.16609	0.173501
$334$	0	0
$335$	−41.4125	−2.26261
$336$	0	0
$337$	30.7720	1.67626	0.838129	−	0.545473i	$-0.183650\pi$
$337$	30.7720	1.67626	0.838129	+	0.545473i	$0.183650\pi$
$338$	0	0
$339$	14.6069	0.793337
$340$	0	0
$341$	12.6769	0.686494
$342$	0	0
$343$	17.4234	0.940773
$344$	0	0
$345$	−9.09212	−0.489503
$346$	0	0
$347$	−25.4367	−1.36551	−0.682757	−	0.730646i	$-0.739219\pi$
$347$	−25.4367	−1.36551	−0.682757	+	0.730646i	$0.739219\pi$
$348$	0	0
$349$	−5.58602	−0.299013	−0.149506	−	0.988761i	$-0.547768\pi$
$349$	−5.58602	−0.299013	−0.149506	+	0.988761i	$0.547768\pi$
$350$	0	0
$351$	−29.4653	−1.57274
$352$	0	0
$353$	−5.47919	−0.291628	−0.145814	−	0.989312i	$-0.546580\pi$
$353$	−5.47919	−0.291628	−0.145814	+	0.989312i	$0.546580\pi$
$354$	0	0
$355$	−27.5010	−1.45960
$356$	0	0
$357$	2.06925	0.109516
$358$	0	0
$359$	−17.0132	−0.897921	−0.448960	−	0.893552i	$-0.648206\pi$
$359$	−17.0132	−0.897921	−0.448960	+	0.893552i	$0.648206\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.13726	0.479582
$364$	0	0
$365$	5.44462	0.284984
$366$	0	0
$367$	−9.20559	−0.480528	−0.240264	−	0.970708i	$-0.577234\pi$
$367$	−9.20559	−0.480528	−0.240264	+	0.970708i	$0.577234\pi$
$368$	0	0
$369$	−0.823780	−0.0428843
$370$	0	0
$371$	−2.78127	−0.144396
$372$	0	0
$373$	6.86681	0.355550	0.177775	−	0.984071i	$-0.443110\pi$
$373$	6.86681	0.355550	0.177775	+	0.984071i	$0.443110\pi$
$374$	0	0
$375$	−5.30596	−0.273998
$376$	0	0
$377$	−58.1672	−2.99576
$378$	0	0
$379$	20.2855	1.04200	0.520998	−	0.853558i	$-0.325560\pi$
$379$	20.2855	1.04200	0.520998	+	0.853558i	$0.325560\pi$
$380$	0	0
$381$	19.1925	0.983261
$382$	0	0
$383$	4.26966	0.218169	0.109085	−	0.994032i	$-0.465208\pi$
$383$	4.26966	0.218169	0.109085	+	0.994032i	$0.465208\pi$
$384$	0	0
$385$	−13.4710	−0.686544
$386$	0	0
$387$	18.0163	0.915820
$388$	0	0
$389$	2.56158	0.129877	0.0649386	−	0.997889i	$-0.479315\pi$
$389$	2.56158	0.129877	0.0649386	+	0.997889i	$0.479315\pi$
$390$	0	0
$391$	1.91788	0.0969912
$392$	0	0
$393$	7.04806	0.355528
$394$	0	0
$395$	−11.1617	−0.561607
$396$	0	0
$397$	−26.6743	−1.33875	−0.669373	−	0.742926i	$-0.733437\pi$
$397$	−26.6743	−1.33875	−0.669373	+	0.742926i	$0.733437\pi$
$398$	0	0
$399$	−2.83442	−0.141898
$400$	0	0
$401$	−1.32390	−0.0661122	−0.0330561	−	0.999453i	$-0.510524\pi$
$401$	−1.32390	−0.0661122	−0.0330561	+	0.999453i	$0.510524\pi$
$402$	0	0
$403$	−51.7984	−2.58026
$404$	0	0
$405$	2.48391	0.123427
$406$	0	0
$407$	−2.30216	−0.114114
$408$	0	0
$409$	20.5225	1.01477	0.507385	−	0.861719i	$-0.330612\pi$
$409$	20.5225	1.01477	0.507385	+	0.861719i	$0.330612\pi$
$410$	0	0
$411$	−11.6508	−0.574692
$412$	0	0
$413$	32.0795	1.57853
$414$	0	0
$415$	−44.3351	−2.17632
$416$	0	0
$417$	4.26389	0.208804
$418$	0	0
$419$	12.6355	0.617283	0.308642	−	0.951178i	$-0.400126\pi$
$419$	12.6355	0.617283	0.308642	+	0.951178i	$0.400126\pi$
$420$	0	0
$421$	32.4459	1.58131	0.790657	−	0.612259i	$-0.209739\pi$
$421$	32.4459	1.58131	0.790657	+	0.612259i	$0.209739\pi$
$422$	0	0
$423$	14.1217	0.686619
$424$	0	0
$425$	4.76945	0.231352
$426$	0	0
$427$	−1.71396	−0.0829444
$428$	0	0
$429$	8.47008	0.408939
$430$	0	0
$431$	19.8630	0.956767	0.478384	−	0.878151i	$-0.341223\pi$
$431$	19.8630	0.956767	0.478384	+	0.878151i	$0.341223\pi$
$432$	0	0
$433$	−6.54023	−0.314304	−0.157152	−	0.987574i	$-0.550231\pi$
$433$	−6.54023	−0.314304	−0.157152	+	0.987574i	$0.550231\pi$
$434$	0	0
$435$	−34.5451	−1.65631
$436$	0	0
$437$	−2.62707	−0.125670
$438$	0	0
$439$	19.8488	0.947332	0.473666	−	0.880705i	$-0.342930\pi$
$439$	19.8488	0.947332	0.473666	+	0.880705i	$0.342930\pi$
$440$	0	0
$441$	−1.44256	−0.0686932
$442$	0	0
$443$	−18.0623	−0.858165	−0.429083	−	0.903265i	$-0.641163\pi$
$443$	−18.0623	−0.858165	−0.429083	+	0.903265i	$0.641163\pi$
$444$	0	0
$445$	5.90418	0.279885
$446$	0	0
$447$	13.1940	0.624057
$448$	0	0
$449$	−41.5384	−1.96032	−0.980159	−	0.198215i	$-0.936486\pi$
$449$	−41.5384	−1.96032	−0.980159	+	0.198215i	$0.936486\pi$
$450$	0	0
$451$	0.598996	0.0282056
$452$	0	0
$453$	−20.8947	−0.981718
$454$	0	0
$455$	55.0428	2.58045
$456$	0	0
$457$	5.17840	0.242235	0.121118	−	0.992638i	$-0.461352\pi$
$457$	5.17840	0.242235	0.121118	+	0.992638i	$0.461352\pi$
$458$	0	0
$459$	3.69126	0.172293
$460$	0	0
$461$	−29.3335	−1.36620	−0.683099	−	0.730326i	$-0.739368\pi$
$461$	−29.3335	−1.36620	−0.683099	+	0.730326i	$0.739368\pi$
$462$	0	0
$463$	4.72241	0.219469	0.109735	−	0.993961i	$-0.465000\pi$
$463$	4.72241	0.219469	0.109735	+	0.993961i	$0.465000\pi$
$464$	0	0
$465$	−30.7627	−1.42659
$466$	0	0
$467$	35.6441	1.64941	0.824705	−	0.565563i	$-0.191341\pi$
$467$	35.6441	1.64941	0.824705	+	0.565563i	$0.191341\pi$
$468$	0	0
$469$	33.9158	1.56608
$470$	0	0
$471$	19.7241	0.908840
$472$	0	0
$473$	−13.1002	−0.602348
$474$	0	0
$475$	−6.53310	−0.299759
$476$	0	0
$477$	−1.96142	−0.0898072
$478$	0	0
$479$	−11.3863	−0.520252	−0.260126	−	0.965575i	$-0.583764\pi$
$479$	−11.3863	−0.520252	−0.260126	+	0.965575i	$0.583764\pi$
$480$	0	0
$481$	9.40672	0.428909
$482$	0	0
$483$	7.44621	0.338814
$484$	0	0
$485$	−43.9839	−1.99721
$486$	0	0
$487$	−36.2347	−1.64195	−0.820975	−	0.570965i	$-0.806569\pi$
$487$	−36.2347	−1.64195	−0.820975	+	0.570965i	$0.806569\pi$
$488$	0	0
$489$	−17.1449	−0.775318
$490$	0	0
$491$	32.8337	1.48176	0.740881	−	0.671636i	$-0.234408\pi$
$491$	32.8337	1.48176	0.740881	+	0.671636i	$0.234408\pi$
$492$	0	0
$493$	7.28689	0.328185
$494$	0	0
$495$	−9.50005	−0.426995
$496$	0	0
$497$	22.5226	1.01028
$498$	0	0
$499$	7.14536	0.319870	0.159935	−	0.987128i	$-0.448871\pi$
$499$	7.14536	0.319870	0.159935	+	0.987128i	$0.448871\pi$
$500$	0	0
$501$	5.44249	0.243153
$502$	0	0
$503$	−24.7089	−1.10171	−0.550857	−	0.834600i	$-0.685699\pi$
$503$	−24.7089	−1.10171	−0.550857	+	0.834600i	$0.685699\pi$
$504$	0	0
$505$	3.31941	0.147712
$506$	0	0
$507$	−21.3606	−0.948659
$508$	0	0
$509$	−4.90294	−0.217319	−0.108659	−	0.994079i	$-0.534656\pi$
$509$	−4.90294	−0.217319	−0.108659	+	0.994079i	$0.534656\pi$
$510$	0	0
$511$	−4.45900	−0.197255
$512$	0	0
$513$	−5.05622	−0.223238
$514$	0	0
$515$	−0.137026	−0.00603809
$516$	0	0
$517$	−10.2683	−0.451600
$518$	0	0
$519$	−21.9967	−0.965546
$520$	0	0
$521$	24.6977	1.08203	0.541013	−	0.841014i	$-0.318041\pi$
$521$	24.6977	1.08203	0.541013	+	0.841014i	$0.318041\pi$
$522$	0	0
$523$	−3.18909	−0.139449	−0.0697246	−	0.997566i	$-0.522212\pi$
$523$	−3.18909	−0.139449	−0.0697246	+	0.997566i	$0.522212\pi$
$524$	0	0
$525$	18.5175	0.808171
$526$	0	0
$527$	6.48904	0.282667
$528$	0	0
$529$	−16.0985	−0.699935
$530$	0	0
$531$	22.6232	0.981763
$532$	0	0
$533$	−2.44752	−0.106014
$534$	0	0
$535$	−42.1952	−1.82426
$536$	0	0
$537$	22.1782	0.957059
$538$	0	0
$539$	1.04893	0.0451805
$540$	0	0
$541$	−44.6386	−1.91916	−0.959582	−	0.281429i	$-0.909192\pi$
$541$	−44.6386	−1.91916	−0.959582	+	0.281429i	$0.909192\pi$
$542$	0	0
$543$	13.7082	0.588274
$544$	0	0
$545$	65.7213	2.81519
$546$	0	0
$547$	1.60025	0.0684216	0.0342108	−	0.999415i	$-0.489108\pi$
$547$	1.60025	0.0684216	0.0342108	+	0.999415i	$0.489108\pi$
$548$	0	0
$549$	−1.20873	−0.0515872
$550$	0	0
$551$	−9.98145	−0.425224
$552$	0	0
$553$	9.14116	0.388722
$554$	0	0
$555$	5.58659	0.237138
$556$	0	0
$557$	−21.2700	−0.901240	−0.450620	−	0.892716i	$-0.648797\pi$
$557$	−21.2700	−0.901240	−0.450620	+	0.892716i	$0.648797\pi$
$558$	0	0
$559$	53.5279	2.26399
$560$	0	0
$561$	−1.06109	−0.0447992
$562$	0	0
$563$	3.10740	0.130961	0.0654806	−	0.997854i	$-0.479142\pi$
$563$	3.10740	0.130961	0.0654806	+	0.997854i	$0.479142\pi$
$564$	0	0
$565$	−48.6755	−2.04779
$566$	0	0
$567$	−2.03426	−0.0854309
$568$	0	0
$569$	−38.5197	−1.61483	−0.807415	−	0.589984i	$-0.799134\pi$
$569$	−38.5197	−1.61483	−0.807415	+	0.589984i	$0.799134\pi$
$570$	0	0
$571$	−14.9193	−0.624354	−0.312177	−	0.950024i	$-0.601058\pi$
$571$	−14.9193	−0.624354	−0.312177	+	0.950024i	$0.601058\pi$
$572$	0	0
$573$	14.8622	0.620877
$574$	0	0
$575$	17.1629	0.715743
$576$	0	0
$577$	−25.9795	−1.08154	−0.540769	−	0.841171i	$-0.681867\pi$
$577$	−25.9795	−1.08154	−0.540769	+	0.841171i	$0.681867\pi$
$578$	0	0
$579$	−15.3348	−0.637291
$580$	0	0
$581$	36.3093	1.50636
$582$	0	0
$583$	1.42621	0.0590675
$584$	0	0
$585$	38.8175	1.60491
$586$	0	0
$587$	−2.61947	−0.108117	−0.0540585	−	0.998538i	$-0.517216\pi$
$587$	−2.61947	−0.108117	−0.0540585	+	0.998538i	$0.517216\pi$
$588$	0	0
$589$	−8.88856	−0.366247
$590$	0	0
$591$	8.71887	0.358647
$592$	0	0
$593$	43.8237	1.79962	0.899811	−	0.436280i	$-0.143704\pi$
$593$	43.8237	1.79962	0.899811	+	0.436280i	$0.143704\pi$
$594$	0	0
$595$	−6.89549	−0.282687
$596$	0	0
$597$	3.92188	0.160512
$598$	0	0
$599$	33.5441	1.37057	0.685287	−	0.728273i	$-0.259677\pi$
$599$	33.5441	1.37057	0.685287	+	0.728273i	$0.259677\pi$
$600$	0	0
$601$	12.7561	0.520331	0.260166	−	0.965564i	$-0.416223\pi$
$601$	12.7561	0.520331	0.260166	+	0.965564i	$0.416223\pi$
$602$	0	0
$603$	23.9182	0.974025
$604$	0	0
$605$	−30.4487	−1.23791
$606$	0	0
$607$	28.2396	1.14621	0.573104	−	0.819482i	$-0.305739\pi$
$607$	28.2396	1.14621	0.573104	+	0.819482i	$0.305739\pi$
$608$	0	0
$609$	28.2916	1.14643
$610$	0	0
$611$	41.9566	1.69738
$612$	0	0
$613$	12.6800	0.512142	0.256071	−	0.966658i	$-0.417572\pi$
$613$	12.6800	0.512142	0.256071	+	0.966658i	$0.417572\pi$
$614$	0	0
$615$	−1.45357	−0.0586134
$616$	0	0
$617$	−34.0381	−1.37032	−0.685161	−	0.728392i	$-0.740268\pi$
$617$	−34.0381	−1.37032	−0.685161	+	0.728392i	$0.740268\pi$
$618$	0	0
$619$	−13.7758	−0.553696	−0.276848	−	0.960914i	$-0.589290\pi$
$619$	−13.7758	−0.553696	−0.276848	+	0.960914i	$0.589290\pi$
$620$	0	0
$621$	13.2831	0.533031
$622$	0	0
$623$	−4.83537	−0.193725
$624$	0	0
$625$	−14.9841	−0.599364
$626$	0	0
$627$	1.45346	0.0580456
$628$	0	0
$629$	−1.17843	−0.0469869
$630$	0	0
$631$	−2.85691	−0.113732	−0.0568658	−	0.998382i	$-0.518111\pi$
$631$	−2.85691	−0.113732	−0.0568658	+	0.998382i	$0.518111\pi$
$632$	0	0
$633$	12.5679	0.499531
$634$	0	0
$635$	−63.9564	−2.53803
$636$	0	0
$637$	−4.28595	−0.169816
$638$	0	0
$639$	15.8835	0.628341
$640$	0	0
$641$	21.2565	0.839580	0.419790	−	0.907621i	$-0.362104\pi$
$641$	21.2565	0.839580	0.419790	+	0.907621i	$0.362104\pi$
$642$	0	0
$643$	9.56151	0.377069	0.188535	−	0.982067i	$-0.439626\pi$
$643$	9.56151	0.377069	0.188535	+	0.982067i	$0.439626\pi$
$644$	0	0
$645$	31.7899	1.25173
$646$	0	0
$647$	3.52419	0.138550	0.0692752	−	0.997598i	$-0.477931\pi$
$647$	3.52419	0.138550	0.0692752	+	0.997598i	$0.477931\pi$
$648$	0	0
$649$	−16.4500	−0.645720
$650$	0	0
$651$	25.1939	0.987426
$652$	0	0
$653$	−17.8690	−0.699269	−0.349634	−	0.936886i	$-0.613694\pi$
$653$	−17.8690	−0.699269	−0.349634	+	0.936886i	$0.613694\pi$
$654$	0	0
$655$	−23.4867	−0.917702
$656$	0	0
$657$	−3.14459	−0.122682
$658$	0	0
$659$	9.89031	0.385272	0.192636	−	0.981270i	$-0.438296\pi$
$659$	9.89031	0.385272	0.192636	+	0.981270i	$0.438296\pi$
$660$	0	0
$661$	−28.7023	−1.11639	−0.558195	−	0.829710i	$-0.688506\pi$
$661$	−28.7023	−1.11639	−0.558195	+	0.829710i	$0.688506\pi$
$662$	0	0
$663$	4.33565	0.168383
$664$	0	0
$665$	9.44531	0.366273
$666$	0	0
$667$	26.2220	1.01532
$668$	0	0
$669$	16.6588	0.644065
$670$	0	0
$671$	0.878902	0.0339296
$672$	0	0
$673$	−10.4268	−0.401924	−0.200962	−	0.979599i	$-0.564407\pi$
$673$	−10.4268	−0.401924	−0.200962	+	0.979599i	$0.564407\pi$
$674$	0	0
$675$	33.0328	1.27143
$676$	0	0
$677$	0.489847	0.0188263	0.00941317	−	0.999956i	$-0.497004\pi$
$677$	0.489847	0.0188263	0.00941317	+	0.999956i	$0.497004\pi$
$678$	0	0
$679$	36.0217	1.38239
$680$	0	0
$681$	−13.5483	−0.519173
$682$	0	0
$683$	−30.6974	−1.17460	−0.587301	−	0.809369i	$-0.699809\pi$
$683$	−30.6974	−1.17460	−0.587301	+	0.809369i	$0.699809\pi$
$684$	0	0
$685$	38.8248	1.48342
$686$	0	0
$687$	−22.4156	−0.855210
$688$	0	0
$689$	−5.82753	−0.222011
$690$	0	0
$691$	29.7021	1.12992	0.564961	−	0.825117i	$-0.308891\pi$
$691$	29.7021	1.12992	0.564961	+	0.825117i	$0.308891\pi$
$692$	0	0
$693$	7.78029	0.295549
$694$	0	0
$695$	−14.2088	−0.538972
$696$	0	0
$697$	0.306612	0.0116138
$698$	0	0
$699$	−8.60019	−0.325289
$700$	0	0
$701$	8.93178	0.337349	0.168674	−	0.985672i	$-0.446051\pi$
$701$	8.93178	0.337349	0.168674	+	0.985672i	$0.446051\pi$
$702$	0	0
$703$	1.61419	0.0608802
$704$	0	0
$705$	24.9178	0.938458
$706$	0	0
$707$	−2.71851	−0.102240
$708$	0	0
$709$	−25.8038	−0.969081	−0.484540	−	0.874769i	$-0.661013\pi$
$709$	−25.8038	−0.969081	−0.484540	+	0.874769i	$0.661013\pi$
$710$	0	0
$711$	6.44656	0.241765
$712$	0	0
$713$	23.3509	0.874498
$714$	0	0
$715$	−28.2254	−1.05557
$716$	0	0
$717$	26.7404	0.998640
$718$	0	0
$719$	−2.11153	−0.0787468	−0.0393734	−	0.999225i	$-0.512536\pi$
$719$	−2.11153	−0.0787468	−0.0393734	+	0.999225i	$0.512536\pi$
$720$	0	0
$721$	0.112221	0.00417932
$722$	0	0
$723$	−2.73352	−0.101661
$724$	0	0
$725$	65.2098	2.42183
$726$	0	0
$727$	43.9116	1.62859	0.814295	−	0.580452i	$-0.197124\pi$
$727$	43.9116	1.62859	0.814295	+	0.580452i	$0.197124\pi$
$728$	0	0
$729$	14.0239	0.519404
$730$	0	0
$731$	−6.70570	−0.248019
$732$	0	0
$733$	−27.2957	−1.00819	−0.504094	−	0.863649i	$-0.668174\pi$
$733$	−27.2957	−1.00819	−0.504094	+	0.863649i	$0.668174\pi$
$734$	0	0
$735$	−2.54540	−0.0938885
$736$	0	0
$737$	−17.3917	−0.640630
$738$	0	0
$739$	6.39207	0.235136	0.117568	−	0.993065i	$-0.462490\pi$
$739$	6.39207	0.235136	0.117568	+	0.993065i	$0.462490\pi$
$740$	0	0
$741$	−5.93889	−0.218170
$742$	0	0
$743$	47.9311	1.75842	0.879211	−	0.476433i	$-0.158070\pi$
$743$	47.9311	1.75842	0.879211	+	0.476433i	$0.158070\pi$
$744$	0	0
$745$	−43.9674	−1.61084
$746$	0	0
$747$	25.6062	0.936880
$748$	0	0
$749$	34.5568	1.26268
$750$	0	0
$751$	−12.8524	−0.468989	−0.234495	−	0.972117i	$-0.575344\pi$
$751$	−12.8524	−0.468989	−0.234495	+	0.972117i	$0.575344\pi$
$752$	0	0
$753$	−1.64829	−0.0600671
$754$	0	0
$755$	69.6287	2.53405
$756$	0	0
$757$	4.52334	0.164404	0.0822018	−	0.996616i	$-0.473805\pi$
$757$	4.52334	0.164404	0.0822018	+	0.996616i	$0.473805\pi$
$758$	0	0
$759$	−3.81834	−0.138597
$760$	0	0
$761$	−33.3495	−1.20892	−0.604459	−	0.796636i	$-0.706611\pi$
$761$	−33.3495	−1.20892	−0.604459	+	0.796636i	$0.706611\pi$
$762$	0	0
$763$	−53.8241	−1.94856
$764$	0	0
$765$	−4.86286	−0.175817
$766$	0	0
$767$	67.2154	2.42701
$768$	0	0
$769$	−39.6251	−1.42892	−0.714458	−	0.699678i	$-0.753327\pi$
$769$	−39.6251	−1.42892	−0.714458	+	0.699678i	$0.753327\pi$
$770$	0	0
$771$	8.65706	0.311776
$772$	0	0
$773$	−32.6481	−1.17427	−0.587135	−	0.809489i	$-0.699744\pi$
$773$	−32.6481	−1.17427	−0.587135	+	0.809489i	$0.699744\pi$
$774$	0	0
$775$	58.0699	2.08593
$776$	0	0
$777$	−4.57527	−0.164137
$778$	0	0
$779$	−0.419992	−0.0150478
$780$	0	0
$781$	−11.5494	−0.413269
$782$	0	0
$783$	50.4684	1.80359
$784$	0	0
$785$	−65.7280	−2.34593
$786$	0	0
$787$	−31.2473	−1.11384	−0.556922	−	0.830564i	$-0.688018\pi$
$787$	−31.2473	−1.11384	−0.556922	+	0.830564i	$0.688018\pi$
$788$	0	0
$789$	−14.2977	−0.509011
$790$	0	0
$791$	39.8640	1.41740
$792$	0	0
$793$	−3.59122	−0.127528
$794$	0	0
$795$	−3.46093	−0.122747
$796$	0	0
$797$	30.2469	1.07140	0.535701	−	0.844408i	$-0.320047\pi$
$797$	30.2469	1.07140	0.535701	+	0.844408i	$0.320047\pi$
$798$	0	0
$799$	−5.25612	−0.185948
$800$	0	0
$801$	−3.41002	−0.120487
$802$	0	0
$803$	2.28653	0.0806899
$804$	0	0
$805$	−24.8135	−0.874561
$806$	0	0
$807$	−20.6106	−0.725527
$808$	0	0
$809$	17.1619	0.603380	0.301690	−	0.953406i	$-0.402449\pi$
$809$	17.1619	0.603380	0.301690	+	0.953406i	$0.402449\pi$
$810$	0	0
$811$	35.2236	1.23687	0.618433	−	0.785837i	$-0.287768\pi$
$811$	35.2236	1.23687	0.618433	+	0.785837i	$0.287768\pi$
$812$	0	0
$813$	−4.65552	−0.163276
$814$	0	0
$815$	57.1330	2.00128
$816$	0	0
$817$	9.18535	0.321355
$818$	0	0
$819$	−31.7905	−1.11085
$820$	0	0
$821$	42.9186	1.49787	0.748935	−	0.662643i	$-0.230565\pi$
$821$	42.9186	1.49787	0.748935	+	0.662643i	$0.230565\pi$
$822$	0	0
$823$	37.9507	1.32288	0.661439	−	0.749999i	$-0.269946\pi$
$823$	37.9507	1.32288	0.661439	+	0.749999i	$0.269946\pi$
$824$	0	0
$825$	−9.49560	−0.330594
$826$	0	0
$827$	−13.9293	−0.484368	−0.242184	−	0.970230i	$-0.577864\pi$
$827$	−13.9293	−0.484368	−0.242184	+	0.970230i	$0.577864\pi$
$828$	0	0
$829$	−30.6787	−1.06551	−0.532757	−	0.846268i	$-0.678844\pi$
$829$	−30.6787	−1.06551	−0.532757	+	0.846268i	$0.678844\pi$
$830$	0	0
$831$	21.3897	0.742000
$832$	0	0
$833$	0.536922	0.0186033
$834$	0	0
$835$	−18.1364	−0.627635
$836$	0	0
$837$	44.9426	1.55344
$838$	0	0
$839$	3.81625	0.131751	0.0658757	−	0.997828i	$-0.479016\pi$
$839$	3.81625	0.131751	0.0658757	+	0.997828i	$0.479016\pi$
$840$	0	0
$841$	70.6293	2.43549
$842$	0	0
$843$	7.63014	0.262796
$844$	0	0
$845$	71.1814	2.44872
$846$	0	0
$847$	24.9367	0.856835
$848$	0	0
$849$	−26.2956	−0.902464
$850$	0	0
$851$	−4.24058	−0.145365
$852$	0	0
$853$	3.74924	0.128372	0.0641858	−	0.997938i	$-0.479555\pi$
$853$	3.74924	0.128372	0.0641858	+	0.997938i	$0.479555\pi$
$854$	0	0
$855$	6.66106	0.227803
$856$	0	0
$857$	48.2740	1.64901	0.824505	−	0.565855i	$-0.191454\pi$
$857$	48.2740	1.64901	0.824505	+	0.565855i	$0.191454\pi$
$858$	0	0
$859$	8.73850	0.298154	0.149077	−	0.988826i	$-0.452370\pi$
$859$	8.73850	0.298154	0.149077	+	0.988826i	$0.452370\pi$
$860$	0	0
$861$	1.19043	0.0405698
$862$	0	0
$863$	18.1458	0.617691	0.308845	−	0.951112i	$-0.400057\pi$
$863$	18.1458	0.617691	0.308845	+	0.951112i	$0.400057\pi$
$864$	0	0
$865$	73.3009	2.49230
$866$	0	0
$867$	16.7817	0.569936
$868$	0	0
$869$	−4.68749	−0.159012
$870$	0	0
$871$	71.0629	2.40788
$872$	0	0
$873$	25.4033	0.859773
$874$	0	0
$875$	−14.4806	−0.489534
$876$	0	0
$877$	44.1816	1.49191	0.745954	−	0.665997i	$-0.231994\pi$
$877$	44.1816	1.49191	0.745954	+	0.665997i	$0.231994\pi$
$878$	0	0
$879$	−21.8550	−0.737149
$880$	0	0
$881$	−49.5363	−1.66892	−0.834461	−	0.551067i	$-0.814221\pi$
$881$	−49.5363	−1.66892	−0.834461	+	0.551067i	$0.814221\pi$
$882$	0	0
$883$	8.87871	0.298793	0.149396	−	0.988777i	$-0.452267\pi$
$883$	8.87871	0.298793	0.149396	+	0.988777i	$0.452267\pi$
$884$	0	0
$885$	39.9188	1.34186
$886$	0	0
$887$	−35.7254	−1.19954	−0.599770	−	0.800172i	$-0.704741\pi$
$887$	−35.7254	−1.19954	−0.599770	+	0.800172i	$0.704741\pi$
$888$	0	0
$889$	52.3786	1.75672
$890$	0	0
$891$	1.04315	0.0349468
$892$	0	0
$893$	7.19973	0.240930
$894$	0	0
$895$	−73.9058	−2.47040
$896$	0	0
$897$	15.6019	0.520932
$898$	0	0
$899$	88.7207	2.95900
$900$	0	0
$901$	0.730044	0.0243213
$902$	0	0
$903$	−26.0351	−0.866394
$904$	0	0
$905$	−45.6807	−1.51848
$906$	0	0
$907$	−26.4802	−0.879262	−0.439631	−	0.898179i	$-0.644891\pi$
$907$	−26.4802	−0.879262	−0.439631	+	0.898179i	$0.644891\pi$
$908$	0	0
$909$	−1.91716	−0.0635882
$910$	0	0
$911$	46.1495	1.52900	0.764500	−	0.644624i	$-0.222986\pi$
$911$	46.1495	1.52900	0.764500	+	0.644624i	$0.222986\pi$
$912$	0	0
$913$	−18.6190	−0.616200
$914$	0	0
$915$	−2.13281	−0.0705084
$916$	0	0
$917$	19.2350	0.635196
$918$	0	0
$919$	−53.2141	−1.75537	−0.877686	−	0.479236i	$-0.840914\pi$
$919$	−53.2141	−1.75537	−0.877686	+	0.479236i	$0.840914\pi$
$920$	0	0
$921$	−18.0478	−0.594697
$922$	0	0
$923$	47.1911	1.55331
$924$	0	0
$925$	−10.5456	−0.346738
$926$	0	0
$927$	0.0791408	0.00259933
$928$	0	0
$929$	−3.69906	−0.121362	−0.0606812	−	0.998157i	$-0.519327\pi$
$929$	−3.69906	−0.121362	−0.0606812	+	0.998157i	$0.519327\pi$
$930$	0	0
$931$	−0.735466	−0.0241039
$932$	0	0
$933$	−0.576227	−0.0188648
$934$	0	0
$935$	3.53594	0.115637
$936$	0	0
$937$	45.8161	1.49675	0.748373	−	0.663278i	$-0.230835\pi$
$937$	45.8161	1.49675	0.748373	+	0.663278i	$0.230835\pi$
$938$	0	0
$939$	12.1772	0.397387
$940$	0	0
$941$	0.237646	0.00774705	0.00387352	−	0.999992i	$-0.498767\pi$
$941$	0.237646	0.00774705	0.00387352	+	0.999992i	$0.498767\pi$
$942$	0	0
$943$	1.10335	0.0359300
$944$	0	0
$945$	−47.7576	−1.55355
$946$	0	0
$947$	23.8032	0.773501	0.386750	−	0.922184i	$-0.373598\pi$
$947$	23.8032	0.773501	0.386750	+	0.922184i	$0.373598\pi$
$948$	0	0
$949$	−9.34284	−0.303281
$950$	0	0
$951$	−21.4281	−0.694853
$952$	0	0
$953$	13.9035	0.450380	0.225190	−	0.974315i	$-0.427700\pi$
$953$	13.9035	0.450380	0.225190	+	0.974315i	$0.427700\pi$
$954$	0	0
$955$	−49.5263	−1.60263
$956$	0	0
$957$	−14.5076	−0.468965
$958$	0	0
$959$	−31.7965	−1.02676
$960$	0	0
$961$	48.0066	1.54860
$962$	0	0
$963$	24.3702	0.785320
$964$	0	0
$965$	51.1010	1.64500
$966$	0	0
$967$	34.3772	1.10550	0.552749	−	0.833348i	$-0.313579\pi$
$967$	34.3772	1.10550	0.552749	+	0.833348i	$0.313579\pi$
$968$	0	0
$969$	0.743994	0.0239005
$970$	0	0
$971$	21.5727	0.692301	0.346151	−	0.938179i	$-0.387489\pi$
$971$	21.5727	0.692301	0.346151	+	0.938179i	$0.387489\pi$
$972$	0	0
$973$	11.6367	0.373055
$974$	0	0
$975$	38.7993	1.24257
$976$	0	0
$977$	−19.5939	−0.626866	−0.313433	−	0.949610i	$-0.601479\pi$
$977$	−19.5939	−0.626866	−0.313433	+	0.949610i	$0.601479\pi$
$978$	0	0
$979$	2.47953	0.0792460
$980$	0	0
$981$	−37.9580	−1.21191
$982$	0	0
$983$	−20.2487	−0.645832	−0.322916	−	0.946428i	$-0.604663\pi$
$983$	−20.2487	−0.645832	−0.322916	+	0.946428i	$0.604663\pi$
$984$	0	0
$985$	−29.0545	−0.925753
$986$	0	0
$987$	−20.4070	−0.649563
$988$	0	0
$989$	−24.1306	−0.767307
$990$	0	0
$991$	−32.4481	−1.03075	−0.515373	−	0.856966i	$-0.672347\pi$
$991$	−32.4481	−1.03075	−0.515373	+	0.856966i	$0.672347\pi$
$992$	0	0
$993$	−9.17390	−0.291125
$994$	0	0
$995$	−13.0691	−0.414320
$996$	0	0
$997$	−32.2942	−1.02277	−0.511384	−	0.859353i	$-0.670867\pi$
$997$	−32.2942	−1.02277	−0.511384	+	0.859353i	$0.670867\pi$
$998$	0	0
$999$	−8.16168	−0.258224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.8	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.8	✓	19	1.1	even	1	trivial

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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.a (trivial)

\(f(q)\)	\(=\)	\(q-1.01911 q^{3} +3.39604 q^{5} -2.78127 q^{7} -1.96142 q^{9} +O(q^{10})\)	\(q-1.01911 q^{3} +3.39604 q^{5} -2.78127 q^{7} -1.96142 q^{9} +1.42621 q^{11} -5.82753 q^{13} -3.46093 q^{15} +0.730044 q^{17} -1.00000 q^{19} +2.83442 q^{21} +2.62707 q^{23} +6.53310 q^{25} +5.05622 q^{27} +9.98145 q^{29} +8.88856 q^{31} -1.45346 q^{33} -9.44531 q^{35} -1.61419 q^{37} +5.93889 q^{39} +0.419992 q^{41} -9.18535 q^{43} -6.66106 q^{45} -7.19973 q^{47} +0.735466 q^{49} -0.743994 q^{51} +1.00000 q^{53} +4.84346 q^{55} +1.01911 q^{57} -11.5341 q^{59} +0.616251 q^{61} +5.45523 q^{63} -19.7905 q^{65} -12.1943 q^{67} -2.67727 q^{69} -8.09796 q^{71} +1.60322 q^{73} -6.65794 q^{75} -3.96667 q^{77} -3.28668 q^{79} +0.731414 q^{81} -13.0549 q^{83} +2.47926 q^{85} -10.1722 q^{87} +1.73855 q^{89} +16.2079 q^{91} -9.05841 q^{93} -3.39604 q^{95} -12.9515 q^{97} -2.79739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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