LMFDB - Embedded newform 4304.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{3} -3.61803 q^{5} +4.23607 q^{7} -2.61803 q^{9} -0.236068 q^{11} -1.00000 q^{13} +2.23607 q^{15} +0.236068 q^{17} -2.00000 q^{19} -2.61803 q^{21} +8.23607 q^{23} +8.09017 q^{25} +3.47214 q^{27} +0.854102 q^{29} -7.47214 q^{31} +0.145898 q^{33} -15.3262 q^{35} -8.70820 q^{37} +0.618034 q^{39} +7.70820 q^{41} -2.23607 q^{43} +9.47214 q^{45} +6.85410 q^{47} +10.9443 q^{49} -0.145898 q^{51} +0.854102 q^{55} +1.23607 q^{57} +6.56231 q^{59} -5.47214 q^{61} -11.0902 q^{63} +3.61803 q^{65} -1.76393 q^{67} -5.09017 q^{69} -9.70820 q^{71} -10.7082 q^{73} -5.00000 q^{75} -1.00000 q^{77} +13.7984 q^{79} +5.70820 q^{81} +5.23607 q^{83} -0.854102 q^{85} -0.527864 q^{87} -9.79837 q^{89} -4.23607 q^{91} +4.61803 q^{93} +7.23607 q^{95} +12.6525 q^{97} +0.618034 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 5 q^{5} + 4 q^{7} - 3 q^{9} + 4 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 3 q^{21} + 12 q^{23} + 5 q^{25} - 2 q^{27} - 5 q^{29} - 6 q^{31} + 7 q^{33} - 15 q^{35} - 4 q^{37} - q^{39} + 2 q^{41}+ \cdots - q^{99}+O(q^{100}) $ 2 * q + q^3 - 5 * q^5 + 4 * q^7 - 3 * q^9 + 4 * q^11 - 2 * q^13 - 4 * q^17 - 4 * q^19 - 3 * q^21 + 12 * q^23 + 5 * q^25 - 2 * q^27 - 5 * q^29 - 6 * q^31 + 7 * q^33 - 15 * q^35 - 4 * q^37 - q^39 + 2 * q^41 + 10 * q^45 + 7 * q^47 + 4 * q^49 - 7 * q^51 - 5 * q^55 - 2 * q^57 - 7 * q^59 - 2 * q^61 - 11 * q^63 + 5 * q^65 - 8 * q^67 + q^69 - 6 * q^71 - 8 * q^73 - 10 * q^75 - 2 * q^77 + 3 * q^79 - 2 * q^81 + 6 * q^83 + 5 * q^85 - 10 * q^87 + 5 * q^89 - 4 * q^91 + 7 * q^93 + 10 * q^95 - 6 * q^97 - q^99

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
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	0	0
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	2.23607	0.577350
$16$	0	0
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−2.61803	−0.571302
$22$	0	0
$23$	8.23607	1.71734	0.858669	−	0.512530i	$-0.171292\pi$
$23$	8.23607	1.71734	0.858669	+	0.512530i	$0.171292\pi$
$24$	0	0
$25$	8.09017	1.61803
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	0.854102	0.158603	0.0793014	−	0.996851i	$-0.474731\pi$
$29$	0.854102	0.158603	0.0793014	+	0.996851i	$0.474731\pi$
$30$	0	0
$31$	−7.47214	−1.34204	−0.671018	−	0.741441i	$-0.734143\pi$
$31$	−7.47214	−1.34204	−0.671018	+	0.741441i	$0.734143\pi$
$32$	0	0
$33$	0.145898	0.0253976
$34$	0	0
$35$	−15.3262	−2.59061
$36$	0	0
$37$	−8.70820	−1.43162	−0.715810	−	0.698295i	$-0.753942\pi$
$37$	−8.70820	−1.43162	−0.715810	+	0.698295i	$0.753942\pi$
$38$	0	0
$39$	0.618034	0.0989646
$40$	0	0
$41$	7.70820	1.20382	0.601910	−	0.798564i	$-0.294407\pi$
$41$	7.70820	1.20382	0.601910	+	0.798564i	$0.294407\pi$
$42$	0	0
$43$	−2.23607	−0.340997	−0.170499	−	0.985358i	$-0.554538\pi$
$43$	−2.23607	−0.340997	−0.170499	+	0.985358i	$0.554538\pi$
$44$	0	0
$45$	9.47214	1.41202
$46$	0	0
$47$	6.85410	0.999774	0.499887	−	0.866091i	$-0.333375\pi$
$47$	6.85410	0.999774	0.499887	+	0.866091i	$0.333375\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	−0.145898	−0.0204298
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0.854102	0.115167
$56$	0	0
$57$	1.23607	0.163721
$58$	0	0
$59$	6.56231	0.854339	0.427170	−	0.904171i	$-0.359511\pi$
$59$	6.56231	0.854339	0.427170	+	0.904171i	$0.359511\pi$
$60$	0	0
$61$	−5.47214	−0.700635	−0.350318	−	0.936631i	$-0.613926\pi$
$61$	−5.47214	−0.700635	−0.350318	+	0.936631i	$0.613926\pi$
$62$	0	0
$63$	−11.0902	−1.39723
$64$	0	0
$65$	3.61803	0.448762
$66$	0	0
$67$	−1.76393	−0.215499	−0.107749	−	0.994178i	$-0.534364\pi$
$67$	−1.76393	−0.215499	−0.107749	+	0.994178i	$0.534364\pi$
$68$	0	0
$69$	−5.09017	−0.612784
$70$	0	0
$71$	−9.70820	−1.15215	−0.576076	−	0.817396i	$-0.695417\pi$
$71$	−9.70820	−1.15215	−0.576076	+	0.817396i	$0.695417\pi$
$72$	0	0
$73$	−10.7082	−1.25330	−0.626650	−	0.779301i	$-0.715575\pi$
$73$	−10.7082	−1.25330	−0.626650	+	0.779301i	$0.715575\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	13.7984	1.55244	0.776219	−	0.630463i	$-0.217135\pi$
$79$	13.7984	1.55244	0.776219	+	0.630463i	$0.217135\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	5.23607	0.574733	0.287367	−	0.957821i	$-0.407220\pi$
$83$	5.23607	0.574733	0.287367	+	0.957821i	$0.407220\pi$
$84$	0	0
$85$	−0.854102	−0.0926404
$86$	0	0
$87$	−0.527864	−0.0565930
$88$	0	0
$89$	−9.79837	−1.03863	−0.519313	−	0.854584i	$-0.673812\pi$
$89$	−9.79837	−1.03863	−0.519313	+	0.854584i	$0.673812\pi$
$90$	0	0
$91$	−4.23607	−0.444061
$92$	0	0
$93$	4.61803	0.478868
$94$	0	0
$95$	7.23607	0.742405
$96$	0	0
$97$	12.6525	1.28466	0.642332	−	0.766426i	$-0.277967\pi$
$97$	12.6525	1.28466	0.642332	+	0.766426i	$0.277967\pi$
$98$	0	0
$99$	0.618034	0.0621148
$100$	0	0
$101$	−5.09017	−0.506491	−0.253245	−	0.967402i	$-0.581498\pi$
$101$	−5.09017	−0.506491	−0.253245	+	0.967402i	$0.581498\pi$
$102$	0	0
$103$	−10.8541	−1.06949	−0.534743	−	0.845015i	$-0.679592\pi$
$103$	−10.8541	−1.06949	−0.534743	+	0.845015i	$0.679592\pi$
$104$	0	0
$105$	9.47214	0.924386
$106$	0	0
$107$	12.2361	1.18291	0.591453	−	0.806340i	$-0.298555\pi$
$107$	12.2361	1.18291	0.591453	+	0.806340i	$0.298555\pi$
$108$	0	0
$109$	−4.70820	−0.450964	−0.225482	−	0.974247i	$-0.572396\pi$
$109$	−4.70820	−0.450964	−0.225482	+	0.974247i	$0.572396\pi$
$110$	0	0
$111$	5.38197	0.510834
$112$	0	0
$113$	−15.3820	−1.44701	−0.723507	−	0.690317i	$-0.757471\pi$
$113$	−15.3820	−1.44701	−0.723507	+	0.690317i	$0.757471\pi$
$114$	0	0
$115$	−29.7984	−2.77871
$116$	0	0
$117$	2.61803	0.242037
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−10.9443	−0.994934
$122$	0	0
$123$	−4.76393	−0.429549
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−20.7984	−1.84556	−0.922779	−	0.385331i	$-0.874087\pi$
$127$	−20.7984	−1.84556	−0.922779	+	0.385331i	$0.874087\pi$
$128$	0	0
$129$	1.38197	0.121675
$130$	0	0
$131$	−2.76393	−0.241486	−0.120743	−	0.992684i	$-0.538528\pi$
$131$	−2.76393	−0.241486	−0.120743	+	0.992684i	$0.538528\pi$
$132$	0	0
$133$	−8.47214	−0.734627
$134$	0	0
$135$	−12.5623	−1.08119
$136$	0	0
$137$	−15.6180	−1.33434	−0.667169	−	0.744906i	$-0.732494\pi$
$137$	−15.6180	−1.33434	−0.667169	+	0.744906i	$0.732494\pi$
$138$	0	0
$139$	−10.5623	−0.895883	−0.447942	−	0.894063i	$-0.647843\pi$
$139$	−10.5623	−0.895883	−0.447942	+	0.894063i	$0.647843\pi$
$140$	0	0
$141$	−4.23607	−0.356741
$142$	0	0
$143$	0.236068	0.0197410
$144$	0	0
$145$	−3.09017	−0.256625
$146$	0	0
$147$	−6.76393	−0.557880
$148$	0	0
$149$	0.326238	0.0267265	0.0133632	−	0.999911i	$-0.495746\pi$
$149$	0.326238	0.0267265	0.0133632	+	0.999911i	$0.495746\pi$
$150$	0	0
$151$	9.94427	0.809253	0.404627	−	0.914482i	$-0.367401\pi$
$151$	9.94427	0.809253	0.404627	+	0.914482i	$0.367401\pi$
$152$	0	0
$153$	−0.618034	−0.0499651
$154$	0	0
$155$	27.0344	2.17146
$156$	0	0
$157$	22.6525	1.80786	0.903932	−	0.427676i	$-0.140668\pi$
$157$	22.6525	1.80786	0.903932	+	0.427676i	$0.140668\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	34.8885	2.74960
$162$	0	0
$163$	−17.3262	−1.35710	−0.678548	−	0.734556i	$-0.737390\pi$
$163$	−17.3262	−1.35710	−0.678548	+	0.734556i	$0.737390\pi$
$164$	0	0
$165$	−0.527864	−0.0410942
$166$	0	0
$167$	−3.61803	−0.279972	−0.139986	−	0.990153i	$-0.544706\pi$
$167$	−3.61803	−0.279972	−0.139986	+	0.990153i	$0.544706\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	5.23607	0.400412
$172$	0	0
$173$	−7.47214	−0.568096	−0.284048	−	0.958810i	$-0.591678\pi$
$173$	−7.47214	−0.568096	−0.284048	+	0.958810i	$0.591678\pi$
$174$	0	0
$175$	34.2705	2.59061
$176$	0	0
$177$	−4.05573	−0.304847
$178$	0	0
$179$	−18.7082	−1.39832	−0.699158	−	0.714967i	$-0.746442\pi$
$179$	−18.7082	−1.39832	−0.699158	+	0.714967i	$0.746442\pi$
$180$	0	0
$181$	19.4164	1.44321	0.721605	−	0.692305i	$-0.243405\pi$
$181$	19.4164	1.44321	0.721605	+	0.692305i	$0.243405\pi$
$182$	0	0
$183$	3.38197	0.250002
$184$	0	0
$185$	31.5066	2.31641
$186$	0	0
$187$	−0.0557281	−0.00407524
$188$	0	0
$189$	14.7082	1.06986
$190$	0	0
$191$	5.05573	0.365820	0.182910	−	0.983130i	$-0.441448\pi$
$191$	5.05573	0.365820	0.182910	+	0.983130i	$0.441448\pi$
$192$	0	0
$193$	−8.23607	−0.592845	−0.296423	−	0.955057i	$-0.595794\pi$
$193$	−8.23607	−0.592845	−0.296423	+	0.955057i	$0.595794\pi$
$194$	0	0
$195$	−2.23607	−0.160128
$196$	0	0
$197$	−22.3820	−1.59465	−0.797325	−	0.603551i	$-0.793752\pi$
$197$	−22.3820	−1.59465	−0.797325	+	0.603551i	$0.793752\pi$
$198$	0	0
$199$	5.38197	0.381517	0.190759	−	0.981637i	$-0.438905\pi$
$199$	5.38197	0.381517	0.190759	+	0.981637i	$0.438905\pi$
$200$	0	0
$201$	1.09017	0.0768947
$202$	0	0
$203$	3.61803	0.253936
$204$	0	0
$205$	−27.8885	−1.94782
$206$	0	0
$207$	−21.5623	−1.49868
$208$	0	0
$209$	0.472136	0.0326583
$210$	0	0
$211$	−0.854102	−0.0587988	−0.0293994	−	0.999568i	$-0.509359\pi$
$211$	−0.854102	−0.0587988	−0.0293994	+	0.999568i	$0.509359\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	8.09017	0.551745
$216$	0	0
$217$	−31.6525	−2.14871
$218$	0	0
$219$	6.61803	0.447205
$220$	0	0
$221$	−0.236068	−0.0158797
$222$	0	0
$223$	5.32624	0.356671	0.178336	−	0.983970i	$-0.442929\pi$
$223$	5.32624	0.356671	0.178336	+	0.983970i	$0.442929\pi$
$224$	0	0
$225$	−21.1803	−1.41202
$226$	0	0
$227$	24.0344	1.59522	0.797611	−	0.603172i	$-0.206097\pi$
$227$	24.0344	1.59522	0.797611	+	0.603172i	$0.206097\pi$
$228$	0	0
$229$	−10.7082	−0.707618	−0.353809	−	0.935318i	$-0.615114\pi$
$229$	−10.7082	−0.707618	−0.353809	+	0.935318i	$0.615114\pi$
$230$	0	0
$231$	0.618034	0.0406637
$232$	0	0
$233$	4.23607	0.277514	0.138757	−	0.990326i	$-0.455689\pi$
$233$	4.23607	0.277514	0.138757	+	0.990326i	$0.455689\pi$
$234$	0	0
$235$	−24.7984	−1.61767
$236$	0	0
$237$	−8.52786	−0.553944
$238$	0	0
$239$	19.2705	1.24651	0.623253	−	0.782020i	$-0.285811\pi$
$239$	19.2705	1.24651	0.623253	+	0.782020i	$0.285811\pi$
$240$	0	0
$241$	−4.70820	−0.303282	−0.151641	−	0.988436i	$-0.548456\pi$
$241$	−4.70820	−0.303282	−0.151641	+	0.988436i	$0.548456\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	−39.5967	−2.52974
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−3.23607	−0.205077
$250$	0	0
$251$	−24.0902	−1.52056	−0.760279	−	0.649597i	$-0.774938\pi$
$251$	−24.0902	−1.52056	−0.760279	+	0.649597i	$0.774938\pi$
$252$	0	0
$253$	−1.94427	−0.122235
$254$	0	0
$255$	0.527864	0.0330561
$256$	0	0
$257$	−5.47214	−0.341342	−0.170671	−	0.985328i	$-0.554594\pi$
$257$	−5.47214	−0.341342	−0.170671	+	0.985328i	$0.554594\pi$
$258$	0	0
$259$	−36.8885	−2.29214
$260$	0	0
$261$	−2.23607	−0.138409
$262$	0	0
$263$	−23.2705	−1.43492	−0.717461	−	0.696599i	$-0.754696\pi$
$263$	−23.2705	−1.43492	−0.717461	+	0.696599i	$0.754696\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.05573	0.370605
$268$	0	0
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	−24.1246	−1.46547	−0.732733	−	0.680516i	$-0.761756\pi$
$271$	−24.1246	−1.46547	−0.732733	+	0.680516i	$0.761756\pi$
$272$	0	0
$273$	2.61803	0.158451
$274$	0	0
$275$	−1.90983	−0.115167
$276$	0	0
$277$	−21.3820	−1.28472	−0.642359	−	0.766404i	$-0.722044\pi$
$277$	−21.3820	−1.28472	−0.642359	+	0.766404i	$0.722044\pi$
$278$	0	0
$279$	19.5623	1.17116
$280$	0	0
$281$	−29.1803	−1.74075	−0.870377	−	0.492387i	$-0.836125\pi$
$281$	−29.1803	−1.74075	−0.870377	+	0.492387i	$0.836125\pi$
$282$	0	0
$283$	−5.76393	−0.342630	−0.171315	−	0.985216i	$-0.554802\pi$
$283$	−5.76393	−0.342630	−0.171315	+	0.985216i	$0.554802\pi$
$284$	0	0
$285$	−4.47214	−0.264906
$286$	0	0
$287$	32.6525	1.92741
$288$	0	0
$289$	−16.9443	−0.996722
$290$	0	0
$291$	−7.81966	−0.458397
$292$	0	0
$293$	7.52786	0.439783	0.219891	−	0.975524i	$-0.429430\pi$
$293$	7.52786	0.439783	0.219891	+	0.975524i	$0.429430\pi$
$294$	0	0
$295$	−23.7426	−1.38235
$296$	0	0
$297$	−0.819660	−0.0475615
$298$	0	0
$299$	−8.23607	−0.476304
$300$	0	0
$301$	−9.47214	−0.545965
$302$	0	0
$303$	3.14590	0.180727
$304$	0	0
$305$	19.7984	1.13365
$306$	0	0
$307$	−32.1246	−1.83345	−0.916724	−	0.399521i	$-0.869177\pi$
$307$	−32.1246	−1.83345	−0.916724	+	0.399521i	$0.869177\pi$
$308$	0	0
$309$	6.70820	0.381616
$310$	0	0
$311$	17.2361	0.977368	0.488684	−	0.872461i	$-0.337477\pi$
$311$	17.2361	0.977368	0.488684	+	0.872461i	$0.337477\pi$
$312$	0	0
$313$	13.1459	0.743050	0.371525	−	0.928423i	$-0.378835\pi$
$313$	13.1459	0.743050	0.371525	+	0.928423i	$0.378835\pi$
$314$	0	0
$315$	40.1246	2.26077
$316$	0	0
$317$	−30.2361	−1.69823	−0.849113	−	0.528211i	$-0.822863\pi$
$317$	−30.2361	−1.69823	−0.849113	+	0.528211i	$0.822863\pi$
$318$	0	0
$319$	−0.201626	−0.0112889
$320$	0	0
$321$	−7.56231	−0.422087
$322$	0	0
$323$	−0.472136	−0.0262703
$324$	0	0
$325$	−8.09017	−0.448762
$326$	0	0
$327$	2.90983	0.160914
$328$	0	0
$329$	29.0344	1.60072
$330$	0	0
$331$	−0.819660	−0.0450526	−0.0225263	−	0.999746i	$-0.507171\pi$
$331$	−0.819660	−0.0450526	−0.0225263	+	0.999746i	$0.507171\pi$
$332$	0	0
$333$	22.7984	1.24934
$334$	0	0
$335$	6.38197	0.348684
$336$	0	0
$337$	−27.9787	−1.52410	−0.762049	−	0.647520i	$-0.775806\pi$
$337$	−27.9787	−1.52410	−0.762049	+	0.647520i	$0.775806\pi$
$338$	0	0
$339$	9.50658	0.516326
$340$	0	0
$341$	1.76393	0.0955223
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	18.4164	0.991506
$346$	0	0
$347$	−13.3607	−0.717239	−0.358619	−	0.933484i	$-0.616752\pi$
$347$	−13.3607	−0.717239	−0.358619	+	0.933484i	$0.616752\pi$
$348$	0	0
$349$	21.2705	1.13858	0.569292	−	0.822135i	$-0.307217\pi$
$349$	21.2705	1.13858	0.569292	+	0.822135i	$0.307217\pi$
$350$	0	0
$351$	−3.47214	−0.185329
$352$	0	0
$353$	−9.85410	−0.524481	−0.262240	−	0.965003i	$-0.584461\pi$
$353$	−9.85410	−0.524481	−0.262240	+	0.965003i	$0.584461\pi$
$354$	0	0
$355$	35.1246	1.86422
$356$	0	0
$357$	−0.618034	−0.0327098
$358$	0	0
$359$	16.9443	0.894284	0.447142	−	0.894463i	$-0.352442\pi$
$359$	16.9443	0.894284	0.447142	+	0.894463i	$0.352442\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	6.76393	0.355014
$364$	0	0
$365$	38.7426	2.02788
$366$	0	0
$367$	9.20163	0.480321	0.240160	−	0.970733i	$-0.422800\pi$
$367$	9.20163	0.480321	0.240160	+	0.970733i	$0.422800\pi$
$368$	0	0
$369$	−20.1803	−1.05055
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6.94427	0.359561	0.179780	−	0.983707i	$-0.442461\pi$
$373$	6.94427	0.359561	0.179780	+	0.983707i	$0.442461\pi$
$374$	0	0
$375$	6.90983	0.356822
$376$	0	0
$377$	−0.854102	−0.0439885
$378$	0	0
$379$	−6.14590	−0.315694	−0.157847	−	0.987464i	$-0.550455\pi$
$379$	−6.14590	−0.315694	−0.157847	+	0.987464i	$0.550455\pi$
$380$	0	0
$381$	12.8541	0.658536
$382$	0	0
$383$	17.3820	0.888177	0.444088	−	0.895983i	$-0.353528\pi$
$383$	17.3820	0.888177	0.444088	+	0.895983i	$0.353528\pi$
$384$	0	0
$385$	3.61803	0.184392
$386$	0	0
$387$	5.85410	0.297581
$388$	0	0
$389$	28.9443	1.46753	0.733766	−	0.679402i	$-0.237761\pi$
$389$	28.9443	1.46753	0.733766	+	0.679402i	$0.237761\pi$
$390$	0	0
$391$	1.94427	0.0983261
$392$	0	0
$393$	1.70820	0.0861675
$394$	0	0
$395$	−49.9230	−2.51190
$396$	0	0
$397$	−34.2705	−1.71999	−0.859994	−	0.510304i	$-0.829533\pi$
$397$	−34.2705	−1.71999	−0.859994	+	0.510304i	$0.829533\pi$
$398$	0	0
$399$	5.23607	0.262131
$400$	0	0
$401$	−4.27051	−0.213259	−0.106630	−	0.994299i	$-0.534006\pi$
$401$	−4.27051	−0.213259	−0.106630	+	0.994299i	$0.534006\pi$
$402$	0	0
$403$	7.47214	0.372214
$404$	0	0
$405$	−20.6525	−1.02623
$406$	0	0
$407$	2.05573	0.101899
$408$	0	0
$409$	−11.6525	−0.576178	−0.288089	−	0.957604i	$-0.593020\pi$
$409$	−11.6525	−0.576178	−0.288089	+	0.957604i	$0.593020\pi$
$410$	0	0
$411$	9.65248	0.476122
$412$	0	0
$413$	27.7984	1.36787
$414$	0	0
$415$	−18.9443	−0.929938
$416$	0	0
$417$	6.52786	0.319671
$418$	0	0
$419$	17.4377	0.851887	0.425944	−	0.904750i	$-0.359942\pi$
$419$	17.4377	0.851887	0.425944	+	0.904750i	$0.359942\pi$
$420$	0	0
$421$	11.4721	0.559118	0.279559	−	0.960129i	$-0.409812\pi$
$421$	11.4721	0.559118	0.279559	+	0.960129i	$0.409812\pi$
$422$	0	0
$423$	−17.9443	−0.872480
$424$	0	0
$425$	1.90983	0.0926404
$426$	0	0
$427$	−23.1803	−1.12178
$428$	0	0
$429$	−0.145898	−0.00704402
$430$	0	0
$431$	15.4721	0.745267	0.372633	−	0.927979i	$-0.378455\pi$
$431$	15.4721	0.745267	0.372633	+	0.927979i	$0.378455\pi$
$432$	0	0
$433$	12.2016	0.586373	0.293186	−	0.956055i	$-0.405284\pi$
$433$	12.2016	0.586373	0.293186	+	0.956055i	$0.405284\pi$
$434$	0	0
$435$	1.90983	0.0915693
$436$	0	0
$437$	−16.4721	−0.787969
$438$	0	0
$439$	−19.1459	−0.913784	−0.456892	−	0.889522i	$-0.651037\pi$
$439$	−19.1459	−0.913784	−0.456892	+	0.889522i	$0.651037\pi$
$440$	0	0
$441$	−28.6525	−1.36440
$442$	0	0
$443$	−33.4164	−1.58766	−0.793831	−	0.608139i	$-0.791916\pi$
$443$	−33.4164	−1.58766	−0.793831	+	0.608139i	$0.791916\pi$
$444$	0	0
$445$	35.4508	1.68053
$446$	0	0
$447$	−0.201626	−0.00953659
$448$	0	0
$449$	2.90983	0.137323	0.0686617	−	0.997640i	$-0.478127\pi$
$449$	2.90983	0.137323	0.0686617	+	0.997640i	$0.478127\pi$
$450$	0	0
$451$	−1.81966	−0.0856844
$452$	0	0
$453$	−6.14590	−0.288759
$454$	0	0
$455$	15.3262	0.718505
$456$	0	0
$457$	26.7984	1.25358	0.626788	−	0.779190i	$-0.284369\pi$
$457$	26.7984	1.25358	0.626788	+	0.779190i	$0.284369\pi$
$458$	0	0
$459$	0.819660	0.0382585
$460$	0	0
$461$	21.1246	0.983871	0.491936	−	0.870632i	$-0.336290\pi$
$461$	21.1246	0.983871	0.491936	+	0.870632i	$0.336290\pi$
$462$	0	0
$463$	20.0902	0.933669	0.466835	−	0.884345i	$-0.345394\pi$
$463$	20.0902	0.933669	0.466835	+	0.884345i	$0.345394\pi$
$464$	0	0
$465$	−16.7082	−0.774824
$466$	0	0
$467$	6.58359	0.304652	0.152326	−	0.988330i	$-0.451324\pi$
$467$	6.58359	0.304652	0.152326	+	0.988330i	$0.451324\pi$
$468$	0	0
$469$	−7.47214	−0.345031
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	0.527864	0.0242712
$474$	0	0
$475$	−16.1803	−0.742405
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.5967	1.67215	0.836074	−	0.548617i	$-0.184845\pi$
$479$	36.5967	1.67215	0.836074	+	0.548617i	$0.184845\pi$
$480$	0	0
$481$	8.70820	0.397060
$482$	0	0
$483$	−21.5623	−0.981119
$484$	0	0
$485$	−45.7771	−2.07863
$486$	0	0
$487$	14.8328	0.672139	0.336070	−	0.941837i	$-0.390902\pi$
$487$	14.8328	0.672139	0.336070	+	0.941837i	$0.390902\pi$
$488$	0	0
$489$	10.7082	0.484242
$490$	0	0
$491$	−6.76393	−0.305252	−0.152626	−	0.988284i	$-0.548773\pi$
$491$	−6.76393	−0.305252	−0.152626	+	0.988284i	$0.548773\pi$
$492$	0	0
$493$	0.201626	0.00908078
$494$	0	0
$495$	−2.23607	−0.100504
$496$	0	0
$497$	−41.1246	−1.84469
$498$	0	0
$499$	−11.9443	−0.534699	−0.267350	−	0.963600i	$-0.586148\pi$
$499$	−11.9443	−0.534699	−0.267350	+	0.963600i	$0.586148\pi$
$500$	0	0
$501$	2.23607	0.0999001
$502$	0	0
$503$	−34.3607	−1.53207	−0.766033	−	0.642801i	$-0.777772\pi$
$503$	−34.3607	−1.53207	−0.766033	+	0.642801i	$0.777772\pi$
$504$	0	0
$505$	18.4164	0.819519
$506$	0	0
$507$	7.41641	0.329374
$508$	0	0
$509$	26.6738	1.18229	0.591147	−	0.806564i	$-0.298675\pi$
$509$	26.6738	1.18229	0.591147	+	0.806564i	$0.298675\pi$
$510$	0	0
$511$	−45.3607	−2.00664
$512$	0	0
$513$	−6.94427	−0.306597
$514$	0	0
$515$	39.2705	1.73047
$516$	0	0
$517$	−1.61803	−0.0711611
$518$	0	0
$519$	4.61803	0.202709
$520$	0	0
$521$	30.4164	1.33257	0.666284	−	0.745699i	$-0.267884\pi$
$521$	30.4164	1.33257	0.666284	+	0.745699i	$0.267884\pi$
$522$	0	0
$523$	−27.6180	−1.20765	−0.603826	−	0.797116i	$-0.706358\pi$
$523$	−27.6180	−1.20765	−0.603826	+	0.797116i	$0.706358\pi$
$524$	0	0
$525$	−21.1803	−0.924386
$526$	0	0
$527$	−1.76393	−0.0768381
$528$	0	0
$529$	44.8328	1.94925
$530$	0	0
$531$	−17.1803	−0.745563
$532$	0	0
$533$	−7.70820	−0.333879
$534$	0	0
$535$	−44.2705	−1.91398
$536$	0	0
$537$	11.5623	0.498950
$538$	0	0
$539$	−2.58359	−0.111283
$540$	0	0
$541$	20.4721	0.880166	0.440083	−	0.897957i	$-0.354949\pi$
$541$	20.4721	0.880166	0.440083	+	0.897957i	$0.354949\pi$
$542$	0	0
$543$	−12.0000	−0.514969
$544$	0	0
$545$	17.0344	0.729675
$546$	0	0
$547$	−3.20163	−0.136892	−0.0684458	−	0.997655i	$-0.521804\pi$
$547$	−3.20163	−0.136892	−0.0684458	+	0.997655i	$0.521804\pi$
$548$	0	0
$549$	14.3262	0.611429
$550$	0	0
$551$	−1.70820	−0.0727719
$552$	0	0
$553$	58.4508	2.48558
$554$	0	0
$555$	−19.4721	−0.826546
$556$	0	0
$557$	28.2918	1.19876	0.599381	−	0.800464i	$-0.295413\pi$
$557$	28.2918	1.19876	0.599381	+	0.800464i	$0.295413\pi$
$558$	0	0
$559$	2.23607	0.0945756
$560$	0	0
$561$	0.0344419	0.00145414
$562$	0	0
$563$	−7.29180	−0.307313	−0.153656	−	0.988124i	$-0.549105\pi$
$563$	−7.29180	−0.307313	−0.153656	+	0.988124i	$0.549105\pi$
$564$	0	0
$565$	55.6525	2.34132
$566$	0	0
$567$	24.1803	1.01548
$568$	0	0
$569$	−5.90983	−0.247753	−0.123876	−	0.992298i	$-0.539533\pi$
$569$	−5.90983	−0.247753	−0.123876	+	0.992298i	$0.539533\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	0	0
$573$	−3.12461	−0.130533
$574$	0	0
$575$	66.6312	2.77871
$576$	0	0
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	0	0
$579$	5.09017	0.211540
$580$	0	0
$581$	22.1803	0.920196
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−9.47214	−0.391625
$586$	0	0
$587$	10.6738	0.440553	0.220277	−	0.975437i	$-0.429304\pi$
$587$	10.6738	0.440553	0.220277	+	0.975437i	$0.429304\pi$
$588$	0	0
$589$	14.9443	0.615768
$590$	0	0
$591$	13.8328	0.569006
$592$	0	0
$593$	−0.583592	−0.0239653	−0.0119826	−	0.999928i	$-0.503814\pi$
$593$	−0.583592	−0.0239653	−0.0119826	+	0.999928i	$0.503814\pi$
$594$	0	0
$595$	−3.61803	−0.148325
$596$	0	0
$597$	−3.32624	−0.136134
$598$	0	0
$599$	15.5967	0.637266	0.318633	−	0.947878i	$-0.396776\pi$
$599$	15.5967	0.637266	0.318633	+	0.947878i	$0.396776\pi$
$600$	0	0
$601$	−25.6869	−1.04779	−0.523896	−	0.851782i	$-0.675522\pi$
$601$	−25.6869	−1.04779	−0.523896	+	0.851782i	$0.675522\pi$
$602$	0	0
$603$	4.61803	0.188061
$604$	0	0
$605$	39.5967	1.60984
$606$	0	0
$607$	−47.5755	−1.93103	−0.965514	−	0.260350i	$-0.916162\pi$
$607$	−47.5755	−1.93103	−0.965514	+	0.260350i	$0.916162\pi$
$608$	0	0
$609$	−2.23607	−0.0906100
$610$	0	0
$611$	−6.85410	−0.277287
$612$	0	0
$613$	−26.5279	−1.07145	−0.535725	−	0.844392i	$-0.679962\pi$
$613$	−26.5279	−1.07145	−0.535725	+	0.844392i	$0.679962\pi$
$614$	0	0
$615$	17.2361	0.695025
$616$	0	0
$617$	36.1591	1.45571	0.727854	−	0.685732i	$-0.240518\pi$
$617$	36.1591	1.45571	0.727854	+	0.685732i	$0.240518\pi$
$618$	0	0
$619$	40.8328	1.64121	0.820605	−	0.571496i	$-0.193637\pi$
$619$	40.8328	1.64121	0.820605	+	0.571496i	$0.193637\pi$
$620$	0	0
$621$	28.5967	1.14755
$622$	0	0
$623$	−41.5066	−1.66293
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.291796	−0.0116532
$628$	0	0
$629$	−2.05573	−0.0819672
$630$	0	0
$631$	−5.20163	−0.207073	−0.103537	−	0.994626i	$-0.533016\pi$
$631$	−5.20163	−0.207073	−0.103537	+	0.994626i	$0.533016\pi$
$632$	0	0
$633$	0.527864	0.0209807
$634$	0	0
$635$	75.2492	2.98617
$636$	0	0
$637$	−10.9443	−0.433628
$638$	0	0
$639$	25.4164	1.00546
$640$	0	0
$641$	3.18034	0.125616	0.0628079	−	0.998026i	$-0.479994\pi$
$641$	3.18034	0.125616	0.0628079	+	0.998026i	$0.479994\pi$
$642$	0	0
$643$	4.14590	0.163498	0.0817491	−	0.996653i	$-0.473949\pi$
$643$	4.14590	0.163498	0.0817491	+	0.996653i	$0.473949\pi$
$644$	0	0
$645$	−5.00000	−0.196875
$646$	0	0
$647$	2.81966	0.110852	0.0554261	−	0.998463i	$-0.482348\pi$
$647$	2.81966	0.110852	0.0554261	+	0.998463i	$0.482348\pi$
$648$	0	0
$649$	−1.54915	−0.0608095
$650$	0	0
$651$	19.5623	0.766707
$652$	0	0
$653$	−27.5410	−1.07776	−0.538882	−	0.842381i	$-0.681153\pi$
$653$	−27.5410	−1.07776	−0.538882	+	0.842381i	$0.681153\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	0	0
$657$	28.0344	1.09373
$658$	0	0
$659$	31.3262	1.22030	0.610148	−	0.792287i	$-0.291110\pi$
$659$	31.3262	1.22030	0.610148	+	0.792287i	$0.291110\pi$
$660$	0	0
$661$	−20.8541	−0.811131	−0.405565	−	0.914066i	$-0.632925\pi$
$661$	−20.8541	−0.811131	−0.405565	+	0.914066i	$0.632925\pi$
$662$	0	0
$663$	0.145898	0.00566621
$664$	0	0
$665$	30.6525	1.18865
$666$	0	0
$667$	7.03444	0.272375
$668$	0	0
$669$	−3.29180	−0.127268
$670$	0	0
$671$	1.29180	0.0498692
$672$	0	0
$673$	10.7082	0.412771	0.206385	−	0.978471i	$-0.433830\pi$
$673$	10.7082	0.412771	0.206385	+	0.978471i	$0.433830\pi$
$674$	0	0
$675$	28.0902	1.08119
$676$	0	0
$677$	−17.4377	−0.670185	−0.335093	−	0.942185i	$-0.608768\pi$
$677$	−17.4377	−0.670185	−0.335093	+	0.942185i	$0.608768\pi$
$678$	0	0
$679$	53.5967	2.05685
$680$	0	0
$681$	−14.8541	−0.569210
$682$	0	0
$683$	21.2361	0.812576	0.406288	−	0.913745i	$-0.366823\pi$
$683$	21.2361	0.812576	0.406288	+	0.913745i	$0.366823\pi$
$684$	0	0
$685$	56.5066	2.15901
$686$	0	0
$687$	6.61803	0.252494
$688$	0	0
$689$	0	0
$690$	0	0
$691$	15.1246	0.575367	0.287684	−	0.957725i	$-0.407115\pi$
$691$	15.1246	0.575367	0.287684	+	0.957725i	$0.407115\pi$
$692$	0	0
$693$	2.61803	0.0994509
$694$	0	0
$695$	38.2148	1.44957
$696$	0	0
$697$	1.81966	0.0689245
$698$	0	0
$699$	−2.61803	−0.0990231
$700$	0	0
$701$	−10.9787	−0.414660	−0.207330	−	0.978271i	$-0.566477\pi$
$701$	−10.9787	−0.414660	−0.207330	+	0.978271i	$0.566477\pi$
$702$	0	0
$703$	17.4164	0.656872
$704$	0	0
$705$	15.3262	0.577220
$706$	0	0
$707$	−21.5623	−0.810934
$708$	0	0
$709$	−37.9443	−1.42503	−0.712514	−	0.701658i	$-0.752443\pi$
$709$	−37.9443	−1.42503	−0.712514	+	0.701658i	$0.752443\pi$
$710$	0	0
$711$	−36.1246	−1.35478
$712$	0	0
$713$	−61.5410	−2.30473
$714$	0	0
$715$	−0.854102	−0.0319416
$716$	0	0
$717$	−11.9098	−0.444781
$718$	0	0
$719$	−42.1803	−1.57306	−0.786531	−	0.617551i	$-0.788125\pi$
$719$	−42.1803	−1.57306	−0.786531	+	0.617551i	$0.788125\pi$
$720$	0	0
$721$	−45.9787	−1.71234
$722$	0	0
$723$	2.90983	0.108218
$724$	0	0
$725$	6.90983	0.256625
$726$	0	0
$727$	44.7082	1.65814	0.829068	−	0.559148i	$-0.188872\pi$
$727$	44.7082	1.65814	0.829068	+	0.559148i	$0.188872\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	−0.527864	−0.0195238
$732$	0	0
$733$	−33.3607	−1.23220	−0.616102	−	0.787666i	$-0.711289\pi$
$733$	−33.3607	−1.23220	−0.616102	+	0.787666i	$0.711289\pi$
$734$	0	0
$735$	24.4721	0.902668
$736$	0	0
$737$	0.416408	0.0153386
$738$	0	0
$739$	−35.0344	−1.28876	−0.644381	−	0.764704i	$-0.722885\pi$
$739$	−35.0344	−1.28876	−0.644381	+	0.764704i	$0.722885\pi$
$740$	0	0
$741$	−1.23607	−0.0454081
$742$	0	0
$743$	9.20163	0.337575	0.168787	−	0.985652i	$-0.446015\pi$
$743$	9.20163	0.337575	0.168787	+	0.985652i	$0.446015\pi$
$744$	0	0
$745$	−1.18034	−0.0432443
$746$	0	0
$747$	−13.7082	−0.501557
$748$	0	0
$749$	51.8328	1.89393
$750$	0	0
$751$	−39.8673	−1.45478	−0.727388	−	0.686226i	$-0.759266\pi$
$751$	−39.8673	−1.45478	−0.727388	+	0.686226i	$0.759266\pi$
$752$	0	0
$753$	14.8885	0.542569
$754$	0	0
$755$	−35.9787	−1.30940
$756$	0	0
$757$	−38.9443	−1.41545	−0.707727	−	0.706486i	$-0.750279\pi$
$757$	−38.9443	−1.41545	−0.707727	+	0.706486i	$0.750279\pi$
$758$	0	0
$759$	1.20163	0.0436163
$760$	0	0
$761$	−1.41641	−0.0513447	−0.0256724	−	0.999670i	$-0.508173\pi$
$761$	−1.41641	−0.0513447	−0.0256724	+	0.999670i	$0.508173\pi$
$762$	0	0
$763$	−19.9443	−0.722031
$764$	0	0
$765$	2.23607	0.0808452
$766$	0	0
$767$	−6.56231	−0.236951
$768$	0	0
$769$	−33.3262	−1.20177	−0.600887	−	0.799334i	$-0.705186\pi$
$769$	−33.3262	−1.20177	−0.600887	+	0.799334i	$0.705186\pi$
$770$	0	0
$771$	3.38197	0.121799
$772$	0	0
$773$	47.5967	1.71194	0.855968	−	0.517029i	$-0.172962\pi$
$773$	47.5967	1.71194	0.855968	+	0.517029i	$0.172962\pi$
$774$	0	0
$775$	−60.4508	−2.17146
$776$	0	0
$777$	22.7984	0.817887
$778$	0	0
$779$	−15.4164	−0.552350
$780$	0	0
$781$	2.29180	0.0820069
$782$	0	0
$783$	2.96556	0.105980
$784$	0	0
$785$	−81.9574	−2.92519
$786$	0	0
$787$	2.34752	0.0836802	0.0418401	−	0.999124i	$-0.486678\pi$
$787$	2.34752	0.0836802	0.0418401	+	0.999124i	$0.486678\pi$
$788$	0	0
$789$	14.3820	0.512012
$790$	0	0
$791$	−65.1591	−2.31679
$792$	0	0
$793$	5.47214	0.194321
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.2016	0.467626	0.233813	−	0.972282i	$-0.424880\pi$
$797$	13.2016	0.467626	0.233813	+	0.972282i	$0.424880\pi$
$798$	0	0
$799$	1.61803	0.0572419
$800$	0	0
$801$	25.6525	0.906386
$802$	0	0
$803$	2.52786	0.0892064
$804$	0	0
$805$	−126.228	−4.44895
$806$	0	0
$807$	−0.618034	−0.0217558
$808$	0	0
$809$	15.5279	0.545931	0.272965	−	0.962024i	$-0.411996\pi$
$809$	15.5279	0.545931	0.272965	+	0.962024i	$0.411996\pi$
$810$	0	0
$811$	47.5967	1.67135	0.835674	−	0.549226i	$-0.185077\pi$
$811$	47.5967	1.67135	0.835674	+	0.549226i	$0.185077\pi$
$812$	0	0
$813$	14.9098	0.522911
$814$	0	0
$815$	62.6869	2.19583
$816$	0	0
$817$	4.47214	0.156460
$818$	0	0
$819$	11.0902	0.387522
$820$	0	0
$821$	38.3607	1.33880	0.669398	−	0.742904i	$-0.266552\pi$
$821$	38.3607	1.33880	0.669398	+	0.742904i	$0.266552\pi$
$822$	0	0
$823$	50.8885	1.77386	0.886932	−	0.461901i	$-0.152832\pi$
$823$	50.8885	1.77386	0.886932	+	0.461901i	$0.152832\pi$
$824$	0	0
$825$	1.18034	0.0410942
$826$	0	0
$827$	37.7426	1.31244	0.656220	−	0.754569i	$-0.272154\pi$
$827$	37.7426	1.31244	0.656220	+	0.754569i	$0.272154\pi$
$828$	0	0
$829$	43.7984	1.52118	0.760590	−	0.649232i	$-0.224910\pi$
$829$	43.7984	1.52118	0.760590	+	0.649232i	$0.224910\pi$
$830$	0	0
$831$	13.2148	0.458416
$832$	0	0
$833$	2.58359	0.0895162
$834$	0	0
$835$	13.0902	0.453004
$836$	0	0
$837$	−25.9443	−0.896765
$838$	0	0
$839$	−14.4508	−0.498899	−0.249449	−	0.968388i	$-0.580250\pi$
$839$	−14.4508	−0.498899	−0.249449	+	0.968388i	$0.580250\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	0	0
$843$	18.0344	0.621139
$844$	0	0
$845$	43.4164	1.49357
$846$	0	0
$847$	−46.3607	−1.59297
$848$	0	0
$849$	3.56231	0.122258
$850$	0	0
$851$	−71.7214	−2.45858
$852$	0	0
$853$	51.8115	1.77399	0.886996	−	0.461776i	$-0.152788\pi$
$853$	51.8115	1.77399	0.886996	+	0.461776i	$0.152788\pi$
$854$	0	0
$855$	−18.9443	−0.647880
$856$	0	0
$857$	−42.0689	−1.43705	−0.718523	−	0.695503i	$-0.755181\pi$
$857$	−42.0689	−1.43705	−0.718523	+	0.695503i	$0.755181\pi$
$858$	0	0
$859$	12.7426	0.434773	0.217387	−	0.976086i	$-0.430247\pi$
$859$	12.7426	0.434773	0.217387	+	0.976086i	$0.430247\pi$
$860$	0	0
$861$	−20.1803	−0.687744
$862$	0	0
$863$	−17.3262	−0.589792	−0.294896	−	0.955529i	$-0.595285\pi$
$863$	−17.3262	−0.589792	−0.294896	+	0.955529i	$0.595285\pi$
$864$	0	0
$865$	27.0344	0.919199
$866$	0	0
$867$	10.4721	0.355652
$868$	0	0
$869$	−3.25735	−0.110498
$870$	0	0
$871$	1.76393	0.0597686
$872$	0	0
$873$	−33.1246	−1.12110
$874$	0	0
$875$	−47.3607	−1.60108
$876$	0	0
$877$	42.5755	1.43767	0.718836	−	0.695180i	$-0.244675\pi$
$877$	42.5755	1.43767	0.718836	+	0.695180i	$0.244675\pi$
$878$	0	0
$879$	−4.65248	−0.156924
$880$	0	0
$881$	−44.5066	−1.49946	−0.749732	−	0.661741i	$-0.769818\pi$
$881$	−44.5066	−1.49946	−0.749732	+	0.661741i	$0.769818\pi$
$882$	0	0
$883$	50.1935	1.68915	0.844573	−	0.535441i	$-0.179854\pi$
$883$	50.1935	1.68915	0.844573	+	0.535441i	$0.179854\pi$
$884$	0	0
$885$	14.6738	0.493253
$886$	0	0
$887$	13.8328	0.464460	0.232230	−	0.972661i	$-0.425398\pi$
$887$	13.8328	0.464460	0.232230	+	0.972661i	$0.425398\pi$
$888$	0	0
$889$	−88.1033	−2.95489
$890$	0	0
$891$	−1.34752	−0.0451438
$892$	0	0
$893$	−13.7082	−0.458728
$894$	0	0
$895$	67.6869	2.26252
$896$	0	0
$897$	5.09017	0.169956
$898$	0	0
$899$	−6.38197	−0.212850
$900$	0	0
$901$	0	0
$902$	0	0
$903$	5.85410	0.194812
$904$	0	0
$905$	−70.2492	−2.33516
$906$	0	0
$907$	14.1246	0.469000	0.234500	−	0.972116i	$-0.424655\pi$
$907$	14.1246	0.469000	0.234500	+	0.972116i	$0.424655\pi$
$908$	0	0
$909$	13.3262	0.442003
$910$	0	0
$911$	−47.3394	−1.56842	−0.784212	−	0.620493i	$-0.786933\pi$
$911$	−47.3394	−1.56842	−0.784212	+	0.620493i	$0.786933\pi$
$912$	0	0
$913$	−1.23607	−0.0409079
$914$	0	0
$915$	−12.2361	−0.404512
$916$	0	0
$917$	−11.7082	−0.386639
$918$	0	0
$919$	5.34752	0.176399	0.0881993	−	0.996103i	$-0.471889\pi$
$919$	5.34752	0.176399	0.0881993	+	0.996103i	$0.471889\pi$
$920$	0	0
$921$	19.8541	0.654215
$922$	0	0
$923$	9.70820	0.319549
$924$	0	0
$925$	−70.4508	−2.31641
$926$	0	0
$927$	28.4164	0.933317
$928$	0	0
$929$	17.8197	0.584644	0.292322	−	0.956320i	$-0.405572\pi$
$929$	17.8197	0.584644	0.292322	+	0.956320i	$0.405572\pi$
$930$	0	0
$931$	−21.8885	−0.717368
$932$	0	0
$933$	−10.6525	−0.348746
$934$	0	0
$935$	0.201626	0.00659388
$936$	0	0
$937$	−33.0902	−1.08101	−0.540504	−	0.841341i	$-0.681767\pi$
$937$	−33.0902	−1.08101	−0.540504	+	0.841341i	$0.681767\pi$
$938$	0	0
$939$	−8.12461	−0.265137
$940$	0	0
$941$	−46.9787	−1.53146	−0.765731	−	0.643161i	$-0.777623\pi$
$941$	−46.9787	−1.53146	−0.765731	+	0.643161i	$0.777623\pi$
$942$	0	0
$943$	63.4853	2.06737
$944$	0	0
$945$	−53.2148	−1.73108
$946$	0	0
$947$	50.2148	1.63176	0.815881	−	0.578220i	$-0.196253\pi$
$947$	50.2148	1.63176	0.815881	+	0.578220i	$0.196253\pi$
$948$	0	0
$949$	10.7082	0.347603
$950$	0	0
$951$	18.6869	0.605965
$952$	0	0
$953$	25.3050	0.819708	0.409854	−	0.912151i	$-0.365580\pi$
$953$	25.3050	0.819708	0.409854	+	0.912151i	$0.365580\pi$
$954$	0	0
$955$	−18.2918	−0.591909
$956$	0	0
$957$	0.124612	0.00402813
$958$	0	0
$959$	−66.1591	−2.13639
$960$	0	0
$961$	24.8328	0.801059
$962$	0	0
$963$	−32.0344	−1.03230
$964$	0	0
$965$	29.7984	0.959244
$966$	0	0
$967$	−1.85410	−0.0596239	−0.0298119	−	0.999556i	$-0.509491\pi$
$967$	−1.85410	−0.0596239	−0.0298119	+	0.999556i	$0.509491\pi$
$968$	0	0
$969$	0.291796	0.00937384
$970$	0	0
$971$	0.652476	0.0209389	0.0104695	−	0.999945i	$-0.496667\pi$
$971$	0.652476	0.0209389	0.0104695	+	0.999945i	$0.496667\pi$
$972$	0	0
$973$	−44.7426	−1.43438
$974$	0	0
$975$	5.00000	0.160128
$976$	0	0
$977$	31.1459	0.996446	0.498223	−	0.867049i	$-0.333986\pi$
$977$	31.1459	0.996446	0.498223	+	0.867049i	$0.333986\pi$
$978$	0	0
$979$	2.31308	0.0739264
$980$	0	0
$981$	12.3262	0.393546
$982$	0	0
$983$	31.4377	1.00271	0.501353	−	0.865243i	$-0.332836\pi$
$983$	31.4377	1.00271	0.501353	+	0.865243i	$0.332836\pi$
$984$	0	0
$985$	80.9787	2.58020
$986$	0	0
$987$	−17.9443	−0.571172
$988$	0	0
$989$	−18.4164	−0.585608
$990$	0	0
$991$	16.3820	0.520390	0.260195	−	0.965556i	$-0.416213\pi$
$991$	16.3820	0.520390	0.260195	+	0.965556i	$0.416213\pi$
$992$	0	0
$993$	0.506578	0.0160758
$994$	0	0
$995$	−19.4721	−0.617308
$996$	0	0
$997$	−17.5836	−0.556878	−0.278439	−	0.960454i	$-0.589817\pi$
$997$	−17.5836	−0.556878	−0.278439	+	0.960454i	$0.589817\pi$
$998$	0	0
$999$	−30.2361	−0.956627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.d.1.1		2
4.3	odd	2		538.2.a.a.1.2	✓	2
12.11	even	2		4842.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.a.1.2	✓	2	4.3	odd	2
4304.2.a.d.1.1		2	1.1	even	1	trivial
4842.2.a.f.1.2		2	12.11	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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} -3.61803 q^{5} +4.23607 q^{7} -2.61803 q^{9} -0.236068 q^{11} -1.00000 q^{13} +2.23607 q^{15} +0.236068 q^{17} -2.00000 q^{19} -2.61803 q^{21} +8.23607 q^{23} +8.09017 q^{25} +3.47214 q^{27} +0.854102 q^{29} -7.47214 q^{31} +0.145898 q^{33} -15.3262 q^{35} -8.70820 q^{37} +0.618034 q^{39} +7.70820 q^{41} -2.23607 q^{43} +9.47214 q^{45} +6.85410 q^{47} +10.9443 q^{49} -0.145898 q^{51} +0.854102 q^{55} +1.23607 q^{57} +6.56231 q^{59} -5.47214 q^{61} -11.0902 q^{63} +3.61803 q^{65} -1.76393 q^{67} -5.09017 q^{69} -9.70820 q^{71} -10.7082 q^{73} -5.00000 q^{75} -1.00000 q^{77} +13.7984 q^{79} +5.70820 q^{81} +5.23607 q^{83} -0.854102 q^{85} -0.527864 q^{87} -9.79837 q^{89} -4.23607 q^{91} +4.61803 q^{93} +7.23607 q^{95} +12.6525 q^{97} +0.618034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 5 q^{5} + 4 q^{7} - 3 q^{9} + 4 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 3 q^{21} + 12 q^{23} + 5 q^{25} - 2 q^{27} - 5 q^{29} - 6 q^{31} + 7 q^{33} - 15 q^{35} - 4 q^{37} - q^{39} + 2 q^{41}+ \cdots - q^{99}+O(q^{100}) \) 2 * q + q^3 - 5 * q^5 + 4 * q^7 - 3 * q^9 + 4 * q^11 - 2 * q^13 - 4 * q^17 - 4 * q^19 - 3 * q^21 + 12 * q^23 + 5 * q^25 - 2 * q^27 - 5 * q^29 - 6 * q^31 + 7 * q^33 - 15 * q^35 - 4 * q^37 - q^39 + 2 * q^41 + 10 * q^45 + 7 * q^47 + 4 * q^49 - 7 * q^51 - 5 * q^55 - 2 * q^57 - 7 * q^59 - 2 * q^61 - 11 * q^63 + 5 * q^65 - 8 * q^67 + q^69 - 6 * q^71 - 8 * q^73 - 10 * q^75 - 2 * q^77 + 3 * q^79 - 2 * q^81 + 6 * q^83 + 5 * q^85 - 10 * q^87 + 5 * q^89 - 4 * q^91 + 7 * q^93 + 10 * q^95 - 6 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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