LMFDB - Embedded newform 4028.2.a.e.1.9

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:	$0$
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} - 3 x^{18} - 35 x^{17} + 103 x^{16} + 501 x^{15} - 1437 x^{14} - 3775 x^{13} + 10450 x^{12} + \cdots - 4 $ x^19 - 3x^18 - 35x^17 + 103x^16 + 501x^15 - 1437x^14 - 3775x^13 + 10450x^12 + 16076x^11 - 42255x^10 - 38701x^9 + 93907x^8 + 49522x^7 - 106284x^6 - 26703x^5 + 50522x^4 - 446x^3 - 6265x^2 + 719x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.00586283$ 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)

$f(q)$	$=$	$q+0.00586283 q^{3} +2.78153 q^{5} -4.08250 q^{7} -2.99997 q^{9} +O(q^{10})$	$q+0.00586283 q^{3} +2.78153 q^{5} -4.08250 q^{7} -2.99997 q^{9} -0.755316 q^{11} -2.25024 q^{13} +0.0163076 q^{15} -2.34381 q^{17} +1.00000 q^{19} -0.0239350 q^{21} +5.02409 q^{23} +2.73691 q^{25} -0.0351768 q^{27} -4.98256 q^{29} +2.57231 q^{31} -0.00442829 q^{33} -11.3556 q^{35} +9.55265 q^{37} -0.0131928 q^{39} -1.74860 q^{41} +11.3506 q^{43} -8.34449 q^{45} +2.44287 q^{47} +9.66679 q^{49} -0.0137414 q^{51} +1.00000 q^{53} -2.10093 q^{55} +0.00586283 q^{57} +12.1287 q^{59} +8.42260 q^{61} +12.2474 q^{63} -6.25912 q^{65} -3.97424 q^{67} +0.0294554 q^{69} -5.56636 q^{71} -9.20513 q^{73} +0.0160460 q^{75} +3.08357 q^{77} +13.6358 q^{79} +8.99969 q^{81} +4.27617 q^{83} -6.51938 q^{85} -0.0292119 q^{87} +3.29001 q^{89} +9.18661 q^{91} +0.0150810 q^{93} +2.78153 q^{95} +9.44726 q^{97} +2.26592 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * 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
$3$	0.00586283	0.00338491	0.00169245	−	0.999999i	$-0.499461\pi$
$3$	0.00586283	0.00338491	0.00169245	+	0.999999i	$0.499461\pi$
$4$	0	0
$5$	2.78153	1.24394	0.621969	−	0.783042i	$-0.286333\pi$
$5$	2.78153	1.24394	0.621969	+	0.783042i	$0.286333\pi$
$6$	0	0
$7$	−4.08250	−1.54304	−0.771520	−	0.636206i	$-0.780503\pi$
$7$	−4.08250	−1.54304	−0.771520	+	0.636206i	$0.780503\pi$
$8$	0	0
$9$	−2.99997	−0.999989
$10$	0	0
$11$	−0.755316	−0.227736	−0.113868	−	0.993496i	$-0.536324\pi$
$11$	−0.755316	−0.227736	−0.113868	+	0.993496i	$0.536324\pi$
$12$	0	0
$13$	−2.25024	−0.624105	−0.312053	−	0.950065i	$-0.601017\pi$
$13$	−2.25024	−0.624105	−0.312053	+	0.950065i	$0.601017\pi$
$14$	0	0
$15$	0.0163076	0.00421061
$16$	0	0
$17$	−2.34381	−0.568458	−0.284229	−	0.958756i	$-0.591738\pi$
$17$	−2.34381	−0.568458	−0.284229	+	0.958756i	$0.591738\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.0239350	−0.00522304
$22$	0	0
$23$	5.02409	1.04760	0.523798	−	0.851843i	$-0.324515\pi$
$23$	5.02409	1.04760	0.523798	+	0.851843i	$0.324515\pi$
$24$	0	0
$25$	2.73691	0.547381
$26$	0	0
$27$	−0.0351768	−0.00676978
$28$	0	0
$29$	−4.98256	−0.925238	−0.462619	−	0.886557i	$-0.653090\pi$
$29$	−4.98256	−0.925238	−0.462619	+	0.886557i	$0.653090\pi$
$30$	0	0
$31$	2.57231	0.462001	0.231001	−	0.972954i	$-0.425800\pi$
$31$	2.57231	0.462001	0.231001	+	0.972954i	$0.425800\pi$
$32$	0	0
$33$	−0.00442829	−0.000770866 0
$34$	0	0
$35$	−11.3556	−1.91944
$36$	0	0
$37$	9.55265	1.57045	0.785223	−	0.619213i	$-0.212548\pi$
$37$	9.55265	1.57045	0.785223	+	0.619213i	$0.212548\pi$
$38$	0	0
$39$	−0.0131928	−0.00211254
$40$	0	0
$41$	−1.74860	−0.273086	−0.136543	−	0.990634i	$-0.543599\pi$
$41$	−1.74860	−0.273086	−0.136543	+	0.990634i	$0.543599\pi$
$42$	0	0
$43$	11.3506	1.73095	0.865476	−	0.500951i	$-0.167016\pi$
$43$	11.3506	1.73095	0.865476	+	0.500951i	$0.167016\pi$
$44$	0	0
$45$	−8.34449	−1.24392
$46$	0	0
$47$	2.44287	0.356329	0.178165	−	0.984001i	$-0.442984\pi$
$47$	2.44287	0.356329	0.178165	+	0.984001i	$0.442984\pi$
$48$	0	0
$49$	9.66679	1.38097
$50$	0	0
$51$	−0.0137414	−0.00192418
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−2.10093	−0.283290
$56$	0	0
$57$	0.00586283	0.000776551 0
$58$	0	0
$59$	12.1287	1.57903	0.789514	−	0.613733i	$-0.210333\pi$
$59$	12.1287	1.57903	0.789514	+	0.613733i	$0.210333\pi$
$60$	0	0
$61$	8.42260	1.07840	0.539202	−	0.842177i	$-0.318726\pi$
$61$	8.42260	1.07840	0.539202	+	0.842177i	$0.318726\pi$
$62$	0	0
$63$	12.2474	1.54302
$64$	0	0
$65$	−6.25912	−0.776348
$66$	0	0
$67$	−3.97424	−0.485530	−0.242765	−	0.970085i	$-0.578054\pi$
$67$	−3.97424	−0.485530	−0.242765	+	0.970085i	$0.578054\pi$
$68$	0	0
$69$	0.0294554	0.00354601
$70$	0	0
$71$	−5.56636	−0.660605	−0.330303	−	0.943875i	$-0.607151\pi$
$71$	−5.56636	−0.660605	−0.330303	+	0.943875i	$0.607151\pi$
$72$	0	0
$73$	−9.20513	−1.07738	−0.538690	−	0.842504i	$-0.681080\pi$
$73$	−9.20513	−1.07738	−0.538690	+	0.842504i	$0.681080\pi$
$74$	0	0
$75$	0.0160460	0.00185284
$76$	0	0
$77$	3.08357	0.351406
$78$	0	0
$79$	13.6358	1.53415	0.767074	−	0.641559i	$-0.221712\pi$
$79$	13.6358	1.53415	0.767074	+	0.641559i	$0.221712\pi$
$80$	0	0
$81$	8.99969	0.999966
$82$	0	0
$83$	4.27617	0.469370	0.234685	−	0.972071i	$-0.424594\pi$
$83$	4.27617	0.469370	0.234685	+	0.972071i	$0.424594\pi$
$84$	0	0
$85$	−6.51938	−0.707126
$86$	0	0
$87$	−0.0292119	−0.00313185
$88$	0	0
$89$	3.29001	0.348740	0.174370	−	0.984680i	$-0.444211\pi$
$89$	3.29001	0.348740	0.174370	+	0.984680i	$0.444211\pi$
$90$	0	0
$91$	9.18661	0.963019
$92$	0	0
$93$	0.0150810	0.00156383
$94$	0	0
$95$	2.78153	0.285379
$96$	0	0
$97$	9.44726	0.959224	0.479612	−	0.877481i	$-0.340777\pi$
$97$	9.44726	0.959224	0.479612	+	0.877481i	$0.340777\pi$
$98$	0	0
$99$	2.26592	0.227734
$100$	0	0
$101$	9.78321	0.973466	0.486733	−	0.873551i	$-0.338188\pi$
$101$	9.78321	0.973466	0.486733	+	0.873551i	$0.338188\pi$
$102$	0	0
$103$	−5.32682	−0.524867	−0.262433	−	0.964950i	$-0.584525\pi$
$103$	−5.32682	−0.524867	−0.262433	+	0.964950i	$0.584525\pi$
$104$	0	0
$105$	−0.0665759	−0.00649714
$106$	0	0
$107$	13.1140	1.26778	0.633889	−	0.773424i	$-0.281458\pi$
$107$	13.1140	1.26778	0.633889	+	0.773424i	$0.281458\pi$
$108$	0	0
$109$	−10.6221	−1.01741	−0.508706	−	0.860940i	$-0.669876\pi$
$109$	−10.6221	−1.01741	−0.508706	+	0.860940i	$0.669876\pi$
$110$	0	0
$111$	0.0560056	0.00531582
$112$	0	0
$113$	15.6539	1.47259	0.736297	−	0.676658i	$-0.236573\pi$
$113$	15.6539	1.47259	0.736297	+	0.676658i	$0.236573\pi$
$114$	0	0
$115$	13.9747	1.30314
$116$	0	0
$117$	6.75065	0.624098
$118$	0	0
$119$	9.56860	0.877152
$120$	0	0
$121$	−10.4295	−0.948136
$122$	0	0
$123$	−0.0102518	−0.000924369 0
$124$	0	0
$125$	−6.29486	−0.563029
$126$	0	0
$127$	−4.80888	−0.426720	−0.213360	−	0.976974i	$-0.568441\pi$
$127$	−4.80888	−0.426720	−0.213360	+	0.976974i	$0.568441\pi$
$128$	0	0
$129$	0.0665467	0.00585911
$130$	0	0
$131$	3.89577	0.340375	0.170187	−	0.985412i	$-0.445563\pi$
$131$	3.89577	0.340375	0.170187	+	0.985412i	$0.445563\pi$
$132$	0	0
$133$	−4.08250	−0.353997
$134$	0	0
$135$	−0.0978453	−0.00842118
$136$	0	0
$137$	16.2278	1.38644	0.693219	−	0.720727i	$-0.256192\pi$
$137$	16.2278	1.38644	0.693219	+	0.720727i	$0.256192\pi$
$138$	0	0
$139$	−7.84856	−0.665706	−0.332853	−	0.942979i	$-0.608011\pi$
$139$	−7.84856	−0.665706	−0.332853	+	0.942979i	$0.608011\pi$
$140$	0	0
$141$	0.0143221	0.00120614
$142$	0	0
$143$	1.69964	0.142131
$144$	0	0
$145$	−13.8591	−1.15094
$146$	0	0
$147$	0.0566747	0.00467445
$148$	0	0
$149$	14.1681	1.16070	0.580349	−	0.814368i	$-0.302916\pi$
$149$	14.1681	1.16070	0.580349	+	0.814368i	$0.302916\pi$
$150$	0	0
$151$	9.86405	0.802725	0.401363	−	0.915919i	$-0.368537\pi$
$151$	9.86405	0.802725	0.401363	+	0.915919i	$0.368537\pi$
$152$	0	0
$153$	7.03135	0.568451
$154$	0	0
$155$	7.15497	0.574701
$156$	0	0
$157$	−23.0585	−1.84027	−0.920136	−	0.391599i	$-0.871922\pi$
$157$	−23.0585	−1.84027	−0.920136	+	0.391599i	$0.871922\pi$
$158$	0	0
$159$	0.00586283	0.000464953 0
$160$	0	0
$161$	−20.5108	−1.61648
$162$	0	0
$163$	1.58916	0.124472	0.0622362	−	0.998061i	$-0.480177\pi$
$163$	1.58916	0.124472	0.0622362	+	0.998061i	$0.480177\pi$
$164$	0	0
$165$	−0.0123174	−0.000958909 0
$166$	0	0
$167$	−10.0414	−0.777029	−0.388514	−	0.921443i	$-0.627012\pi$
$167$	−10.0414	−0.777029	−0.388514	+	0.921443i	$0.627012\pi$
$168$	0	0
$169$	−7.93640	−0.610493
$170$	0	0
$171$	−2.99997	−0.229413
$172$	0	0
$173$	−17.2794	−1.31373	−0.656866	−	0.754008i	$-0.728118\pi$
$173$	−17.2794	−1.31373	−0.656866	+	0.754008i	$0.728118\pi$
$174$	0	0
$175$	−11.1734	−0.844631
$176$	0	0
$177$	0.0711088	0.00534486
$178$	0	0
$179$	11.4809	0.858124	0.429062	−	0.903275i	$-0.358844\pi$
$179$	11.4809	0.858124	0.429062	+	0.903275i	$0.358844\pi$
$180$	0	0
$181$	−15.4309	−1.14697	−0.573483	−	0.819217i	$-0.694408\pi$
$181$	−15.4309	−1.14697	−0.573483	+	0.819217i	$0.694408\pi$
$182$	0	0
$183$	0.0493803	0.00365030
$184$	0	0
$185$	26.5710	1.95354
$186$	0	0
$187$	1.77032	0.129458
$188$	0	0
$189$	0.143609	0.0104460
$190$	0	0
$191$	4.83743	0.350024	0.175012	−	0.984566i	$-0.444004\pi$
$191$	4.83743	0.350024	0.175012	+	0.984566i	$0.444004\pi$
$192$	0	0
$193$	−8.79617	−0.633162	−0.316581	−	0.948566i	$-0.602535\pi$
$193$	−8.79617	−0.633162	−0.316581	+	0.948566i	$0.602535\pi$
$194$	0	0
$195$	−0.0366962	−0.00262787
$196$	0	0
$197$	−19.2018	−1.36807	−0.684035	−	0.729449i	$-0.739777\pi$
$197$	−19.2018	−1.36807	−0.684035	+	0.729449i	$0.739777\pi$
$198$	0	0
$199$	13.9541	0.989180	0.494590	−	0.869126i	$-0.335318\pi$
$199$	13.9541	0.989180	0.494590	+	0.869126i	$0.335318\pi$
$200$	0	0
$201$	−0.0233003	−0.00164347
$202$	0	0
$203$	20.3413	1.42768
$204$	0	0
$205$	−4.86379	−0.339701
$206$	0	0
$207$	−15.0721	−1.04758
$208$	0	0
$209$	−0.755316	−0.0522463
$210$	0	0
$211$	−9.66277	−0.665212	−0.332606	−	0.943066i	$-0.607928\pi$
$211$	−9.66277	−0.665212	−0.332606	+	0.943066i	$0.607928\pi$
$212$	0	0
$213$	−0.0326346	−0.00223609
$214$	0	0
$215$	31.5720	2.15320
$216$	0	0
$217$	−10.5015	−0.712886
$218$	0	0
$219$	−0.0539682	−0.00364683
$220$	0	0
$221$	5.27415	0.354777
$222$	0	0
$223$	17.2260	1.15354	0.576769	−	0.816907i	$-0.304313\pi$
$223$	17.2260	1.15354	0.576769	+	0.816907i	$0.304313\pi$
$224$	0	0
$225$	−8.21063	−0.547375
$226$	0	0
$227$	−10.2694	−0.681604	−0.340802	−	0.940135i	$-0.610699\pi$
$227$	−10.2694	−0.681604	−0.340802	+	0.940135i	$0.610699\pi$
$228$	0	0
$229$	3.57182	0.236032	0.118016	−	0.993012i	$-0.462347\pi$
$229$	3.57182	0.236032	0.118016	+	0.993012i	$0.462347\pi$
$230$	0	0
$231$	0.0180785	0.00118948
$232$	0	0
$233$	−13.7721	−0.902241	−0.451120	−	0.892463i	$-0.648975\pi$
$233$	−13.7721	−0.902241	−0.451120	+	0.892463i	$0.648975\pi$
$234$	0	0
$235$	6.79491	0.443251
$236$	0	0
$237$	0.0799444	0.00519295
$238$	0	0
$239$	18.7520	1.21297	0.606484	−	0.795096i	$-0.292579\pi$
$239$	18.7520	1.21297	0.606484	+	0.795096i	$0.292579\pi$
$240$	0	0
$241$	−25.1340	−1.61902	−0.809512	−	0.587103i	$-0.800268\pi$
$241$	−25.1340	−1.61902	−0.809512	+	0.587103i	$0.800268\pi$
$242$	0	0
$243$	0.158294	0.0101546
$244$	0	0
$245$	26.8885	1.71784
$246$	0	0
$247$	−2.25024	−0.143180
$248$	0	0
$249$	0.0250705	0.00158878
$250$	0	0
$251$	−15.3690	−0.970083	−0.485042	−	0.874491i	$-0.661196\pi$
$251$	−15.3690	−0.970083	−0.485042	+	0.874491i	$0.661196\pi$
$252$	0	0
$253$	−3.79477	−0.238575
$254$	0	0
$255$	−0.0382220	−0.00239356
$256$	0	0
$257$	−1.81353	−0.113125	−0.0565624	−	0.998399i	$-0.518014\pi$
$257$	−1.81353	−0.113125	−0.0565624	+	0.998399i	$0.518014\pi$
$258$	0	0
$259$	−38.9987	−2.42326
$260$	0	0
$261$	14.9475	0.925227
$262$	0	0
$263$	4.08105	0.251649	0.125824	−	0.992053i	$-0.459842\pi$
$263$	4.08105	0.251649	0.125824	+	0.992053i	$0.459842\pi$
$264$	0	0
$265$	2.78153	0.170868
$266$	0	0
$267$	0.0192888	0.00118045
$268$	0	0
$269$	19.5208	1.19021	0.595103	−	0.803649i	$-0.297111\pi$
$269$	19.5208	1.19021	0.595103	+	0.803649i	$0.297111\pi$
$270$	0	0
$271$	20.3423	1.23571	0.617854	−	0.786293i	$-0.288002\pi$
$271$	20.3423	1.23571	0.617854	+	0.786293i	$0.288002\pi$
$272$	0	0
$273$	0.0538596	0.00325973
$274$	0	0
$275$	−2.06723	−0.124659
$276$	0	0
$277$	7.84882	0.471590	0.235795	−	0.971803i	$-0.424231\pi$
$277$	7.84882	0.471590	0.235795	+	0.971803i	$0.424231\pi$
$278$	0	0
$279$	−7.71685	−0.461996
$280$	0	0
$281$	−9.27503	−0.553302	−0.276651	−	0.960970i	$-0.589225\pi$
$281$	−9.27503	−0.553302	−0.276651	+	0.960970i	$0.589225\pi$
$282$	0	0
$283$	22.6886	1.34870	0.674348	−	0.738413i	$-0.264425\pi$
$283$	22.6886	1.34870	0.674348	+	0.738413i	$0.264425\pi$
$284$	0	0
$285$	0.0163076	0.000965981 0
$286$	0	0
$287$	7.13866	0.421382
$288$	0	0
$289$	−11.5066	−0.676856
$290$	0	0
$291$	0.0553877	0.00324689
$292$	0	0
$293$	−14.5326	−0.849002	−0.424501	−	0.905427i	$-0.639550\pi$
$293$	−14.5326	−0.849002	−0.424501	+	0.905427i	$0.639550\pi$
$294$	0	0
$295$	33.7365	1.96421
$296$	0	0
$297$	0.0265696	0.00154172
$298$	0	0
$299$	−11.3054	−0.653810
$300$	0	0
$301$	−46.3388	−2.67093
$302$	0	0
$303$	0.0573573	0.00329509
$304$	0	0
$305$	23.4277	1.34147
$306$	0	0
$307$	−10.2488	−0.584928	−0.292464	−	0.956277i	$-0.594475\pi$
$307$	−10.2488	−0.584928	−0.292464	+	0.956277i	$0.594475\pi$
$308$	0	0
$309$	−0.0312302	−0.00177663
$310$	0	0
$311$	−5.07396	−0.287718	−0.143859	−	0.989598i	$-0.545951\pi$
$311$	−5.07396	−0.287718	−0.143859	+	0.989598i	$0.545951\pi$
$312$	0	0
$313$	1.92725	0.108934	0.0544672	−	0.998516i	$-0.482654\pi$
$313$	1.92725	0.108934	0.0544672	+	0.998516i	$0.482654\pi$
$314$	0	0
$315$	34.0664	1.91942
$316$	0	0
$317$	6.83403	0.383838	0.191919	−	0.981411i	$-0.438529\pi$
$317$	6.83403	0.383838	0.191919	+	0.981411i	$0.438529\pi$
$318$	0	0
$319$	3.76341	0.210710
$320$	0	0
$321$	0.0768851	0.00429131
$322$	0	0
$323$	−2.34381	−0.130413
$324$	0	0
$325$	−6.15871	−0.341624
$326$	0	0
$327$	−0.0622755	−0.00344384
$328$	0	0
$329$	−9.97301	−0.549830
$330$	0	0
$331$	12.2403	0.672787	0.336394	−	0.941721i	$-0.390793\pi$
$331$	12.2403	0.672787	0.336394	+	0.941721i	$0.390793\pi$
$332$	0	0
$333$	−28.6576	−1.57043
$334$	0	0
$335$	−11.0545	−0.603969
$336$	0	0
$337$	29.2949	1.59580	0.797898	−	0.602793i	$-0.205945\pi$
$337$	29.2949	1.59580	0.797898	+	0.602793i	$0.205945\pi$
$338$	0	0
$339$	0.0917761	0.00498460
$340$	0	0
$341$	−1.94291	−0.105214
$342$	0	0
$343$	−10.8871	−0.587851
$344$	0	0
$345$	0.0819311	0.00441102
$346$	0	0
$347$	1.49244	0.0801183	0.0400592	−	0.999197i	$-0.487245\pi$
$347$	1.49244	0.0801183	0.0400592	+	0.999197i	$0.487245\pi$
$348$	0	0
$349$	23.9337	1.28114	0.640571	−	0.767899i	$-0.278698\pi$
$349$	23.9337	1.28114	0.640571	+	0.767899i	$0.278698\pi$
$350$	0	0
$351$	0.0791563	0.00422505
$352$	0	0
$353$	23.4661	1.24897	0.624487	−	0.781036i	$-0.285308\pi$
$353$	23.4661	1.24897	0.624487	+	0.781036i	$0.285308\pi$
$354$	0	0
$355$	−15.4830	−0.821752
$356$	0	0
$357$	0.0560991	0.00296908
$358$	0	0
$359$	−1.27386	−0.0672320	−0.0336160	−	0.999435i	$-0.510702\pi$
$359$	−1.27386	−0.0672320	−0.0336160	+	0.999435i	$0.510702\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.0611464	−0.00320935
$364$	0	0
$365$	−25.6044	−1.34019
$366$	0	0
$367$	−0.627811	−0.0327715	−0.0163857	−	0.999866i	$-0.505216\pi$
$367$	−0.627811	−0.0327715	−0.0163857	+	0.999866i	$0.505216\pi$
$368$	0	0
$369$	5.24574	0.273082
$370$	0	0
$371$	−4.08250	−0.211953
$372$	0	0
$373$	28.7832	1.49034	0.745168	−	0.666876i	$-0.232369\pi$
$373$	28.7832	1.49034	0.745168	+	0.666876i	$0.232369\pi$
$374$	0	0
$375$	−0.0369057	−0.00190580
$376$	0	0
$377$	11.2120	0.577446
$378$	0	0
$379$	0.314260	0.0161425	0.00807124	−	0.999967i	$-0.497431\pi$
$379$	0.314260	0.0161425	0.00807124	+	0.999967i	$0.497431\pi$
$380$	0	0
$381$	−0.0281937	−0.00144441
$382$	0	0
$383$	10.5371	0.538419	0.269209	−	0.963082i	$-0.413238\pi$
$383$	10.5371	0.538419	0.269209	+	0.963082i	$0.413238\pi$
$384$	0	0
$385$	8.57705	0.437127
$386$	0	0
$387$	−34.0514	−1.73093
$388$	0	0
$389$	−10.3864	−0.526614	−0.263307	−	0.964712i	$-0.584813\pi$
$389$	−10.3864	−0.526614	−0.263307	+	0.964712i	$0.584813\pi$
$390$	0	0
$391$	−11.7755	−0.595514
$392$	0	0
$393$	0.0228402	0.00115214
$394$	0	0
$395$	37.9284	1.90838
$396$	0	0
$397$	−10.9088	−0.547499	−0.273750	−	0.961801i	$-0.588264\pi$
$397$	−10.9088	−0.547499	−0.273750	+	0.961801i	$0.588264\pi$
$398$	0	0
$399$	−0.0239350	−0.00119825
$400$	0	0
$401$	29.6639	1.48135	0.740673	−	0.671865i	$-0.234507\pi$
$401$	29.6639	1.48135	0.740673	+	0.671865i	$0.234507\pi$
$402$	0	0
$403$	−5.78833	−0.288337
$404$	0	0
$405$	25.0329	1.24390
$406$	0	0
$407$	−7.21527	−0.357648
$408$	0	0
$409$	19.4051	0.959520	0.479760	−	0.877400i	$-0.340724\pi$
$409$	19.4051	0.959520	0.479760	+	0.877400i	$0.340724\pi$
$410$	0	0
$411$	0.0951410	0.00469296
$412$	0	0
$413$	−49.5156	−2.43650
$414$	0	0
$415$	11.8943	0.583868
$416$	0	0
$417$	−0.0460148	−0.00225335
$418$	0	0
$419$	9.34682	0.456622	0.228311	−	0.973588i	$-0.426680\pi$
$419$	9.34682	0.456622	0.228311	+	0.973588i	$0.426680\pi$
$420$	0	0
$421$	−18.9486	−0.923499	−0.461750	−	0.887010i	$-0.652778\pi$
$421$	−18.9486	−0.923499	−0.461750	+	0.887010i	$0.652778\pi$
$422$	0	0
$423$	−7.32852	−0.356325
$424$	0	0
$425$	−6.41479	−0.311163
$426$	0	0
$427$	−34.3853	−1.66402
$428$	0	0
$429$	0.00996473	0.000481102 0
$430$	0	0
$431$	−29.9876	−1.44445	−0.722226	−	0.691657i	$-0.756881\pi$
$431$	−29.9876	−1.44445	−0.722226	+	0.691657i	$0.756881\pi$
$432$	0	0
$433$	4.28924	0.206128	0.103064	−	0.994675i	$-0.467135\pi$
$433$	4.28924	0.206128	0.103064	+	0.994675i	$0.467135\pi$
$434$	0	0
$435$	−0.0812538	−0.00389582
$436$	0	0
$437$	5.02409	0.240335
$438$	0	0
$439$	−5.55883	−0.265308	−0.132654	−	0.991162i	$-0.542350\pi$
$439$	−5.55883	−0.265308	−0.132654	+	0.991162i	$0.542350\pi$
$440$	0	0
$441$	−29.0000	−1.38095
$442$	0	0
$443$	4.67625	0.222176	0.111088	−	0.993811i	$-0.464567\pi$
$443$	4.67625	0.222176	0.111088	+	0.993811i	$0.464567\pi$
$444$	0	0
$445$	9.15125	0.433811
$446$	0	0
$447$	0.0830653	0.00392885
$448$	0	0
$449$	9.79484	0.462247	0.231124	−	0.972924i	$-0.425760\pi$
$449$	9.79484	0.462247	0.231124	+	0.972924i	$0.425760\pi$
$450$	0	0
$451$	1.32075	0.0621915
$452$	0	0
$453$	0.0578313	0.00271715
$454$	0	0
$455$	25.5528	1.19794
$456$	0	0
$457$	34.2988	1.60443	0.802214	−	0.597036i	$-0.203655\pi$
$457$	34.2988	1.60443	0.802214	+	0.597036i	$0.203655\pi$
$458$	0	0
$459$	0.0824477	0.00384833
$460$	0	0
$461$	19.7206	0.918480	0.459240	−	0.888312i	$-0.348122\pi$
$461$	19.7206	0.918480	0.459240	+	0.888312i	$0.348122\pi$
$462$	0	0
$463$	28.1823	1.30974	0.654872	−	0.755740i	$-0.272722\pi$
$463$	28.1823	1.30974	0.654872	+	0.755740i	$0.272722\pi$
$464$	0	0
$465$	0.0419484	0.00194531
$466$	0	0
$467$	29.0763	1.34549	0.672744	−	0.739875i	$-0.265115\pi$
$467$	29.0763	1.34549	0.672744	+	0.739875i	$0.265115\pi$
$468$	0	0
$469$	16.2248	0.749192
$470$	0	0
$471$	−0.135188	−0.00622915
$472$	0	0
$473$	−8.57329	−0.394200
$474$	0	0
$475$	2.73691	0.125578
$476$	0	0
$477$	−2.99997	−0.137359
$478$	0	0
$479$	12.8523	0.587236	0.293618	−	0.955923i	$-0.405141\pi$
$479$	12.8523	0.587236	0.293618	+	0.955923i	$0.405141\pi$
$480$	0	0
$481$	−21.4958	−0.980124
$482$	0	0
$483$	−0.120252	−0.00547164
$484$	0	0
$485$	26.2778	1.19322
$486$	0	0
$487$	40.0963	1.81694	0.908469	−	0.417953i	$-0.137252\pi$
$487$	40.0963	1.81694	0.908469	+	0.417953i	$0.137252\pi$
$488$	0	0
$489$	0.00931696	0.000421328 0
$490$	0	0
$491$	−26.0504	−1.17564	−0.587820	−	0.808992i	$-0.700014\pi$
$491$	−26.0504	−1.17564	−0.587820	+	0.808992i	$0.700014\pi$
$492$	0	0
$493$	11.6782	0.525959
$494$	0	0
$495$	6.30273	0.283286
$496$	0	0
$497$	22.7246	1.01934
$498$	0	0
$499$	16.2862	0.729072	0.364536	−	0.931189i	$-0.381228\pi$
$499$	16.2862	0.729072	0.364536	+	0.931189i	$0.381228\pi$
$500$	0	0
$501$	−0.0588712	−0.00263017
$502$	0	0
$503$	−18.3919	−0.820056	−0.410028	−	0.912073i	$-0.634481\pi$
$503$	−18.3919	−0.820056	−0.410028	+	0.912073i	$0.634481\pi$
$504$	0	0
$505$	27.2123	1.21093
$506$	0	0
$507$	−0.0465298	−0.00206646
$508$	0	0
$509$	−6.11411	−0.271003	−0.135502	−	0.990777i	$-0.543265\pi$
$509$	−6.11411	−0.271003	−0.135502	+	0.990777i	$0.543265\pi$
$510$	0	0
$511$	37.5799	1.66244
$512$	0	0
$513$	−0.0351768	−0.00155309
$514$	0	0
$515$	−14.8167	−0.652902
$516$	0	0
$517$	−1.84514	−0.0811490
$518$	0	0
$519$	−0.101306	−0.00444686
$520$	0	0
$521$	23.1537	1.01438	0.507192	−	0.861833i	$-0.330683\pi$
$521$	23.1537	1.01438	0.507192	+	0.861833i	$0.330683\pi$
$522$	0	0
$523$	1.42721	0.0624077	0.0312039	−	0.999513i	$-0.490066\pi$
$523$	1.42721	0.0624077	0.0312039	+	0.999513i	$0.490066\pi$
$524$	0	0
$525$	−0.0655079	−0.00285900
$526$	0	0
$527$	−6.02902	−0.262628
$528$	0	0
$529$	2.24149	0.0974561
$530$	0	0
$531$	−36.3858	−1.57901
$532$	0	0
$533$	3.93478	0.170434
$534$	0	0
$535$	36.4770	1.57704
$536$	0	0
$537$	0.0673107	0.00290467
$538$	0	0
$539$	−7.30147	−0.314497
$540$	0	0
$541$	5.74893	0.247166	0.123583	−	0.992334i	$-0.460562\pi$
$541$	5.74893	0.247166	0.123583	+	0.992334i	$0.460562\pi$
$542$	0	0
$543$	−0.0904685	−0.00388238
$544$	0	0
$545$	−29.5457	−1.26560
$546$	0	0
$547$	12.4548	0.532529	0.266264	−	0.963900i	$-0.414211\pi$
$547$	12.4548	0.532529	0.266264	+	0.963900i	$0.414211\pi$
$548$	0	0
$549$	−25.2675	−1.07839
$550$	0	0
$551$	−4.98256	−0.212264
$552$	0	0
$553$	−55.6681	−2.36725
$554$	0	0
$555$	0.155781	0.00661254
$556$	0	0
$557$	20.5820	0.872087	0.436043	−	0.899926i	$-0.356379\pi$
$557$	20.5820	0.872087	0.436043	+	0.899926i	$0.356379\pi$
$558$	0	0
$559$	−25.5416	−1.08030
$560$	0	0
$561$	0.0103791	0.000438205 0
$562$	0	0
$563$	−42.3645	−1.78545	−0.892724	−	0.450603i	$-0.851209\pi$
$563$	−42.3645	−1.78545	−0.892724	+	0.450603i	$0.851209\pi$
$564$	0	0
$565$	43.5418	1.83182
$566$	0	0
$567$	−36.7412	−1.54299
$568$	0	0
$569$	22.4670	0.941865	0.470933	−	0.882169i	$-0.343918\pi$
$569$	22.4670	0.941865	0.470933	+	0.882169i	$0.343918\pi$
$570$	0	0
$571$	−33.1835	−1.38869	−0.694343	−	0.719644i	$-0.744305\pi$
$571$	−33.1835	−1.38869	−0.694343	+	0.719644i	$0.744305\pi$
$572$	0	0
$573$	0.0283610	0.00118480
$574$	0	0
$575$	13.7505	0.573434
$576$	0	0
$577$	29.0978	1.21136	0.605679	−	0.795709i	$-0.292901\pi$
$577$	29.0978	1.21136	0.605679	+	0.795709i	$0.292901\pi$
$578$	0	0
$579$	−0.0515704	−0.00214319
$580$	0	0
$581$	−17.4574	−0.724257
$582$	0	0
$583$	−0.755316	−0.0312820
$584$	0	0
$585$	18.7771	0.776339
$586$	0	0
$587$	1.64545	0.0679149	0.0339574	−	0.999423i	$-0.489189\pi$
$587$	1.64545	0.0679149	0.0339574	+	0.999423i	$0.489189\pi$
$588$	0	0
$589$	2.57231	0.105990
$590$	0	0
$591$	−0.112577	−0.00463079
$592$	0	0
$593$	−15.5718	−0.639457	−0.319728	−	0.947509i	$-0.603592\pi$
$593$	−15.5718	−0.639457	−0.319728	+	0.947509i	$0.603592\pi$
$594$	0	0
$595$	26.6153	1.09112
$596$	0	0
$597$	0.0818106	0.00334828
$598$	0	0
$599$	−10.7870	−0.440744	−0.220372	−	0.975416i	$-0.570727\pi$
$599$	−10.7870	−0.440744	−0.220372	+	0.975416i	$0.570727\pi$
$600$	0	0
$601$	−7.02598	−0.286596	−0.143298	−	0.989680i	$-0.545771\pi$
$601$	−7.02598	−0.286596	−0.143298	+	0.989680i	$0.545771\pi$
$602$	0	0
$603$	11.9226	0.485525
$604$	0	0
$605$	−29.0100	−1.17942
$606$	0	0
$607$	14.4058	0.584712	0.292356	−	0.956310i	$-0.405561\pi$
$607$	14.4058	0.584712	0.292356	+	0.956310i	$0.405561\pi$
$608$	0	0
$609$	0.119258	0.00483256
$610$	0	0
$611$	−5.49705	−0.222387
$612$	0	0
$613$	0.773237	0.0312307	0.0156154	−	0.999878i	$-0.495029\pi$
$613$	0.773237	0.0312307	0.0156154	+	0.999878i	$0.495029\pi$
$614$	0	0
$615$	−0.0285156	−0.00114986
$616$	0	0
$617$	−49.3984	−1.98870	−0.994352	−	0.106132i	$-0.966153\pi$
$617$	−49.3984	−1.98870	−0.994352	+	0.106132i	$0.966153\pi$
$618$	0	0
$619$	−38.7341	−1.55685	−0.778427	−	0.627736i	$-0.783982\pi$
$619$	−38.7341	−1.55685	−0.778427	+	0.627736i	$0.783982\pi$
$620$	0	0
$621$	−0.176731	−0.00709199
$622$	0	0
$623$	−13.4314	−0.538119
$624$	0	0
$625$	−31.1939	−1.24776
$626$	0	0
$627$	−0.00442829	−0.000176849 0
$628$	0	0
$629$	−22.3896	−0.892732
$630$	0	0
$631$	0.0350292	0.00139449	0.000697246	−	1.00000i	$-0.499778\pi$
$631$	0.0350292	0.00139449	0.000697246		1.00000i	$0.499778\pi$
$632$	0	0
$633$	−0.0566512	−0.00225168
$634$	0	0
$635$	−13.3761	−0.530813
$636$	0	0
$637$	−21.7526	−0.861870
$638$	0	0
$639$	16.6989	0.660598
$640$	0	0
$641$	−21.6860	−0.856546	−0.428273	−	0.903650i	$-0.640878\pi$
$641$	−21.6860	−0.856546	−0.428273	+	0.903650i	$0.640878\pi$
$642$	0	0
$643$	41.0641	1.61941	0.809705	−	0.586836i	$-0.199627\pi$
$643$	41.0641	1.61941	0.809705	+	0.586836i	$0.199627\pi$
$644$	0	0
$645$	0.185102	0.00728837
$646$	0	0
$647$	18.1761	0.714578	0.357289	−	0.933994i	$-0.383701\pi$
$647$	18.1761	0.714578	0.357289	+	0.933994i	$0.383701\pi$
$648$	0	0
$649$	−9.16103	−0.359602
$650$	0	0
$651$	−0.0615683	−0.00241305
$652$	0	0
$653$	−3.50144	−0.137022	−0.0685110	−	0.997650i	$-0.521825\pi$
$653$	−3.50144	−0.137022	−0.0685110	+	0.997650i	$0.521825\pi$
$654$	0	0
$655$	10.8362	0.423405
$656$	0	0
$657$	27.6151	1.07737
$658$	0	0
$659$	−0.946157	−0.0368571	−0.0184285	−	0.999830i	$-0.505866\pi$
$659$	−0.946157	−0.0368571	−0.0184285	+	0.999830i	$0.505866\pi$
$660$	0	0
$661$	−43.8148	−1.70420	−0.852099	−	0.523381i	$-0.824671\pi$
$661$	−43.8148	−1.70420	−0.852099	+	0.523381i	$0.824671\pi$
$662$	0	0
$663$	0.0309214	0.00120089
$664$	0	0
$665$	−11.3556	−0.440351
$666$	0	0
$667$	−25.0328	−0.969275
$668$	0	0
$669$	0.100993	0.00390462
$670$	0	0
$671$	−6.36172	−0.245592
$672$	0	0
$673$	39.0538	1.50541	0.752707	−	0.658356i	$-0.228748\pi$
$673$	39.0538	1.50541	0.752707	+	0.658356i	$0.228748\pi$
$674$	0	0
$675$	−0.0962756	−0.00370565
$676$	0	0
$677$	−31.2185	−1.19982	−0.599912	−	0.800066i	$-0.704798\pi$
$677$	−31.2185	−1.19982	−0.599912	+	0.800066i	$0.704798\pi$
$678$	0	0
$679$	−38.5684	−1.48012
$680$	0	0
$681$	−0.0602078	−0.00230717
$682$	0	0
$683$	29.2042	1.11747	0.558734	−	0.829347i	$-0.311287\pi$
$683$	29.2042	1.11747	0.558734	+	0.829347i	$0.311287\pi$
$684$	0	0
$685$	45.1382	1.72464
$686$	0	0
$687$	0.0209410	0.000798948 0
$688$	0	0
$689$	−2.25024	−0.0857275
$690$	0	0
$691$	−25.5895	−0.973471	−0.486736	−	0.873549i	$-0.661812\pi$
$691$	−25.5895	−0.973471	−0.486736	+	0.873549i	$0.661812\pi$
$692$	0	0
$693$	−9.25062	−0.351402
$694$	0	0
$695$	−21.8310	−0.828097
$696$	0	0
$697$	4.09839	0.155238
$698$	0	0
$699$	−0.0807435	−0.00305400
$700$	0	0
$701$	7.89262	0.298100	0.149050	−	0.988830i	$-0.452378\pi$
$701$	7.89262	0.298100	0.149050	+	0.988830i	$0.452378\pi$
$702$	0	0
$703$	9.55265	0.360285
$704$	0	0
$705$	0.0398374	0.00150036
$706$	0	0
$707$	−39.9399	−1.50210
$708$	0	0
$709$	−34.8589	−1.30915	−0.654576	−	0.755996i	$-0.727153\pi$
$709$	−34.8589	−1.30915	−0.654576	+	0.755996i	$0.727153\pi$
$710$	0	0
$711$	−40.9069	−1.53413
$712$	0	0
$713$	12.9235	0.483990
$714$	0	0
$715$	4.72761	0.176803
$716$	0	0
$717$	0.109940	0.00410579
$718$	0	0
$719$	45.7788	1.70726	0.853631	−	0.520879i	$-0.174396\pi$
$719$	45.7788	1.70726	0.853631	+	0.520879i	$0.174396\pi$
$720$	0	0
$721$	21.7467	0.809890
$722$	0	0
$723$	−0.147356	−0.00548025
$724$	0	0
$725$	−13.6368	−0.506458
$726$	0	0
$727$	−41.3733	−1.53445	−0.767225	−	0.641379i	$-0.778363\pi$
$727$	−41.3733	−1.53445	−0.767225	+	0.641379i	$0.778363\pi$
$728$	0	0
$729$	−26.9981	−0.999931
$730$	0	0
$731$	−26.6037	−0.983972
$732$	0	0
$733$	40.4851	1.49535	0.747675	−	0.664065i	$-0.231170\pi$
$733$	40.4851	1.49535	0.747675	+	0.664065i	$0.231170\pi$
$734$	0	0
$735$	0.157642	0.00581473
$736$	0	0
$737$	3.00180	0.110573
$738$	0	0
$739$	−23.7603	−0.874037	−0.437019	−	0.899452i	$-0.643966\pi$
$739$	−23.7603	−0.874037	−0.437019	+	0.899452i	$0.643966\pi$
$740$	0	0
$741$	−0.0131928	−0.000484650 0
$742$	0	0
$743$	28.1082	1.03119	0.515595	−	0.856832i	$-0.327571\pi$
$743$	28.1082	1.03119	0.515595	+	0.856832i	$0.327571\pi$
$744$	0	0
$745$	39.4091	1.44384
$746$	0	0
$747$	−12.8284	−0.469365
$748$	0	0
$749$	−53.5378	−1.95623
$750$	0	0
$751$	−4.83275	−0.176349	−0.0881747	−	0.996105i	$-0.528103\pi$
$751$	−4.83275	−0.176349	−0.0881747	+	0.996105i	$0.528103\pi$
$752$	0	0
$753$	−0.0901059	−0.00328364
$754$	0	0
$755$	27.4372	0.998540
$756$	0	0
$757$	−31.2588	−1.13612	−0.568061	−	0.822986i	$-0.692306\pi$
$757$	−31.2588	−1.13612	−0.568061	+	0.822986i	$0.692306\pi$
$758$	0	0
$759$	−0.0222481	−0.000807556 0
$760$	0	0
$761$	−25.8426	−0.936794	−0.468397	−	0.883518i	$-0.655168\pi$
$761$	−25.8426	−0.936794	−0.468397	+	0.883518i	$0.655168\pi$
$762$	0	0
$763$	43.3647	1.56991
$764$	0	0
$765$	19.5579	0.707118
$766$	0	0
$767$	−27.2926	−0.985480
$768$	0	0
$769$	−31.0014	−1.11794	−0.558970	−	0.829188i	$-0.688803\pi$
$769$	−31.0014	−1.11794	−0.558970	+	0.829188i	$0.688803\pi$
$770$	0	0
$771$	−0.0106324	−0.000382917 0
$772$	0	0
$773$	−5.46412	−0.196531	−0.0982654	−	0.995160i	$-0.531329\pi$
$773$	−5.46412	−0.196531	−0.0982654	+	0.995160i	$0.531329\pi$
$774$	0	0
$775$	7.04018	0.252891
$776$	0	0
$777$	−0.228643	−0.00820251
$778$	0	0
$779$	−1.74860	−0.0626501
$780$	0	0
$781$	4.20436	0.150444
$782$	0	0
$783$	0.175270	0.00626365
$784$	0	0
$785$	−64.1380	−2.28918
$786$	0	0
$787$	−13.3741	−0.476734	−0.238367	−	0.971175i	$-0.576612\pi$
$787$	−13.3741	−0.476734	−0.238367	+	0.971175i	$0.576612\pi$
$788$	0	0
$789$	0.0239265	0.000851807 0
$790$	0	0
$791$	−63.9070	−2.27227
$792$	0	0
$793$	−18.9529	−0.673038
$794$	0	0
$795$	0.0163076	0.000578372 0
$796$	0	0
$797$	31.0111	1.09847	0.549234	−	0.835669i	$-0.314920\pi$
$797$	31.0111	1.09847	0.549234	+	0.835669i	$0.314920\pi$
$798$	0	0
$799$	−5.72562	−0.202558
$800$	0	0
$801$	−9.86991	−0.348736
$802$	0	0
$803$	6.95278	0.245358
$804$	0	0
$805$	−57.0515	−2.01080
$806$	0	0
$807$	0.114447	0.00402874
$808$	0	0
$809$	32.7566	1.15166	0.575830	−	0.817569i	$-0.304679\pi$
$809$	32.7566	1.15166	0.575830	+	0.817569i	$0.304679\pi$
$810$	0	0
$811$	−51.9694	−1.82489	−0.912447	−	0.409195i	$-0.865809\pi$
$811$	−51.9694	−1.82489	−0.912447	+	0.409195i	$0.865809\pi$
$812$	0	0
$813$	0.119264	0.00418275
$814$	0	0
$815$	4.42029	0.154836
$816$	0	0
$817$	11.3506	0.397107
$818$	0	0
$819$	−27.5595	−0.963008
$820$	0	0
$821$	−23.8146	−0.831135	−0.415568	−	0.909562i	$-0.636417\pi$
$821$	−23.8146	−0.831135	−0.415568	+	0.909562i	$0.636417\pi$
$822$	0	0
$823$	−13.3713	−0.466095	−0.233047	−	0.972465i	$-0.574870\pi$
$823$	−13.3713	−0.466095	−0.233047	+	0.972465i	$0.574870\pi$
$824$	0	0
$825$	−0.0121198	−0.000421958 0
$826$	0	0
$827$	18.3120	0.636771	0.318386	−	0.947961i	$-0.396859\pi$
$827$	18.3120	0.636771	0.318386	+	0.947961i	$0.396859\pi$
$828$	0	0
$829$	−10.1093	−0.351109	−0.175554	−	0.984470i	$-0.556172\pi$
$829$	−10.1093	−0.351109	−0.175554	+	0.984470i	$0.556172\pi$
$830$	0	0
$831$	0.0460163	0.00159629
$832$	0	0
$833$	−22.6571	−0.785023
$834$	0	0
$835$	−27.9305	−0.966576
$836$	0	0
$837$	−0.0904857	−0.00312764
$838$	0	0
$839$	2.05796	0.0710485	0.0355243	−	0.999369i	$-0.488690\pi$
$839$	2.05796	0.0710485	0.0355243	+	0.999369i	$0.488690\pi$
$840$	0	0
$841$	−4.17410	−0.143934
$842$	0	0
$843$	−0.0543779	−0.00187288
$844$	0	0
$845$	−22.0753	−0.759415
$846$	0	0
$847$	42.5784	1.46301
$848$	0	0
$849$	0.133019	0.00456521
$850$	0	0
$851$	47.9934	1.64519
$852$	0	0
$853$	−0.317193	−0.0108605	−0.00543025	−	0.999985i	$-0.501729\pi$
$853$	−0.317193	−0.0108605	−0.00543025	+	0.999985i	$0.501729\pi$
$854$	0	0
$855$	−8.34449	−0.285376
$856$	0	0
$857$	28.3266	0.967617	0.483809	−	0.875174i	$-0.339253\pi$
$857$	28.3266	0.967617	0.483809	+	0.875174i	$0.339253\pi$
$858$	0	0
$859$	0.642069	0.0219071	0.0109536	−	0.999940i	$-0.496513\pi$
$859$	0.642069	0.0219071	0.0109536	+	0.999940i	$0.496513\pi$
$860$	0	0
$861$	0.0418528	0.00142634
$862$	0	0
$863$	31.1563	1.06057	0.530286	−	0.847819i	$-0.322085\pi$
$863$	31.1563	1.06057	0.530286	+	0.847819i	$0.322085\pi$
$864$	0	0
$865$	−48.0633	−1.63420
$866$	0	0
$867$	−0.0674610	−0.00229109
$868$	0	0
$869$	−10.2993	−0.349381
$870$	0	0
$871$	8.94300	0.303022
$872$	0	0
$873$	−28.3415	−0.959213
$874$	0	0
$875$	25.6987	0.868776
$876$	0	0
$877$	35.0086	1.18216	0.591078	−	0.806615i	$-0.298703\pi$
$877$	35.0086	1.18216	0.591078	+	0.806615i	$0.298703\pi$
$878$	0	0
$879$	−0.0852020	−0.00287379
$880$	0	0
$881$	−8.05568	−0.271403	−0.135701	−	0.990750i	$-0.543329\pi$
$881$	−8.05568	−0.271403	−0.135701	+	0.990750i	$0.543329\pi$
$882$	0	0
$883$	−0.132307	−0.00445248	−0.00222624	−	0.999998i	$-0.500709\pi$
$883$	−0.132307	−0.00445248	−0.00222624	+	0.999998i	$0.500709\pi$
$884$	0	0
$885$	0.197791	0.00664868
$886$	0	0
$887$	−36.6813	−1.23164	−0.615819	−	0.787888i	$-0.711175\pi$
$887$	−36.6813	−1.23164	−0.615819	+	0.787888i	$0.711175\pi$
$888$	0	0
$889$	19.6323	0.658445
$890$	0	0
$891$	−6.79761	−0.227728
$892$	0	0
$893$	2.44287	0.0817475
$894$	0	0
$895$	31.9345	1.06745
$896$	0	0
$897$	−0.0662818	−0.00221309
$898$	0	0
$899$	−12.8167	−0.427461
$900$	0	0
$901$	−2.34381	−0.0780837
$902$	0	0
$903$	−0.271677	−0.00904083
$904$	0	0
$905$	−42.9214	−1.42676
$906$	0	0
$907$	−17.6441	−0.585863	−0.292931	−	0.956133i	$-0.594631\pi$
$907$	−17.6441	−0.585863	−0.292931	+	0.956133i	$0.594631\pi$
$908$	0	0
$909$	−29.3493	−0.973455
$910$	0	0
$911$	12.7625	0.422841	0.211421	−	0.977395i	$-0.432191\pi$
$911$	12.7625	0.422841	0.211421	+	0.977395i	$0.432191\pi$
$912$	0	0
$913$	−3.22986	−0.106893
$914$	0	0
$915$	0.137353	0.00454074
$916$	0	0
$917$	−15.9045	−0.525212
$918$	0	0
$919$	5.66800	0.186970	0.0934850	−	0.995621i	$-0.470199\pi$
$919$	5.66800	0.186970	0.0934850	+	0.995621i	$0.470199\pi$
$920$	0	0
$921$	−0.0600868	−0.00197993
$922$	0	0
$923$	12.5257	0.412287
$924$	0	0
$925$	26.1447	0.859633
$926$	0	0
$927$	15.9803	0.524861
$928$	0	0
$929$	−16.9248	−0.555285	−0.277642	−	0.960684i	$-0.589553\pi$
$929$	−16.9248	−0.555285	−0.277642	+	0.960684i	$0.589553\pi$
$930$	0	0
$931$	9.66679	0.316816
$932$	0	0
$933$	−0.0297478	−0.000973898 0
$934$	0	0
$935$	4.92419	0.161038
$936$	0	0
$937$	11.0193	0.359985	0.179992	−	0.983668i	$-0.442393\pi$
$937$	11.0193	0.359985	0.179992	+	0.983668i	$0.442393\pi$
$938$	0	0
$939$	0.0112991	0.000368733 0
$940$	0	0
$941$	49.1217	1.60132	0.800661	−	0.599117i	$-0.204482\pi$
$941$	49.1217	1.60132	0.800661	+	0.599117i	$0.204482\pi$
$942$	0	0
$943$	−8.78513	−0.286083
$944$	0	0
$945$	0.399453	0.0129942
$946$	0	0
$947$	−6.52345	−0.211984	−0.105992	−	0.994367i	$-0.533802\pi$
$947$	−6.52345	−0.211984	−0.105992	+	0.994367i	$0.533802\pi$
$948$	0	0
$949$	20.7138	0.672398
$950$	0	0
$951$	0.0400668	0.00129925
$952$	0	0
$953$	57.5392	1.86388	0.931938	−	0.362617i	$-0.118117\pi$
$953$	57.5392	1.86388	0.931938	+	0.362617i	$0.118117\pi$
$954$	0	0
$955$	13.4555	0.435408
$956$	0	0
$957$	0.0220642	0.000713235 0
$958$	0	0
$959$	−66.2501	−2.13933
$960$	0	0
$961$	−24.3832	−0.786555
$962$	0	0
$963$	−39.3415	−1.26776
$964$	0	0
$965$	−24.4668	−0.787614
$966$	0	0
$967$	−22.8801	−0.735774	−0.367887	−	0.929870i	$-0.619919\pi$
$967$	−22.8801	−0.735774	−0.367887	+	0.929870i	$0.619919\pi$
$968$	0	0
$969$	−0.0137414	−0.000441436 0
$970$	0	0
$971$	12.4806	0.400523	0.200261	−	0.979743i	$-0.435821\pi$
$971$	12.4806	0.400523	0.200261	+	0.979743i	$0.435821\pi$
$972$	0	0
$973$	32.0417	1.02721
$974$	0	0
$975$	−0.0361075	−0.00115636
$976$	0	0
$977$	−12.5962	−0.402988	−0.201494	−	0.979490i	$-0.564580\pi$
$977$	−12.5962	−0.402988	−0.201494	+	0.979490i	$0.564580\pi$
$978$	0	0
$979$	−2.48499	−0.0794207
$980$	0	0
$981$	31.8659	1.01740
$982$	0	0
$983$	−14.2558	−0.454689	−0.227344	−	0.973814i	$-0.573004\pi$
$983$	−14.2558	−0.454689	−0.227344	+	0.973814i	$0.573004\pi$
$984$	0	0
$985$	−53.4103	−1.70179
$986$	0	0
$987$	−0.0584701	−0.00186112
$988$	0	0
$989$	57.0265	1.81334
$990$	0	0
$991$	−32.8141	−1.04237	−0.521187	−	0.853443i	$-0.674511\pi$
$991$	−32.8141	−1.04237	−0.521187	+	0.853443i	$0.674511\pi$
$992$	0	0
$993$	0.0717627	0.00227732
$994$	0	0
$995$	38.8138	1.23048
$996$	0	0
$997$	22.4881	0.712206	0.356103	−	0.934447i	$-0.384105\pi$
$997$	22.4881	0.712206	0.356103	+	0.934447i	$0.384105\pi$
$998$	0	0
$999$	−0.336032	−0.0106316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.e.1.9	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.e.1.9	✓	19	1.1	even	1	trivial

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.a (trivial)

\(f(q)\)	\(=\)	\(q+0.00586283 q^{3} +2.78153 q^{5} -4.08250 q^{7} -2.99997 q^{9} +O(q^{10})\)	\(q+0.00586283 q^{3} +2.78153 q^{5} -4.08250 q^{7} -2.99997 q^{9} -0.755316 q^{11} -2.25024 q^{13} +0.0163076 q^{15} -2.34381 q^{17} +1.00000 q^{19} -0.0239350 q^{21} +5.02409 q^{23} +2.73691 q^{25} -0.0351768 q^{27} -4.98256 q^{29} +2.57231 q^{31} -0.00442829 q^{33} -11.3556 q^{35} +9.55265 q^{37} -0.0131928 q^{39} -1.74860 q^{41} +11.3506 q^{43} -8.34449 q^{45} +2.44287 q^{47} +9.66679 q^{49} -0.0137414 q^{51} +1.00000 q^{53} -2.10093 q^{55} +0.00586283 q^{57} +12.1287 q^{59} +8.42260 q^{61} +12.2474 q^{63} -6.25912 q^{65} -3.97424 q^{67} +0.0294554 q^{69} -5.56636 q^{71} -9.20513 q^{73} +0.0160460 q^{75} +3.08357 q^{77} +13.6358 q^{79} +8.99969 q^{81} +4.27617 q^{83} -6.51938 q^{85} -0.0292119 q^{87} +3.29001 q^{89} +9.18661 q^{91} +0.0150810 q^{93} +2.78153 q^{95} +9.44726 q^{97} +2.26592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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