LMFDB - Embedded newform 304.3.e.c.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,3,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not 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:	$8.28340003655$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.3.e.c.113.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{3} -1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.82843i q^{3} -1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -16.9706i q^{13} +2.82843i q^{15} -25.0000 q^{17} -19.0000 q^{19} +14.1421i q^{21} +10.0000 q^{23} -24.0000 q^{25} -28.2843i q^{27} +42.4264i q^{29} +42.4264i q^{31} +14.1421i q^{33} +5.00000 q^{35} -25.4558i q^{37} -48.0000 q^{39} -42.4264i q^{41} -5.00000 q^{43} -1.00000 q^{45} -5.00000 q^{47} -24.0000 q^{49} +70.7107i q^{51} -25.4558i q^{53} +5.00000 q^{55} +53.7401i q^{57} -84.8528i q^{59} +95.0000 q^{61} -5.00000 q^{63} +16.9706i q^{65} -110.309i q^{67} -28.2843i q^{69} -25.0000 q^{73} +67.8823i q^{75} +25.0000 q^{77} -42.4264i q^{79} -71.0000 q^{81} +130.000 q^{83} +25.0000 q^{85} +120.000 q^{87} -127.279i q^{89} +84.8528i q^{91} +120.000 q^{93} +19.0000 q^{95} +16.9706i q^{97} -5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 10 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 10 * q^7 + 2 * q^9	$ 2 q - 2 q^{5} - 10 q^{7} + 2 q^{9} - 10 q^{11} - 50 q^{17} - 38 q^{19} + 20 q^{23} - 48 q^{25} + 10 q^{35} - 96 q^{39} - 10 q^{43} - 2 q^{45} - 10 q^{47} - 48 q^{49} + 10 q^{55} + 190 q^{61} - 10 q^{63} - 50 q^{73} + 50 q^{77} - 142 q^{81} + 260 q^{83} + 50 q^{85} + 240 q^{87} + 240 q^{93} + 38 q^{95} - 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 10 * q^7 + 2 * q^9 - 10 * q^11 - 50 * q^17 - 38 * q^19 + 20 * q^23 - 48 * q^25 + 10 * q^35 - 96 * q^39 - 10 * q^43 - 2 * q^45 - 10 * q^47 - 48 * q^49 + 10 * q^55 + 190 * q^61 - 10 * q^63 - 50 * q^73 + 50 * q^77 - 142 * q^81 + 260 * q^83 + 50 * q^85 + 240 * q^87 + 240 * q^93 + 38 * q^95 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 2.82843i	− 0.942809i	−0.881917	−	0.471405i	$-0.843747\pi$
$3$	− 2.82843i	− 0.942809i	0.881917	−	0.471405i	$-0.156253\pi$
$4$	0	0
$5$	−1.00000	−0.200000	−0.100000	−	0.994987i	$-0.531884\pi$
$5$	−1.00000	−0.200000	−0.100000	+	0.994987i	$0.531884\pi$
$6$	0	0
$7$	−5.00000	−0.714286	−0.357143	−	0.934050i	$-0.616249\pi$
$7$	−5.00000	−0.714286	−0.357143	+	0.934050i	$0.616249\pi$
$8$	0	0
$9$	1.00000	0.111111
$10$	0	0
$11$	−5.00000	−0.454545	−0.227273	−	0.973831i	$-0.572981\pi$
$11$	−5.00000	−0.454545	−0.227273	+	0.973831i	$0.572981\pi$
$12$	0	0
$13$	− 16.9706i	− 1.30543i	−0.757604	−	0.652714i	$-0.773630\pi$
$13$	− 16.9706i	− 1.30543i	0.757604	−	0.652714i	$-0.226370\pi$
$14$	0	0
$15$	2.82843i	0.188562i
$16$	0	0
$17$	−25.0000	−1.47059	−0.735294	−	0.677748i	$-0.762956\pi$
$17$	−25.0000	−1.47059	−0.735294	+	0.677748i	$0.762956\pi$
$18$	0	0
$19$	−19.0000	−1.00000
$19$	−19.0000	−1.00000
$20$	0	0
$21$	14.1421i	0.673435i
$22$	0	0
$23$	10.0000	0.434783	0.217391	−	0.976085i	$-0.430245\pi$
$23$	10.0000	0.434783	0.217391	+	0.976085i	$0.430245\pi$
$24$	0	0
$25$	−24.0000	−0.960000
$26$	0	0
$27$	− 28.2843i	− 1.04757i
$28$	0	0
$29$	42.4264i	1.46298i	0.681852	+	0.731490i	$0.261175\pi$
$29$	42.4264i	1.46298i	−0.681852	+	0.731490i	$0.738825\pi$
$30$	0	0
$31$	42.4264i	1.36859i	0.729204	+	0.684297i	$0.239891\pi$
$31$	42.4264i	1.36859i	−0.729204	+	0.684297i	$0.760109\pi$
$32$	0	0
$33$	14.1421i	0.428550i
$34$	0	0
$35$	5.00000	0.142857
$36$	0	0
$37$	− 25.4558i	− 0.687996i	−0.938970	−	0.343998i	$-0.888219\pi$
$37$	− 25.4558i	− 0.687996i	0.938970	−	0.343998i	$-0.111781\pi$
$38$	0	0
$39$	−48.0000	−1.23077
$40$	0	0
$41$	− 42.4264i	− 1.03479i	−0.855747	−	0.517395i	$-0.826902\pi$
$41$	− 42.4264i	− 1.03479i	0.855747	−	0.517395i	$-0.173098\pi$
$42$	0	0
$43$	−5.00000	−0.116279	−0.0581395	−	0.998308i	$-0.518517\pi$
$43$	−5.00000	−0.116279	−0.0581395	+	0.998308i	$0.518517\pi$
$44$	0	0
$45$	−1.00000	−0.0222222
$46$	0	0
$47$	−5.00000	−0.106383	−0.0531915	−	0.998584i	$-0.516939\pi$
$47$	−5.00000	−0.106383	−0.0531915	+	0.998584i	$0.516939\pi$
$48$	0	0
$49$	−24.0000	−0.489796
$50$	0	0
$51$	70.7107i	1.38648i
$52$	0	0
$53$	− 25.4558i	− 0.480299i	−0.970736	−	0.240149i	$-0.922804\pi$
$53$	− 25.4558i	− 0.480299i	0.970736	−	0.240149i	$-0.0771965\pi$
$54$	0	0
$55$	5.00000	0.0909091
$56$	0	0
$57$	53.7401i	0.942809i
$58$	0	0
$59$	− 84.8528i	− 1.43818i	−0.694915	−	0.719092i	$-0.744558\pi$
$59$	− 84.8528i	− 1.43818i	0.694915	−	0.719092i	$-0.255442\pi$
$60$	0	0
$61$	95.0000	1.55738	0.778689	−	0.627411i	$-0.215885\pi$
$61$	95.0000	1.55738	0.778689	+	0.627411i	$0.215885\pi$
$62$	0	0
$63$	−5.00000	−0.0793651
$64$	0	0
$65$	16.9706i	0.261086i
$66$	0	0
$67$	− 110.309i	− 1.64640i	−0.567753	−	0.823199i	$-0.692187\pi$
$67$	− 110.309i	− 1.64640i	0.567753	−	0.823199i	$-0.307813\pi$
$68$	0	0
$69$	− 28.2843i	− 0.409917i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−25.0000	−0.342466	−0.171233	−	0.985231i	$-0.554775\pi$
$73$	−25.0000	−0.342466	−0.171233	+	0.985231i	$0.554775\pi$
$74$	0	0
$75$	67.8823i	0.905097i
$76$	0	0
$77$	25.0000	0.324675
$78$	0	0
$79$	− 42.4264i	− 0.537043i	−0.963274	−	0.268522i	$-0.913465\pi$
$79$	− 42.4264i	− 0.537043i	0.963274	−	0.268522i	$-0.0865351\pi$
$80$	0	0
$81$	−71.0000	−0.876543
$82$	0	0
$83$	130.000	1.56627	0.783133	−	0.621855i	$-0.213621\pi$
$83$	130.000	1.56627	0.783133	+	0.621855i	$0.213621\pi$
$84$	0	0
$85$	25.0000	0.294118
$86$	0	0
$87$	120.000	1.37931
$88$	0	0
$89$	− 127.279i	− 1.43010i	−0.699071	−	0.715052i	$-0.746403\pi$
$89$	− 127.279i	− 1.43010i	0.699071	−	0.715052i	$-0.253597\pi$
$90$	0	0
$91$	84.8528i	0.932449i
$92$	0	0
$93$	120.000	1.29032
$94$	0	0
$95$	19.0000	0.200000
$96$	0	0
$97$	16.9706i	0.174954i	0.996167	+	0.0874771i	$0.0278805\pi$
$97$	16.9706i	0.174954i	−0.996167	+	0.0874771i	$0.972120\pi$
$98$	0	0
$99$	−5.00000	−0.0505051
$100$	0	0
$101$	50.0000	0.495050	0.247525	−	0.968882i	$-0.420383\pi$
$101$	50.0000	0.495050	0.247525	+	0.968882i	$0.420383\pi$
$102$	0	0
$103$	16.9706i	0.164763i	0.996601	+	0.0823814i	$0.0262526\pi$
$103$	16.9706i	0.164763i	−0.996601	+	0.0823814i	$0.973747\pi$
$104$	0	0
$105$	− 14.1421i	− 0.134687i
$106$	0	0
$107$	101.823i	0.951620i	0.879548	+	0.475810i	$0.157845\pi$
$107$	101.823i	0.951620i	−0.879548	+	0.475810i	$0.842155\pi$
$108$	0	0
$109$	127.279i	1.16770i	0.811862	+	0.583850i	$0.198454\pi$
$109$	127.279i	1.16770i	−0.811862	+	0.583850i	$0.801546\pi$
$110$	0	0
$111$	−72.0000	−0.648649
$112$	0	0
$113$	110.309i	0.976183i	0.872793	+	0.488091i	$0.162307\pi$
$113$	110.309i	0.976183i	−0.872793	+	0.488091i	$0.837693\pi$
$114$	0	0
$115$	−10.0000	−0.0869565
$116$	0	0
$117$	− 16.9706i	− 0.145048i
$118$	0	0
$119$	125.000	1.05042
$120$	0	0
$121$	−96.0000	−0.793388
$122$	0	0
$123$	−120.000	−0.975610
$124$	0	0
$125$	49.0000	0.392000
$126$	0	0
$127$	− 229.103i	− 1.80396i	−0.431780	−	0.901979i	$-0.642114\pi$
$127$	− 229.103i	− 1.80396i	0.431780	−	0.901979i	$-0.357886\pi$
$128$	0	0
$129$	14.1421i	0.109629i
$130$	0	0
$131$	163.000	1.24427	0.622137	−	0.782908i	$-0.286265\pi$
$131$	163.000	1.24427	0.622137	+	0.782908i	$0.286265\pi$
$132$	0	0
$133$	95.0000	0.714286
$134$	0	0
$135$	28.2843i	0.209513i
$136$	0	0
$137$	95.0000	0.693431	0.346715	−	0.937970i	$-0.387297\pi$
$137$	95.0000	0.693431	0.346715	+	0.937970i	$0.387297\pi$
$138$	0	0
$139$	−125.000	−0.899281	−0.449640	−	0.893210i	$-0.648448\pi$
$139$	−125.000	−0.899281	−0.449640	+	0.893210i	$0.648448\pi$
$140$	0	0
$141$	14.1421i	0.100299i
$142$	0	0
$143$	84.8528i	0.593376i
$144$	0	0
$145$	− 42.4264i	− 0.292596i
$146$	0	0
$147$	67.8823i	0.461784i
$148$	0	0
$149$	215.000	1.44295	0.721477	−	0.692439i	$-0.243464\pi$
$149$	215.000	1.44295	0.721477	+	0.692439i	$0.243464\pi$
$150$	0	0
$151$	84.8528i	0.561939i	0.959717	+	0.280970i	$0.0906560\pi$
$151$	84.8528i	0.561939i	−0.959717	+	0.280970i	$0.909344\pi$
$152$	0	0
$153$	−25.0000	−0.163399
$154$	0	0
$155$	− 42.4264i	− 0.273719i
$156$	0	0
$157$	−190.000	−1.21019	−0.605096	−	0.796153i	$-0.706865\pi$
$157$	−190.000	−1.21019	−0.605096	+	0.796153i	$0.706865\pi$
$158$	0	0
$159$	−72.0000	−0.452830
$160$	0	0
$161$	−50.0000	−0.310559
$162$	0	0
$163$	−110.000	−0.674847	−0.337423	−	0.941353i	$-0.609555\pi$
$163$	−110.000	−0.674847	−0.337423	+	0.941353i	$0.609555\pi$
$164$	0	0
$165$	− 14.1421i	− 0.0857099i
$166$	0	0
$167$	− 59.3970i	− 0.355670i	−0.984060	−	0.177835i	$-0.943091\pi$
$167$	− 59.3970i	− 0.355670i	0.984060	−	0.177835i	$-0.0569094\pi$
$168$	0	0
$169$	−119.000	−0.704142
$170$	0	0
$171$	−19.0000	−0.111111
$172$	0	0
$173$	− 186.676i	− 1.07905i	−0.841969	−	0.539527i	$-0.818603\pi$
$173$	− 186.676i	− 1.07905i	0.841969	−	0.539527i	$-0.181397\pi$
$174$	0	0
$175$	120.000	0.685714
$176$	0	0
$177$	−240.000	−1.35593
$178$	0	0
$179$	127.279i	0.711057i	0.934665	+	0.355529i	$0.115699\pi$
$179$	127.279i	0.711057i	−0.934665	+	0.355529i	$0.884301\pi$
$180$	0	0
$181$	− 254.558i	− 1.40640i	−0.710992	−	0.703200i	$-0.751754\pi$
$181$	− 254.558i	− 1.40640i	0.710992	−	0.703200i	$-0.248246\pi$
$182$	0	0
$183$	− 268.701i	− 1.46831i
$184$	0	0
$185$	25.4558i	0.137599i
$186$	0	0
$187$	125.000	0.668449
$188$	0	0
$189$	141.421i	0.748261i
$190$	0	0
$191$	−293.000	−1.53403	−0.767016	−	0.641628i	$-0.778259\pi$
$191$	−293.000	−1.53403	−0.767016	+	0.641628i	$0.778259\pi$
$192$	0	0
$193$	59.3970i	0.307756i	0.988090	+	0.153878i	$0.0491763\pi$
$193$	59.3970i	0.307756i	−0.988090	+	0.153878i	$0.950824\pi$
$194$	0	0
$195$	48.0000	0.246154
$196$	0	0
$197$	−70.0000	−0.355330	−0.177665	−	0.984091i	$-0.556854\pi$
$197$	−70.0000	−0.355330	−0.177665	+	0.984091i	$0.556854\pi$
$198$	0	0
$199$	−173.000	−0.869347	−0.434673	−	0.900588i	$-0.643136\pi$
$199$	−173.000	−0.869347	−0.434673	+	0.900588i	$0.643136\pi$
$200$	0	0
$201$	−312.000	−1.55224
$202$	0	0
$203$	− 212.132i	− 1.04499i
$204$	0	0
$205$	42.4264i	0.206958i
$206$	0	0
$207$	10.0000	0.0483092
$208$	0	0
$209$	95.0000	0.454545
$210$	0	0
$211$	− 84.8528i	− 0.402146i	−0.979576	−	0.201073i	$-0.935557\pi$
$211$	− 84.8528i	− 0.402146i	0.979576	−	0.201073i	$-0.0644429\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.00000	0.0232558
$216$	0	0
$217$	− 212.132i	− 0.977567i
$218$	0	0
$219$	70.7107i	0.322880i
$220$	0	0
$221$	424.264i	1.91975i
$222$	0	0
$223$	364.867i	1.63618i	0.575094	+	0.818088i	$0.304966\pi$
$223$	364.867i	1.63618i	−0.575094	+	0.818088i	$0.695034\pi$
$224$	0	0
$225$	−24.0000	−0.106667
$226$	0	0
$227$	− 67.8823i	− 0.299041i	−0.988759	−	0.149520i	$-0.952227\pi$
$227$	− 67.8823i	− 0.299041i	0.988759	−	0.149520i	$-0.0477730\pi$
$228$	0	0
$229$	−145.000	−0.633188	−0.316594	−	0.948561i	$-0.602539\pi$
$229$	−145.000	−0.633188	−0.316594	+	0.948561i	$0.602539\pi$
$230$	0	0
$231$	− 70.7107i	− 0.306107i
$232$	0	0
$233$	335.000	1.43777	0.718884	−	0.695130i	$-0.244653\pi$
$233$	335.000	1.43777	0.718884	+	0.695130i	$0.244653\pi$
$234$	0	0
$235$	5.00000	0.0212766
$236$	0	0
$237$	−120.000	−0.506329
$238$	0	0
$239$	−197.000	−0.824268	−0.412134	−	0.911123i	$-0.635216\pi$
$239$	−197.000	−0.824268	−0.412134	+	0.911123i	$0.635216\pi$
$240$	0	0
$241$	296.985i	1.23230i	0.787628	+	0.616151i	$0.211309\pi$
$241$	296.985i	1.23230i	−0.787628	+	0.616151i	$0.788691\pi$
$242$	0	0
$243$	− 53.7401i	− 0.221153i
$244$	0	0
$245$	24.0000	0.0979592
$246$	0	0
$247$	322.441i	1.30543i
$248$	0	0
$249$	− 367.696i	− 1.47669i
$250$	0	0
$251$	−173.000	−0.689243	−0.344622	−	0.938742i	$-0.611993\pi$
$251$	−173.000	−0.689243	−0.344622	+	0.938742i	$0.611993\pi$
$252$	0	0
$253$	−50.0000	−0.197628
$254$	0	0
$255$	− 70.7107i	− 0.277297i
$256$	0	0
$257$	− 67.8823i	− 0.264133i	−0.991241	−	0.132067i	$-0.957839\pi$
$257$	− 67.8823i	− 0.264133i	0.991241	−	0.132067i	$-0.0421613\pi$
$258$	0	0
$259$	127.279i	0.491426i
$260$	0	0
$261$	42.4264i	0.162553i
$262$	0	0
$263$	355.000	1.34981	0.674905	−	0.737905i	$-0.264185\pi$
$263$	355.000	1.34981	0.674905	+	0.737905i	$0.264185\pi$
$264$	0	0
$265$	25.4558i	0.0960598i
$266$	0	0
$267$	−360.000	−1.34831
$268$	0	0
$269$	− 381.838i	− 1.41947i	−0.704468	−	0.709735i	$-0.748814\pi$
$269$	− 381.838i	− 1.41947i	0.704468	−	0.709735i	$-0.251186\pi$
$270$	0	0
$271$	−110.000	−0.405904	−0.202952	−	0.979189i	$-0.565054\pi$
$271$	−110.000	−0.405904	−0.202952	+	0.979189i	$0.565054\pi$
$272$	0	0
$273$	240.000	0.879121
$274$	0	0
$275$	120.000	0.436364
$276$	0	0
$277$	−265.000	−0.956679	−0.478339	−	0.878175i	$-0.658761\pi$
$277$	−265.000	−0.956679	−0.478339	+	0.878175i	$0.658761\pi$
$278$	0	0
$279$	42.4264i	0.152066i
$280$	0	0
$281$	424.264i	1.50984i	0.655819	+	0.754918i	$0.272323\pi$
$281$	424.264i	1.50984i	−0.655819	+	0.754918i	$0.727677\pi$
$282$	0	0
$283$	−125.000	−0.441696	−0.220848	−	0.975308i	$-0.570882\pi$
$283$	−125.000	−0.441696	−0.220848	+	0.975308i	$0.570882\pi$
$284$	0	0
$285$	− 53.7401i	− 0.188562i
$286$	0	0
$287$	212.132i	0.739136i
$288$	0	0
$289$	336.000	1.16263
$290$	0	0
$291$	48.0000	0.164948
$292$	0	0
$293$	186.676i	0.637120i	0.947903	+	0.318560i	$0.103199\pi$
$293$	186.676i	0.637120i	−0.947903	+	0.318560i	$0.896801\pi$
$294$	0	0
$295$	84.8528i	0.287637i
$296$	0	0
$297$	141.421i	0.476166i
$298$	0	0
$299$	− 169.706i	− 0.567577i
$300$	0	0
$301$	25.0000	0.0830565
$302$	0	0
$303$	− 141.421i	− 0.466737i
$304$	0	0
$305$	−95.0000	−0.311475
$306$	0	0
$307$	− 280.014i	− 0.912099i	−0.889955	−	0.456049i	$-0.849264\pi$
$307$	− 280.014i	− 0.912099i	0.889955	−	0.456049i	$-0.150736\pi$
$308$	0	0
$309$	48.0000	0.155340
$310$	0	0
$311$	235.000	0.755627	0.377814	−	0.925882i	$-0.376676\pi$
$311$	235.000	0.755627	0.377814	+	0.925882i	$0.376676\pi$
$312$	0	0
$313$	−310.000	−0.990415	−0.495208	−	0.868775i	$-0.664908\pi$
$313$	−310.000	−0.990415	−0.495208	+	0.868775i	$0.664908\pi$
$314$	0	0
$315$	5.00000	0.0158730
$316$	0	0
$317$	− 186.676i	− 0.588884i	−0.955669	−	0.294442i	$-0.904866\pi$
$317$	− 186.676i	− 0.588884i	0.955669	−	0.294442i	$-0.0951338\pi$
$318$	0	0
$319$	− 212.132i	− 0.664991i
$320$	0	0
$321$	288.000	0.897196
$322$	0	0
$323$	475.000	1.47059
$324$	0	0
$325$	407.294i	1.25321i
$326$	0	0
$327$	360.000	1.10092
$328$	0	0
$329$	25.0000	0.0759878
$330$	0	0
$331$	− 296.985i	− 0.897235i	−0.893724	−	0.448618i	$-0.851917\pi$
$331$	− 296.985i	− 0.897235i	0.893724	−	0.448618i	$-0.148083\pi$
$332$	0	0
$333$	− 25.4558i	− 0.0764440i
$334$	0	0
$335$	110.309i	0.329280i
$336$	0	0
$337$	− 526.087i	− 1.56109i	−0.625099	−	0.780545i	$-0.714942\pi$
$337$	− 526.087i	− 1.56109i	0.625099	−	0.780545i	$-0.285058\pi$
$338$	0	0
$339$	312.000	0.920354
$340$	0	0
$341$	− 212.132i	− 0.622088i
$342$	0	0
$343$	365.000	1.06414
$344$	0	0
$345$	28.2843i	0.0819834i
$346$	0	0
$347$	−125.000	−0.360231	−0.180115	−	0.983646i	$-0.557647\pi$
$347$	−125.000	−0.360231	−0.180115	+	0.983646i	$0.557647\pi$
$348$	0	0
$349$	23.0000	0.0659026	0.0329513	−	0.999457i	$-0.489509\pi$
$349$	23.0000	0.0659026	0.0329513	+	0.999457i	$0.489509\pi$
$350$	0	0
$351$	−480.000	−1.36752
$352$	0	0
$353$	410.000	1.16147	0.580737	−	0.814092i	$-0.302765\pi$
$353$	410.000	1.16147	0.580737	+	0.814092i	$0.302765\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 353.553i	− 0.990346i
$358$	0	0
$359$	475.000	1.32312	0.661560	−	0.749892i	$-0.269895\pi$
$359$	475.000	1.32312	0.661560	+	0.749892i	$0.269895\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	271.529i	0.748014i
$364$	0	0
$365$	25.0000	0.0684932
$366$	0	0
$367$	−230.000	−0.626703	−0.313351	−	0.949637i	$-0.601452\pi$
$367$	−230.000	−0.626703	−0.313351	+	0.949637i	$0.601452\pi$
$368$	0	0
$369$	− 42.4264i	− 0.114977i
$370$	0	0
$371$	127.279i	0.343071i
$372$	0	0
$373$	− 67.8823i	− 0.181990i	−0.995851	−	0.0909950i	$-0.970995\pi$
$373$	− 67.8823i	− 0.181990i	0.995851	−	0.0909950i	$-0.0290047\pi$
$374$	0	0
$375$	− 138.593i	− 0.369581i
$376$	0	0
$377$	720.000	1.90981
$378$	0	0
$379$	254.558i	0.671658i	0.941923	+	0.335829i	$0.109016\pi$
$379$	254.558i	0.671658i	−0.941923	+	0.335829i	$0.890984\pi$
$380$	0	0
$381$	−648.000	−1.70079
$382$	0	0
$383$	144.250i	0.376631i	0.982109	+	0.188316i	$0.0603028\pi$
$383$	144.250i	0.376631i	−0.982109	+	0.188316i	$0.939697\pi$
$384$	0	0
$385$	−25.0000	−0.0649351
$386$	0	0
$387$	−5.00000	−0.0129199
$388$	0	0
$389$	−553.000	−1.42159	−0.710797	−	0.703397i	$-0.751666\pi$
$389$	−553.000	−1.42159	−0.710797	+	0.703397i	$0.751666\pi$
$390$	0	0
$391$	−250.000	−0.639386
$392$	0	0
$393$	− 461.034i	− 1.17311i
$394$	0	0
$395$	42.4264i	0.107409i
$396$	0	0
$397$	335.000	0.843829	0.421914	−	0.906636i	$-0.361358\pi$
$397$	335.000	0.843829	0.421914	+	0.906636i	$0.361358\pi$
$398$	0	0
$399$	− 268.701i	− 0.673435i
$400$	0	0
$401$	212.132i	0.529008i	0.964385	+	0.264504i	$0.0852082\pi$
$401$	212.132i	0.529008i	−0.964385	+	0.264504i	$0.914792\pi$
$402$	0	0
$403$	720.000	1.78660
$404$	0	0
$405$	71.0000	0.175309
$406$	0	0
$407$	127.279i	0.312725i
$408$	0	0
$409$	721.249i	1.76344i	0.471769	+	0.881722i	$0.343616\pi$
$409$	721.249i	1.76344i	−0.471769	+	0.881722i	$0.656384\pi$
$410$	0	0
$411$	− 268.701i	− 0.653773i
$412$	0	0
$413$	424.264i	1.02727i
$414$	0	0
$415$	−130.000	−0.313253
$416$	0	0
$417$	353.553i	0.847850i
$418$	0	0
$419$	−62.0000	−0.147971	−0.0739857	−	0.997259i	$-0.523572\pi$
$419$	−62.0000	−0.147971	−0.0739857	+	0.997259i	$0.523572\pi$
$420$	0	0
$421$	296.985i	0.705427i	0.935731	+	0.352714i	$0.114741\pi$
$421$	296.985i	0.705427i	−0.935731	+	0.352714i	$0.885259\pi$
$422$	0	0
$423$	−5.00000	−0.0118203
$424$	0	0
$425$	600.000	1.41176
$426$	0	0
$427$	−475.000	−1.11241
$428$	0	0
$429$	240.000	0.559441
$430$	0	0
$431$	509.117i	1.18125i	0.806948	+	0.590623i	$0.201118\pi$
$431$	509.117i	1.18125i	−0.806948	+	0.590623i	$0.798882\pi$
$432$	0	0
$433$	− 229.103i	− 0.529105i	−0.964371	−	0.264553i	$-0.914776\pi$
$433$	− 229.103i	− 0.529105i	0.964371	−	0.264553i	$-0.0852243\pi$
$434$	0	0
$435$	−120.000	−0.275862
$436$	0	0
$437$	−190.000	−0.434783
$438$	0	0
$439$	− 806.102i	− 1.83622i	−0.396323	−	0.918111i	$-0.629714\pi$
$439$	− 806.102i	− 1.83622i	0.396323	−	0.918111i	$-0.370286\pi$
$440$	0	0
$441$	−24.0000	−0.0544218
$442$	0	0
$443$	−365.000	−0.823928	−0.411964	−	0.911200i	$-0.635157\pi$
$443$	−365.000	−0.823928	−0.411964	+	0.911200i	$0.635157\pi$
$444$	0	0
$445$	127.279i	0.286021i
$446$	0	0
$447$	− 608.112i	− 1.36043i
$448$	0	0
$449$	− 763.675i	− 1.70084i	−0.526108	−	0.850418i	$-0.676349\pi$
$449$	− 763.675i	− 1.70084i	0.526108	−	0.850418i	$-0.323651\pi$
$450$	0	0
$451$	212.132i	0.470359i
$452$	0	0
$453$	240.000	0.529801
$454$	0	0
$455$	− 84.8528i	− 0.186490i
$456$	0	0
$457$	−265.000	−0.579869	−0.289934	−	0.957047i	$-0.593633\pi$
$457$	−265.000	−0.579869	−0.289934	+	0.957047i	$0.593633\pi$
$458$	0	0
$459$	707.107i	1.54054i
$460$	0	0
$461$	−553.000	−1.19957	−0.599783	−	0.800163i	$-0.704746\pi$
$461$	−553.000	−1.19957	−0.599783	+	0.800163i	$0.704746\pi$
$462$	0	0
$463$	−485.000	−1.04752	−0.523758	−	0.851867i	$-0.675470\pi$
$463$	−485.000	−1.04752	−0.523758	+	0.851867i	$0.675470\pi$
$464$	0	0
$465$	−120.000	−0.258065
$466$	0	0
$467$	115.000	0.246253	0.123126	−	0.992391i	$-0.460708\pi$
$467$	115.000	0.246253	0.123126	+	0.992391i	$0.460708\pi$
$468$	0	0
$469$	551.543i	1.17600i
$470$	0	0
$471$	537.401i	1.14098i
$472$	0	0
$473$	25.0000	0.0528541
$474$	0	0
$475$	456.000	0.960000
$476$	0	0
$477$	− 25.4558i	− 0.0533665i
$478$	0	0
$479$	490.000	1.02296	0.511482	−	0.859294i	$-0.329097\pi$
$479$	490.000	1.02296	0.511482	+	0.859294i	$0.329097\pi$
$480$	0	0
$481$	−432.000	−0.898129
$482$	0	0
$483$	141.421i	0.292798i
$484$	0	0
$485$	− 16.9706i	− 0.0349909i
$486$	0	0
$487$	610.940i	1.25450i	0.778819	+	0.627249i	$0.215819\pi$
$487$	610.940i	1.25450i	−0.778819	+	0.627249i	$0.784181\pi$
$488$	0	0
$489$	311.127i	0.636252i
$490$	0	0
$491$	82.0000	0.167006	0.0835031	−	0.996508i	$-0.473389\pi$
$491$	82.0000	0.167006	0.0835031	+	0.996508i	$0.473389\pi$
$492$	0	0
$493$	− 1060.66i	− 2.15144i
$494$	0	0
$495$	5.00000	0.0101010
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−485.000	−0.971944	−0.485972	−	0.873974i	$-0.661534\pi$
$499$	−485.000	−0.971944	−0.485972	+	0.873974i	$0.661534\pi$
$500$	0	0
$501$	−168.000	−0.335329
$502$	0	0
$503$	250.000	0.497018	0.248509	−	0.968630i	$-0.420059\pi$
$503$	250.000	0.497018	0.248509	+	0.968630i	$0.420059\pi$
$504$	0	0
$505$	−50.0000	−0.0990099
$506$	0	0
$507$	336.583i	0.663871i
$508$	0	0
$509$	− 169.706i	− 0.333410i	−0.986007	−	0.166705i	$-0.946687\pi$
$509$	− 169.706i	− 0.333410i	0.986007	−	0.166705i	$-0.0533127\pi$
$510$	0	0
$511$	125.000	0.244618
$512$	0	0
$513$	537.401i	1.04757i
$514$	0	0
$515$	− 16.9706i	− 0.0329525i
$516$	0	0
$517$	25.0000	0.0483559
$518$	0	0
$519$	−528.000	−1.01734
$520$	0	0
$521$	− 127.279i	− 0.244298i	−0.992512	−	0.122149i	$-0.961021\pi$
$521$	− 127.279i	− 0.244298i	0.992512	−	0.122149i	$-0.0389786\pi$
$522$	0	0
$523$	− 356.382i	− 0.681418i	−0.940169	−	0.340709i	$-0.889333\pi$
$523$	− 356.382i	− 0.681418i	0.940169	−	0.340709i	$-0.110667\pi$
$524$	0	0
$525$	− 339.411i	− 0.646498i
$526$	0	0
$527$	− 1060.66i	− 2.01264i
$528$	0	0
$529$	−429.000	−0.810964
$530$	0	0
$531$	− 84.8528i	− 0.159798i
$532$	0	0
$533$	−720.000	−1.35084
$534$	0	0
$535$	− 101.823i	− 0.190324i
$536$	0	0
$537$	360.000	0.670391
$538$	0	0
$539$	120.000	0.222635
$540$	0	0
$541$	−25.0000	−0.0462107	−0.0231054	−	0.999733i	$-0.507355\pi$
$541$	−25.0000	−0.0462107	−0.0231054	+	0.999733i	$0.507355\pi$
$542$	0	0
$543$	−720.000	−1.32597
$544$	0	0
$545$	− 127.279i	− 0.233540i
$546$	0	0
$547$	− 16.9706i	− 0.0310248i	−0.999880	−	0.0155124i	$-0.995062\pi$
$547$	− 16.9706i	− 0.0310248i	0.999880	−	0.0155124i	$-0.00493795\pi$
$548$	0	0
$549$	95.0000	0.173042
$550$	0	0
$551$	− 806.102i	− 1.46298i
$552$	0	0
$553$	212.132i	0.383602i
$554$	0	0
$555$	72.0000	0.129730
$556$	0	0
$557$	−745.000	−1.33752	−0.668761	−	0.743477i	$-0.733175\pi$
$557$	−745.000	−1.33752	−0.668761	+	0.743477i	$0.733175\pi$
$558$	0	0
$559$	84.8528i	0.151794i
$560$	0	0
$561$	− 353.553i	− 0.630220i
$562$	0	0
$563$	− 313.955i	− 0.557647i	−0.960342	−	0.278824i	$-0.910056\pi$
$563$	− 313.955i	− 0.557647i	0.960342	−	0.278824i	$-0.0899445\pi$
$564$	0	0
$565$	− 110.309i	− 0.195237i
$566$	0	0
$567$	355.000	0.626102
$568$	0	0
$569$	424.264i	0.745631i	0.927905	+	0.372816i	$0.121608\pi$
$569$	424.264i	0.745631i	−0.927905	+	0.372816i	$0.878392\pi$
$570$	0	0
$571$	−1070.00	−1.87391	−0.936953	−	0.349456i	$-0.886366\pi$
$571$	−1070.00	−1.87391	−0.936953	+	0.349456i	$0.886366\pi$
$572$	0	0
$573$	828.729i	1.44630i
$574$	0	0
$575$	−240.000	−0.417391
$576$	0	0
$577$	−25.0000	−0.0433276	−0.0216638	−	0.999765i	$-0.506896\pi$
$577$	−25.0000	−0.0433276	−0.0216638	+	0.999765i	$0.506896\pi$
$578$	0	0
$579$	168.000	0.290155
$580$	0	0
$581$	−650.000	−1.11876
$582$	0	0
$583$	127.279i	0.218318i
$584$	0	0
$585$	16.9706i	0.0290095i
$586$	0	0
$587$	−725.000	−1.23509	−0.617547	−	0.786534i	$-0.711873\pi$
$587$	−725.000	−1.23509	−0.617547	+	0.786534i	$0.711873\pi$
$588$	0	0
$589$	− 806.102i	− 1.36859i
$590$	0	0
$591$	197.990i	0.335008i
$592$	0	0
$593$	650.000	1.09612	0.548061	−	0.836439i	$-0.315366\pi$
$593$	650.000	1.09612	0.548061	+	0.836439i	$0.315366\pi$
$594$	0	0
$595$	−125.000	−0.210084
$596$	0	0
$597$	489.318i	0.819628i
$598$	0	0
$599$	296.985i	0.495801i	0.968785	+	0.247901i	$0.0797406\pi$
$599$	296.985i	0.495801i	−0.968785	+	0.247901i	$0.920259\pi$
$600$	0	0
$601$	− 848.528i	− 1.41186i	−0.708281	−	0.705930i	$-0.750529\pi$
$601$	− 848.528i	− 1.41186i	0.708281	−	0.705930i	$-0.249471\pi$
$602$	0	0
$603$	− 110.309i	− 0.182933i
$604$	0	0
$605$	96.0000	0.158678
$606$	0	0
$607$	271.529i	0.447329i	0.974666	+	0.223665i	$0.0718021\pi$
$607$	271.529i	0.447329i	−0.974666	+	0.223665i	$0.928198\pi$
$608$	0	0
$609$	−600.000	−0.985222
$610$	0	0
$611$	84.8528i	0.138875i
$612$	0	0
$613$	1055.00	1.72104	0.860522	−	0.509413i	$-0.170138\pi$
$613$	1055.00	1.72104	0.860522	+	0.509413i	$0.170138\pi$
$614$	0	0
$615$	120.000	0.195122
$616$	0	0
$617$	−505.000	−0.818476	−0.409238	−	0.912428i	$-0.634206\pi$
$617$	−505.000	−0.818476	−0.409238	+	0.912428i	$0.634206\pi$
$618$	0	0
$619$	130.000	0.210016	0.105008	−	0.994471i	$-0.466513\pi$
$619$	130.000	0.210016	0.105008	+	0.994471i	$0.466513\pi$
$620$	0	0
$621$	− 282.843i	− 0.455463i
$622$	0	0
$623$	636.396i	1.02150i
$624$	0	0
$625$	551.000	0.881600
$626$	0	0
$627$	− 268.701i	− 0.428550i
$628$	0	0
$629$	636.396i	1.01176i
$630$	0	0
$631$	475.000	0.752773	0.376387	−	0.926463i	$-0.377166\pi$
$631$	475.000	0.752773	0.376387	+	0.926463i	$0.377166\pi$
$632$	0	0
$633$	−240.000	−0.379147
$634$	0	0
$635$	229.103i	0.360791i
$636$	0	0
$637$	407.294i	0.639393i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 848.528i	− 1.32376i	−0.749611	−	0.661878i	$-0.769760\pi$
$641$	− 848.528i	− 1.32376i	0.749611	−	0.661878i	$-0.230240\pi$
$642$	0	0
$643$	955.000	1.48523	0.742613	−	0.669721i	$-0.233586\pi$
$643$	955.000	1.48523	0.742613	+	0.669721i	$0.233586\pi$
$644$	0	0
$645$	− 14.1421i	− 0.0219258i
$646$	0	0
$647$	−965.000	−1.49150	−0.745750	−	0.666226i	$-0.767908\pi$
$647$	−965.000	−1.49150	−0.745750	+	0.666226i	$0.767908\pi$
$648$	0	0
$649$	424.264i	0.653720i
$650$	0	0
$651$	−600.000	−0.921659
$652$	0	0
$653$	935.000	1.43185	0.715926	−	0.698176i	$-0.246005\pi$
$653$	935.000	1.43185	0.715926	+	0.698176i	$0.246005\pi$
$654$	0	0
$655$	−163.000	−0.248855
$656$	0	0
$657$	−25.0000	−0.0380518
$658$	0	0
$659$	84.8528i	0.128760i	0.997925	+	0.0643800i	$0.0205070\pi$
$659$	84.8528i	0.128760i	−0.997925	+	0.0643800i	$0.979493\pi$
$660$	0	0
$661$	− 678.823i	− 1.02696i	−0.858101	−	0.513481i	$-0.828356\pi$
$661$	− 678.823i	− 1.02696i	0.858101	−	0.513481i	$-0.171644\pi$
$662$	0	0
$663$	1200.00	1.80995
$664$	0	0
$665$	−95.0000	−0.142857
$666$	0	0
$667$	424.264i	0.636078i
$668$	0	0
$669$	1032.00	1.54260
$670$	0	0
$671$	−475.000	−0.707899
$672$	0	0
$673$	− 186.676i	− 0.277379i	−0.990336	−	0.138690i	$-0.955711\pi$
$673$	− 186.676i	− 0.277379i	0.990336	−	0.138690i	$-0.0442890\pi$
$674$	0	0
$675$	678.823i	1.00566i
$676$	0	0
$677$	907.925i	1.34110i	0.741864	+	0.670550i	$0.233942\pi$
$677$	907.925i	1.34110i	−0.741864	+	0.670550i	$0.766058\pi$
$678$	0	0
$679$	− 84.8528i	− 0.124967i
$680$	0	0
$681$	−192.000	−0.281938
$682$	0	0
$683$	− 1120.06i	− 1.63991i	−0.572429	−	0.819954i	$-0.693999\pi$
$683$	− 1120.06i	− 1.63991i	0.572429	−	0.819954i	$-0.306001\pi$
$684$	0	0
$685$	−95.0000	−0.138686
$686$	0	0
$687$	410.122i	0.596975i
$688$	0	0
$689$	−432.000	−0.626996
$690$	0	0
$691$	715.000	1.03473	0.517366	−	0.855764i	$-0.326913\pi$
$691$	715.000	1.03473	0.517366	+	0.855764i	$0.326913\pi$
$692$	0	0
$693$	25.0000	0.0360750
$694$	0	0
$695$	125.000	0.179856
$696$	0	0
$697$	1060.66i	1.52175i
$698$	0	0
$699$	− 947.523i	− 1.35554i
$700$	0	0
$701$	−430.000	−0.613409	−0.306705	−	0.951805i	$-0.599226\pi$
$701$	−430.000	−0.613409	−0.306705	+	0.951805i	$0.599226\pi$
$702$	0	0
$703$	483.661i	0.687996i
$704$	0	0
$705$	− 14.1421i	− 0.0200598i
$706$	0	0
$707$	−250.000	−0.353607
$708$	0	0
$709$	−382.000	−0.538787	−0.269394	−	0.963030i	$-0.586823\pi$
$709$	−382.000	−0.538787	−0.269394	+	0.963030i	$0.586823\pi$
$710$	0	0
$711$	− 42.4264i	− 0.0596715i
$712$	0	0
$713$	424.264i	0.595041i
$714$	0	0
$715$	− 84.8528i	− 0.118675i
$716$	0	0
$717$	557.200i	0.777127i
$718$	0	0
$719$	115.000	0.159944	0.0799722	−	0.996797i	$-0.474517\pi$
$719$	115.000	0.159944	0.0799722	+	0.996797i	$0.474517\pi$
$720$	0	0
$721$	− 84.8528i	− 0.117688i
$722$	0	0
$723$	840.000	1.16183
$724$	0	0
$725$	− 1018.23i	− 1.40446i
$726$	0	0
$727$	1075.00	1.47868	0.739340	−	0.673333i	$-0.235138\pi$
$727$	1075.00	1.47868	0.739340	+	0.673333i	$0.235138\pi$
$728$	0	0
$729$	−791.000	−1.08505
$730$	0	0
$731$	125.000	0.170999
$732$	0	0
$733$	530.000	0.723056	0.361528	−	0.932361i	$-0.382255\pi$
$733$	530.000	0.723056	0.361528	+	0.932361i	$0.382255\pi$
$734$	0	0
$735$	− 67.8823i	− 0.0923568i
$736$	0	0
$737$	551.543i	0.748363i
$738$	0	0
$739$	547.000	0.740189	0.370095	−	0.928994i	$-0.379325\pi$
$739$	547.000	0.740189	0.370095	+	0.928994i	$0.379325\pi$
$740$	0	0
$741$	912.000	1.23077
$742$	0	0
$743$	958.837i	1.29049i	0.763974	+	0.645247i	$0.223245\pi$
$743$	958.837i	1.29049i	−0.763974	+	0.645247i	$0.776755\pi$
$744$	0	0
$745$	−215.000	−0.288591
$746$	0	0
$747$	130.000	0.174029
$748$	0	0
$749$	− 509.117i	− 0.679729i
$750$	0	0
$751$	169.706i	0.225973i	0.993597	+	0.112986i	$0.0360417\pi$
$751$	169.706i	0.225973i	−0.993597	+	0.112986i	$0.963958\pi$
$752$	0	0
$753$	489.318i	0.649825i
$754$	0	0
$755$	− 84.8528i	− 0.112388i
$756$	0	0
$757$	1055.00	1.39366	0.696830	−	0.717237i	$-0.254593\pi$
$757$	1055.00	1.39366	0.696830	+	0.717237i	$0.254593\pi$
$758$	0	0
$759$	141.421i	0.186326i
$760$	0	0
$761$	215.000	0.282523	0.141261	−	0.989972i	$-0.454884\pi$
$761$	215.000	0.282523	0.141261	+	0.989972i	$0.454884\pi$
$762$	0	0
$763$	− 636.396i	− 0.834071i
$764$	0	0
$765$	25.0000	0.0326797
$766$	0	0
$767$	−1440.00	−1.87744
$768$	0	0
$769$	−145.000	−0.188557	−0.0942783	−	0.995546i	$-0.530054\pi$
$769$	−145.000	−0.188557	−0.0942783	+	0.995546i	$0.530054\pi$
$770$	0	0
$771$	−192.000	−0.249027
$772$	0	0
$773$	407.294i	0.526900i	0.964673	+	0.263450i	$0.0848604\pi$
$773$	407.294i	0.526900i	−0.964673	+	0.263450i	$0.915140\pi$
$774$	0	0
$775$	− 1018.23i	− 1.31385i
$776$	0	0
$777$	360.000	0.463320
$778$	0	0
$779$	806.102i	1.03479i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1200.00	1.53257
$784$	0	0
$785$	190.000	0.242038
$786$	0	0
$787$	− 186.676i	− 0.237200i	−0.992942	−	0.118600i	$-0.962159\pi$
$787$	− 186.676i	− 0.237200i	0.992942	−	0.118600i	$-0.0378406\pi$
$788$	0	0
$789$	− 1004.09i	− 1.27261i
$790$	0	0
$791$	− 551.543i	− 0.697273i
$792$	0	0
$793$	− 1612.20i	− 2.03304i
$794$	0	0
$795$	72.0000	0.0905660
$796$	0	0
$797$	− 704.278i	− 0.883662i	−0.897098	−	0.441831i	$-0.854329\pi$
$797$	− 704.278i	− 0.883662i	0.897098	−	0.441831i	$-0.145671\pi$
$798$	0	0
$799$	125.000	0.156446
$800$	0	0
$801$	− 127.279i	− 0.158900i
$802$	0	0
$803$	125.000	0.155666
$804$	0	0
$805$	50.0000	0.0621118
$806$	0	0
$807$	−1080.00	−1.33829
$808$	0	0
$809$	−457.000	−0.564895	−0.282447	−	0.959283i	$-0.591146\pi$
$809$	−457.000	−0.564895	−0.282447	+	0.959283i	$0.591146\pi$
$810$	0	0
$811$	− 509.117i	− 0.627764i	−0.949462	−	0.313882i	$-0.898370\pi$
$811$	− 509.117i	− 0.627764i	0.949462	−	0.313882i	$-0.101630\pi$
$812$	0	0
$813$	311.127i	0.382690i
$814$	0	0
$815$	110.000	0.134969
$816$	0	0
$817$	95.0000	0.116279
$818$	0	0
$819$	84.8528i	0.103605i
$820$	0	0
$821$	167.000	0.203410	0.101705	−	0.994815i	$-0.467570\pi$
$821$	167.000	0.203410	0.101705	+	0.994815i	$0.467570\pi$
$822$	0	0
$823$	1315.00	1.59781	0.798906	−	0.601455i	$-0.205412\pi$
$823$	1315.00	1.59781	0.798906	+	0.601455i	$0.205412\pi$
$824$	0	0
$825$	− 339.411i	− 0.411408i
$826$	0	0
$827$	534.573i	0.646400i	0.946331	+	0.323200i	$0.104759\pi$
$827$	534.573i	0.646400i	−0.946331	+	0.323200i	$0.895241\pi$
$828$	0	0
$829$	763.675i	0.921201i	0.887608	+	0.460600i	$0.152366\pi$
$829$	763.675i	0.921201i	−0.887608	+	0.460600i	$0.847634\pi$
$830$	0	0
$831$	749.533i	0.901965i
$832$	0	0
$833$	600.000	0.720288
$834$	0	0
$835$	59.3970i	0.0711341i
$836$	0	0
$837$	1200.00	1.43369
$838$	0	0
$839$	339.411i	0.404543i	0.979330	+	0.202271i	$0.0648323\pi$
$839$	339.411i	0.404543i	−0.979330	+	0.202271i	$0.935168\pi$
$840$	0	0
$841$	−959.000	−1.14031
$842$	0	0
$843$	1200.00	1.42349
$844$	0	0
$845$	119.000	0.140828
$846$	0	0
$847$	480.000	0.566706
$848$	0	0
$849$	353.553i	0.416435i
$850$	0	0
$851$	− 254.558i	− 0.299129i
$852$	0	0
$853$	770.000	0.902696	0.451348	−	0.892348i	$-0.350943\pi$
$853$	770.000	0.902696	0.451348	+	0.892348i	$0.350943\pi$
$854$	0	0
$855$	19.0000	0.0222222
$856$	0	0
$857$	− 1255.82i	− 1.46537i	−0.680568	−	0.732685i	$-0.738267\pi$
$857$	− 1255.82i	− 1.46537i	0.680568	−	0.732685i	$-0.261733\pi$
$858$	0	0
$859$	−557.000	−0.648428	−0.324214	−	0.945984i	$-0.605100\pi$
$859$	−557.000	−0.648428	−0.324214	+	0.945984i	$0.605100\pi$
$860$	0	0
$861$	600.000	0.696864
$862$	0	0
$863$	− 992.778i	− 1.15038i	−0.818020	−	0.575190i	$-0.804928\pi$
$863$	− 992.778i	− 1.15038i	0.818020	−	0.575190i	$-0.195072\pi$
$864$	0	0
$865$	186.676i	0.215811i
$866$	0	0
$867$	− 950.352i	− 1.09614i
$868$	0	0
$869$	212.132i	0.244111i
$870$	0	0
$871$	−1872.00	−2.14925
$872$	0	0
$873$	16.9706i	0.0194394i
$874$	0	0
$875$	−245.000	−0.280000
$876$	0	0
$877$	186.676i	0.212858i	0.994320	+	0.106429i	$0.0339416\pi$
$877$	186.676i	0.212858i	−0.994320	+	0.106429i	$0.966058\pi$
$878$	0	0
$879$	528.000	0.600683
$880$	0	0
$881$	−25.0000	−0.0283768	−0.0141884	−	0.999899i	$-0.504516\pi$
$881$	−25.0000	−0.0283768	−0.0141884	+	0.999899i	$0.504516\pi$
$882$	0	0
$883$	−965.000	−1.09287	−0.546433	−	0.837503i	$-0.684015\pi$
$883$	−965.000	−1.09287	−0.546433	+	0.837503i	$0.684015\pi$
$884$	0	0
$885$	240.000	0.271186
$886$	0	0
$887$	780.646i	0.880097i	0.897974	+	0.440048i	$0.145039\pi$
$887$	780.646i	0.880097i	−0.897974	+	0.440048i	$0.854961\pi$
$888$	0	0
$889$	1145.51i	1.28854i
$890$	0	0
$891$	355.000	0.398429
$892$	0	0
$893$	95.0000	0.106383
$894$	0	0
$895$	− 127.279i	− 0.142211i
$896$	0	0
$897$	−480.000	−0.535117
$898$	0	0
$899$	−1800.00	−2.00222
$900$	0	0
$901$	636.396i	0.706322i
$902$	0	0
$903$	− 70.7107i	− 0.0783064i
$904$	0	0
$905$	254.558i	0.281280i
$906$	0	0
$907$	313.955i	0.346147i	0.984909	+	0.173074i	$0.0553698\pi$
$907$	313.955i	0.346147i	−0.984909	+	0.173074i	$0.944630\pi$
$908$	0	0
$909$	50.0000	0.0550055
$910$	0	0
$911$	− 933.381i	− 1.02457i	−0.858816	−	0.512284i	$-0.828800\pi$
$911$	− 933.381i	− 1.02457i	0.858816	−	0.512284i	$-0.171200\pi$
$912$	0	0
$913$	−650.000	−0.711939
$914$	0	0
$915$	268.701i	0.293662i
$916$	0	0
$917$	−815.000	−0.888768
$918$	0	0
$919$	538.000	0.585419	0.292709	−	0.956201i	$-0.405443\pi$
$919$	538.000	0.585419	0.292709	+	0.956201i	$0.405443\pi$
$920$	0	0
$921$	−792.000	−0.859935
$922$	0	0
$923$	0	0
$924$	0	0
$925$	610.940i	0.660476i
$926$	0	0
$927$	16.9706i	0.0183070i
$928$	0	0
$929$	−742.000	−0.798708	−0.399354	−	0.916797i	$-0.630766\pi$
$929$	−742.000	−0.798708	−0.399354	+	0.916797i	$0.630766\pi$
$930$	0	0
$931$	456.000	0.489796
$932$	0	0
$933$	− 664.680i	− 0.712412i
$934$	0	0
$935$	−125.000	−0.133690
$936$	0	0
$937$	335.000	0.357524	0.178762	−	0.983892i	$-0.442791\pi$
$937$	335.000	0.357524	0.178762	+	0.983892i	$0.442791\pi$
$938$	0	0
$939$	876.812i	0.933773i
$940$	0	0
$941$	− 424.264i	− 0.450865i	−0.974259	−	0.225433i	$-0.927620\pi$
$941$	− 424.264i	− 0.450865i	0.974259	−	0.225433i	$-0.0723795\pi$
$942$	0	0
$943$	− 424.264i	− 0.449909i
$944$	0	0
$945$	− 141.421i	− 0.149652i
$946$	0	0
$947$	1210.00	1.27772	0.638860	−	0.769323i	$-0.279406\pi$
$947$	1210.00	1.27772	0.638860	+	0.769323i	$0.279406\pi$
$948$	0	0
$949$	424.264i	0.447064i
$950$	0	0
$951$	−528.000	−0.555205
$952$	0	0
$953$	992.778i	1.04174i	0.853636	+	0.520870i	$0.174392\pi$
$953$	992.778i	1.04174i	−0.853636	+	0.520870i	$0.825608\pi$
$954$	0	0
$955$	293.000	0.306806
$956$	0	0
$957$	−600.000	−0.626959
$958$	0	0
$959$	−475.000	−0.495308
$960$	0	0
$961$	−839.000	−0.873049
$962$	0	0
$963$	101.823i	0.105736i
$964$	0	0
$965$	− 59.3970i	− 0.0615513i
$966$	0	0
$967$	−350.000	−0.361944	−0.180972	−	0.983488i	$-0.557924\pi$
$967$	−350.000	−0.361944	−0.180972	+	0.983488i	$0.557924\pi$
$968$	0	0
$969$	− 1343.50i	− 1.38648i
$970$	0	0
$971$	− 254.558i	− 0.262161i	−0.991372	−	0.131081i	$-0.958155\pi$
$971$	− 254.558i	− 0.262161i	0.991372	−	0.131081i	$-0.0418447\pi$
$972$	0	0
$973$	625.000	0.642343
$974$	0	0
$975$	1152.00	1.18154
$976$	0	0
$977$	− 398.808i	− 0.408197i	−0.978950	−	0.204098i	$-0.934574\pi$
$977$	− 398.808i	− 0.408197i	0.978950	−	0.204098i	$-0.0654262\pi$
$978$	0	0
$979$	636.396i	0.650047i
$980$	0	0
$981$	127.279i	0.129744i
$982$	0	0
$983$	695.793i	0.707826i	0.935278	+	0.353913i	$0.115149\pi$
$983$	695.793i	0.707826i	−0.935278	+	0.353913i	$0.884851\pi$
$984$	0	0
$985$	70.0000	0.0710660
$986$	0	0
$987$	− 70.7107i	− 0.0716420i
$988$	0	0
$989$	−50.0000	−0.0505561
$990$	0	0
$991$	− 381.838i	− 0.385305i	−0.981267	−	0.192653i	$-0.938291\pi$
$991$	− 381.838i	− 0.385305i	0.981267	−	0.192653i	$-0.0617091\pi$
$992$	0	0
$993$	−840.000	−0.845921
$994$	0	0
$995$	173.000	0.173869
$996$	0	0
$997$	−265.000	−0.265797	−0.132899	−	0.991130i	$-0.542428\pi$
$997$	−265.000	−0.265797	−0.132899	+	0.991130i	$0.542428\pi$
$998$	0	0
$999$	−720.000	−0.720721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.3.e.c.113.1		2
3.2	odd	2		2736.3.o.h.721.1		2
4.3	odd	2		38.3.b.a.37.2	yes	2
8.3	odd	2		1216.3.e.j.1025.1		2
8.5	even	2		1216.3.e.i.1025.2		2
12.11	even	2		342.3.d.a.37.1		2
19.18	odd	2	inner	304.3.e.c.113.2		2
20.3	even	4		950.3.d.a.949.3		4
20.7	even	4		950.3.d.a.949.2		4
20.19	odd	2		950.3.c.a.151.1		2
57.56	even	2		2736.3.o.h.721.2		2
76.75	even	2		38.3.b.a.37.1	✓	2
152.37	odd	2		1216.3.e.i.1025.1		2
152.75	even	2		1216.3.e.j.1025.2		2
228.227	odd	2		342.3.d.a.37.2		2
380.227	odd	4		950.3.d.a.949.4		4
380.303	odd	4		950.3.d.a.949.1		4
380.379	even	2		950.3.c.a.151.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.b.a.37.1	✓	2	76.75	even	2
38.3.b.a.37.2	yes	2	4.3	odd	2
304.3.e.c.113.1		2	1.1	even	1	trivial
304.3.e.c.113.2		2	19.18	odd	2	inner
342.3.d.a.37.1		2	12.11	even	2
342.3.d.a.37.2		2	228.227	odd	2
950.3.c.a.151.1		2	20.19	odd	2
950.3.c.a.151.2		2	380.379	even	2
950.3.d.a.949.1		4	380.303	odd	4
950.3.d.a.949.2		4	20.7	even	4
950.3.d.a.949.3		4	20.3	even	4
950.3.d.a.949.4		4	380.227	odd	4
1216.3.e.i.1025.1		2	152.37	odd	2
1216.3.e.i.1025.2		2	8.5	even	2
1216.3.e.j.1025.1		2	8.3	odd	2
1216.3.e.j.1025.2		2	152.75	even	2
2736.3.o.h.721.1		2	3.2	odd	2
2736.3.o.h.721.2		2	57.56	even	2

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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{3} -1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.82843i q^{3} -1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -16.9706i q^{13} +2.82843i q^{15} -25.0000 q^{17} -19.0000 q^{19} +14.1421i q^{21} +10.0000 q^{23} -24.0000 q^{25} -28.2843i q^{27} +42.4264i q^{29} +42.4264i q^{31} +14.1421i q^{33} +5.00000 q^{35} -25.4558i q^{37} -48.0000 q^{39} -42.4264i q^{41} -5.00000 q^{43} -1.00000 q^{45} -5.00000 q^{47} -24.0000 q^{49} +70.7107i q^{51} -25.4558i q^{53} +5.00000 q^{55} +53.7401i q^{57} -84.8528i q^{59} +95.0000 q^{61} -5.00000 q^{63} +16.9706i q^{65} -110.309i q^{67} -28.2843i q^{69} -25.0000 q^{73} +67.8823i q^{75} +25.0000 q^{77} -42.4264i q^{79} -71.0000 q^{81} +130.000 q^{83} +25.0000 q^{85} +120.000 q^{87} -127.279i q^{89} +84.8528i q^{91} +120.000 q^{93} +19.0000 q^{95} +16.9706i q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 10 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 10 * q^7 + 2 * q^9	\( 2 q - 2 q^{5} - 10 q^{7} + 2 q^{9} - 10 q^{11} - 50 q^{17} - 38 q^{19} + 20 q^{23} - 48 q^{25} + 10 q^{35} - 96 q^{39} - 10 q^{43} - 2 q^{45} - 10 q^{47} - 48 q^{49} + 10 q^{55} + 190 q^{61} - 10 q^{63} - 50 q^{73} + 50 q^{77} - 142 q^{81} + 260 q^{83} + 50 q^{85} + 240 q^{87} + 240 q^{93} + 38 q^{95} - 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 10 * q^7 + 2 * q^9 - 10 * q^11 - 50 * q^17 - 38 * q^19 + 20 * q^23 - 48 * q^25 + 10 * q^35 - 96 * q^39 - 10 * q^43 - 2 * q^45 - 10 * q^47 - 48 * q^49 + 10 * q^55 + 190 * q^61 - 10 * q^63 - 50 * q^73 + 50 * q^77 - 142 * q^81 + 260 * q^83 + 50 * q^85 + 240 * q^87 + 240 * q^93 + 38 * q^95 - 10 * q^99

Introduction

L-functions

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