LMFDB - Embedded newform 4014.2.d.a.4013.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(4013,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4014.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.3
Character	$\chi$	$=$	4014.4013
Dual form			4014.2.d.a.4013.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -0.969046 q^{5} -3.46872 q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -0.969046 q^{5} -3.46872 q^{7} +1.00000i q^{8} +0.969046i q^{10} -2.55954 q^{11} +6.72007i q^{13} +3.46872i q^{14} +1.00000 q^{16} +1.07269i q^{17} +2.07962 q^{19} +0.969046 q^{20} +2.55954i q^{22} -3.34775 q^{23} -4.06095 q^{25} +6.72007 q^{26} +3.46872 q^{28} +1.25255i q^{29} +3.16125 q^{31} -1.00000i q^{32} +1.07269 q^{34} +3.36135 q^{35} +1.73443 q^{37} -2.07962i q^{38} -0.969046i q^{40} +2.41214i q^{41} -1.61201 q^{43} +2.55954 q^{44} +3.34775i q^{46} -10.7575i q^{47} +5.03201 q^{49} +4.06095i q^{50} -6.72007i q^{52} +1.87835i q^{53} +2.48031 q^{55} -3.46872i q^{56} +1.25255 q^{58} +6.77146 q^{59} -7.49500i q^{61} -3.16125i q^{62} -1.00000 q^{64} -6.51206i q^{65} +2.86045i q^{67} -1.07269i q^{68} -3.36135i q^{70} -2.10243 q^{71} +0.281139 q^{73} -1.73443i q^{74} -2.07962 q^{76} +8.87833 q^{77} -1.49716i q^{79} -0.969046 q^{80} +2.41214 q^{82} -9.35419i q^{83} -1.03949i q^{85} +1.61201i q^{86} -2.55954i q^{88} -3.81081i q^{89} -23.3100i q^{91} +3.34775 q^{92} -10.7575 q^{94} -2.01524 q^{95} +13.0886i q^{97} -5.03201i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) $ 72 * q - 72 * q^4 + 16 * q^7	$ 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) $ 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−0.969046			−0.433370			−0.216685	−	0.976242i	$-0.569525\pi$
$5$	−0.969046			−0.433370			−0.216685	+	0.976242i	$0.569525\pi$
$6$		0			0
$7$	−3.46872			−1.31105			−0.655526	−	0.755172i	$-0.727553\pi$
$7$	−3.46872			−1.31105			−0.655526	+	0.755172i	$0.727553\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$			0.969046i			0.306439i
$11$	−2.55954			−0.771731			−0.385865	−	0.922555i	$-0.626097\pi$
$11$	−2.55954			−0.771731			−0.385865	+	0.922555i	$0.626097\pi$
$12$		0			0
$13$			6.72007i			1.86381i	0.362698	+	0.931907i	$0.381856\pi$
$13$			6.72007i			1.86381i	−0.362698	+	0.931907i	$0.618144\pi$
$14$			3.46872i			0.927054i
$15$		0			0
$16$	1.00000			0.250000
$17$			1.07269i			0.260166i	0.991503	+	0.130083i	$0.0415245\pi$
$17$			1.07269i			0.260166i	−0.991503	+	0.130083i	$0.958476\pi$
$18$		0			0
$19$	2.07962			0.477096			0.238548	−	0.971131i	$-0.423328\pi$
$19$	2.07962			0.477096			0.238548	+	0.971131i	$0.423328\pi$
$20$	0.969046			0.216685
$21$		0			0
$22$			2.55954i			0.545696i
$23$	−3.34775			−0.698054			−0.349027	−	0.937113i	$-0.613488\pi$
$23$	−3.34775			−0.698054			−0.349027	+	0.937113i	$0.613488\pi$
$24$		0			0
$25$	−4.06095			−0.812190
$26$	6.72007			1.31791
$27$		0			0
$28$	3.46872			0.655526
$29$			1.25255i			0.232593i	0.993215	+	0.116296i	$0.0371022\pi$
$29$			1.25255i			0.232593i	−0.993215	+	0.116296i	$0.962898\pi$
$30$		0			0
$31$	3.16125			0.567777			0.283888	−	0.958857i	$-0.408376\pi$
$31$	3.16125			0.567777			0.283888	+	0.958857i	$0.408376\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	1.07269			0.183965
$35$	3.36135			0.568171
$36$		0			0
$37$	1.73443			0.285139			0.142569	−	0.989785i	$-0.454464\pi$
$37$	1.73443			0.285139			0.142569	+	0.989785i	$0.454464\pi$
$38$		−	2.07962i		−	0.337358i
$39$		0			0
$40$		−	0.969046i		−	0.153220i
$41$			2.41214i			0.376714i	0.982101	+	0.188357i	$0.0603161\pi$
$41$			2.41214i			0.376714i	−0.982101	+	0.188357i	$0.939684\pi$
$42$		0			0
$43$	−1.61201			−0.245829			−0.122915	−	0.992417i	$-0.539224\pi$
$43$	−1.61201			−0.245829			−0.122915	+	0.992417i	$0.539224\pi$
$44$	2.55954			0.385865
$45$		0			0
$46$			3.34775i			0.493599i
$47$		−	10.7575i		−	1.56914i	−0.620039	−	0.784571i	$-0.712883\pi$
$47$		−	10.7575i		−	1.56914i	0.620039	−	0.784571i	$-0.287117\pi$
$48$		0			0
$49$	5.03201			0.718858
$50$			4.06095i			0.574305i
$51$		0			0
$52$		−	6.72007i		−	0.931907i
$53$			1.87835i			0.258011i	0.991644	+	0.129006i	$0.0411786\pi$
$53$			1.87835i			0.258011i	−0.991644	+	0.129006i	$0.958821\pi$
$54$		0			0
$55$	2.48031			0.334445
$56$		−	3.46872i		−	0.463527i
$57$		0			0
$58$	1.25255			0.164468
$59$	6.77146			0.881569			0.440785	−	0.897613i	$-0.354700\pi$
$59$	6.77146			0.881569			0.440785	+	0.897613i	$0.354700\pi$
$60$		0			0
$61$		−	7.49500i		−	0.959636i	−0.877368	−	0.479818i	$-0.840703\pi$
$61$		−	7.49500i		−	0.959636i	0.877368	−	0.479818i	$-0.159297\pi$
$62$		−	3.16125i		−	0.401479i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	6.51206i		−	0.807721i
$66$		0			0
$67$			2.86045i			0.349460i	0.984616	+	0.174730i	$0.0559053\pi$
$67$			2.86045i			0.349460i	−0.984616	+	0.174730i	$0.944095\pi$
$68$		−	1.07269i		−	0.130083i
$69$		0			0
$70$		−	3.36135i		−	0.401758i
$71$	−2.10243			−0.249512			−0.124756	−	0.992187i	$-0.539815\pi$
$71$	−2.10243			−0.249512			−0.124756	+	0.992187i	$0.539815\pi$
$72$		0			0
$73$	0.281139			0.0329049			0.0164524	−	0.999865i	$-0.494763\pi$
$73$	0.281139			0.0329049			0.0164524	+	0.999865i	$0.494763\pi$
$74$		−	1.73443i		−	0.201623i
$75$		0			0
$76$	−2.07962			−0.238548
$77$	8.87833			1.01178
$78$		0			0
$79$		−	1.49716i		−	0.168443i	−0.996447	−	0.0842216i	$-0.973160\pi$
$79$		−	1.49716i		−	0.168443i	0.996447	−	0.0842216i	$-0.0268403\pi$
$80$	−0.969046			−0.108343
$81$		0			0
$82$	2.41214			0.266377
$83$		−	9.35419i		−	1.02676i	−0.858163	−	0.513378i	$-0.828394\pi$
$83$		−	9.35419i		−	1.02676i	0.858163	−	0.513378i	$-0.171606\pi$
$84$		0			0
$85$		−	1.03949i		−	0.112748i
$86$			1.61201i			0.173828i
$87$		0			0
$88$		−	2.55954i		−	0.272848i
$89$		−	3.81081i		−	0.403945i	−0.979391	−	0.201973i	$-0.935265\pi$
$89$		−	3.81081i		−	0.403945i	0.979391	−	0.201973i	$-0.0647352\pi$
$90$		0			0
$91$		−	23.3100i		−	2.44356i
$92$	3.34775			0.349027
$93$		0			0
$94$	−10.7575			−1.10955
$95$	−2.01524			−0.206759
$96$		0			0
$97$			13.0886i			1.32895i	0.747310	+	0.664475i	$0.231345\pi$
$97$			13.0886i			1.32895i	−0.747310	+	0.664475i	$0.768655\pi$
$98$		−	5.03201i		−	0.508310i
$99$		0			0
$100$	4.06095			0.406095
$101$		−	10.3809i		−	1.03294i	−0.856306	−	0.516469i	$-0.827246\pi$
$101$		−	10.3809i		−	1.03294i	0.856306	−	0.516469i	$-0.172754\pi$
$102$		0			0
$103$			3.40046i			0.335057i	0.985867	+	0.167528i	$0.0535786\pi$
$103$			3.40046i			0.335057i	−0.985867	+	0.167528i	$0.946421\pi$
$104$	−6.72007			−0.658957
$105$		0			0
$106$	1.87835			0.182442
$107$	13.0004			1.25680			0.628399	−	0.777891i	$-0.283711\pi$
$107$	13.0004			1.25680			0.628399	+	0.777891i	$0.283711\pi$
$108$		0			0
$109$	−15.8716			−1.52022			−0.760110	−	0.649794i	$-0.774855\pi$
$109$	−15.8716			−1.52022			−0.760110	+	0.649794i	$0.774855\pi$
$110$		−	2.48031i		−	0.236489i
$111$		0			0
$112$	−3.46872			−0.327763
$113$	−0.213882			−0.0201203			−0.0100602	−	0.999949i	$-0.503202\pi$
$113$	−0.213882			−0.0201203			−0.0100602	+	0.999949i	$0.503202\pi$
$114$		0			0
$115$	3.24412			0.302516
$116$		−	1.25255i		−	0.116296i
$117$		0			0
$118$		−	6.77146i		−	0.623363i
$119$		−	3.72087i		−	0.341092i
$120$		0			0
$121$	−4.44875			−0.404431
$122$	−7.49500			−0.678565
$123$		0			0
$124$	−3.16125			−0.283888
$125$	8.78047			0.785350
$126$		0			0
$127$	−2.76173			−0.245064			−0.122532	−	0.992465i	$-0.539101\pi$
$127$	−2.76173			−0.245064			−0.122532	+	0.992465i	$0.539101\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−6.51206			−0.571145
$131$		−	11.6965i		−	1.02193i	−0.859602	−	0.510964i	$-0.829289\pi$
$131$		−	11.6965i		−	1.02193i	0.859602	−	0.510964i	$-0.170711\pi$
$132$		0			0
$133$	−7.21360			−0.625498
$134$	2.86045			0.247106
$135$		0			0
$136$	−1.07269			−0.0919827
$137$	12.9587			1.10714			0.553569	−	0.832804i	$-0.313266\pi$
$137$	12.9587			1.10714			0.553569	+	0.832804i	$0.313266\pi$
$138$		0			0
$139$	−11.9531			−1.01385			−0.506924	−	0.861991i	$-0.669218\pi$
$139$	−11.9531			−1.01385			−0.506924	+	0.861991i	$0.669218\pi$
$140$	−3.36135			−0.284086
$141$		0			0
$142$			2.10243i			0.176432i
$143$		−	17.2003i		−	1.43836i
$144$		0			0
$145$		−	1.21378i		−	0.100799i
$146$		−	0.281139i		−	0.0232673i
$147$		0			0
$148$	−1.73443			−0.142569
$149$	−13.5939			−1.11366			−0.556829	−	0.830627i	$-0.687982\pi$
$149$	−13.5939			−1.11366			−0.556829	+	0.830627i	$0.687982\pi$
$150$		0			0
$151$			1.07562i			0.0875326i	0.999042	+	0.0437663i	$0.0139357\pi$
$151$			1.07562i			0.0875326i	−0.999042	+	0.0437663i	$0.986064\pi$
$152$			2.07962i			0.168679i
$153$		0			0
$154$		−	8.87833i		−	0.715436i
$155$	−3.06339			−0.246058
$156$		0			0
$157$			3.39354i			0.270834i	0.990789	+	0.135417i	$0.0432374\pi$
$157$			3.39354i			0.270834i	−0.990789	+	0.135417i	$0.956763\pi$
$158$	−1.49716			−0.119107
$159$		0			0
$160$			0.969046i			0.0766098i
$161$	11.6124			0.915185
$162$		0			0
$163$			6.09521i			0.477414i	0.971092	+	0.238707i	$0.0767235\pi$
$163$			6.09521i			0.477414i	−0.971092	+	0.238707i	$0.923276\pi$
$164$		−	2.41214i		−	0.188357i
$165$		0			0
$166$	−9.35419			−0.726026
$167$	−7.12890			−0.551651			−0.275825	−	0.961208i	$-0.588951\pi$
$167$	−7.12890			−0.551651			−0.275825	+	0.961208i	$0.588951\pi$
$168$		0			0
$169$	−32.1594			−2.47380
$170$	−1.03949			−0.0797252
$171$		0			0
$172$	1.61201			0.122915
$173$	12.3333			0.937685			0.468843	−	0.883282i	$-0.344671\pi$
$173$	12.3333			0.937685			0.468843	+	0.883282i	$0.344671\pi$
$174$		0			0
$175$	14.0863			1.06482
$176$	−2.55954			−0.192933
$177$		0			0
$178$	−3.81081			−0.285633
$179$		−	20.1718i		−	1.50771i	−0.657041	−	0.753855i	$-0.728192\pi$
$179$		−	20.1718i		−	1.50771i	0.657041	−	0.753855i	$-0.271808\pi$
$180$		0			0
$181$	−2.88222			−0.214233			−0.107117	−	0.994246i	$-0.534162\pi$
$181$	−2.88222			−0.214233			−0.107117	+	0.994246i	$0.534162\pi$
$182$	−23.3100			−1.72786
$183$		0			0
$184$		−	3.34775i		−	0.246799i
$185$	−1.68074			−0.123571
$186$		0			0
$187$		−	2.74560i		−	0.200778i
$188$			10.7575i			0.784571i
$189$		0			0
$190$			2.01524i			0.146201i
$191$	11.6916			0.845978			0.422989	−	0.906135i	$-0.360981\pi$
$191$	11.6916			0.845978			0.422989	+	0.906135i	$0.360981\pi$
$192$		0			0
$193$		−	0.0831582i		−	0.00598586i	−0.999996	−	0.00299293i	$-0.999047\pi$
$193$		−	0.0831582i		−	0.00598586i	0.999996	−	0.00299293i	$-0.000952680\pi$
$194$	13.0886			0.939710
$195$		0			0
$196$	−5.03201			−0.359429
$197$		−	9.84566i		−	0.701474i	−0.936474	−	0.350737i	$-0.885931\pi$
$197$		−	9.84566i		−	0.701474i	0.936474	−	0.350737i	$-0.114069\pi$
$198$		0			0
$199$	−12.3477			−0.875305			−0.437652	−	0.899144i	$-0.644190\pi$
$199$	−12.3477			−0.875305			−0.437652	+	0.899144i	$0.644190\pi$
$200$		−	4.06095i		−	0.287153i
$201$		0			0
$202$	−10.3809			−0.730398
$203$		−	4.34474i		−	0.304941i
$204$		0			0
$205$		−	2.33748i		−	0.163257i
$206$	3.40046			0.236921
$207$		0			0
$208$			6.72007i			0.465953i
$209$	−5.32286			−0.368190
$210$		0			0
$211$	22.7720			1.56769			0.783845	−	0.620957i	$-0.213256\pi$
$211$	22.7720			1.56769			0.783845	+	0.620957i	$0.213256\pi$
$212$		−	1.87835i		−	0.129006i
$213$		0			0
$214$		−	13.0004i		−	0.888690i
$215$	1.56211			0.106535
$216$		0			0
$217$	−10.9655			−0.744385
$218$			15.8716i			1.07496i
$219$		0			0
$220$	−2.48031			−0.167223
$221$	−7.20858			−0.484902
$222$		0			0
$223$	13.3682	−	6.65524i	0.895199	−	0.445667i
$223$	13.3682	−	6.65524i	0.895199	−	0.445667i
$224$			3.46872i			0.231764i
$225$		0			0
$226$			0.213882i			0.0142272i
$227$		−	16.1068i		−	1.06905i	−0.845154	−	0.534523i	$-0.820491\pi$
$227$		−	16.1068i		−	1.06905i	0.845154	−	0.534523i	$-0.179509\pi$
$228$		0			0
$229$		−	23.1904i		−	1.53246i	−0.642565	−	0.766231i	$-0.722130\pi$
$229$		−	23.1904i		−	1.53246i	0.642565	−	0.766231i	$-0.277870\pi$
$230$		−	3.24412i		−	0.213911i
$231$		0			0
$232$	−1.25255			−0.0822339
$233$	19.4967			1.27727			0.638637	−	0.769508i	$-0.279499\pi$
$233$	19.4967			1.27727			0.638637	+	0.769508i	$0.279499\pi$
$234$		0			0
$235$			10.4245i			0.680020i
$236$	−6.77146			−0.440785
$237$		0			0
$238$	−3.72087			−0.241188
$239$			7.31789i			0.473355i	0.971588	+	0.236677i	$0.0760585\pi$
$239$			7.31789i			0.473355i	−0.971588	+	0.236677i	$0.923942\pi$
$240$		0			0
$241$	4.68387			0.301715			0.150857	−	0.988556i	$-0.451797\pi$
$241$	4.68387			0.301715			0.150857	+	0.988556i	$0.451797\pi$
$242$			4.44875i			0.285976i
$243$		0			0
$244$			7.49500i			0.479818i
$245$	−4.87625			−0.311532
$246$		0			0
$247$			13.9752i			0.889219i
$248$			3.16125i			0.200739i
$249$		0			0
$250$		−	8.78047i		−	0.555326i
$251$		−	12.7990i		−	0.807865i	−0.914789	−	0.403932i	$-0.867643\pi$
$251$		−	12.7990i		−	0.807865i	0.914789	−	0.403932i	$-0.132357\pi$
$252$		0			0
$253$	8.56870			0.538710
$254$			2.76173i			0.173286i
$255$		0			0
$256$	1.00000			0.0625000
$257$			2.15971i			0.134719i	0.997729	+	0.0673596i	$0.0214575\pi$
$257$			2.15971i			0.134719i	−0.997729	+	0.0673596i	$0.978543\pi$
$258$		0			0
$259$	−6.01625			−0.373832
$260$			6.51206i			0.403861i
$261$		0			0
$262$	−11.6965			−0.722612
$263$	11.3470			0.699688			0.349844	−	0.936808i	$-0.386235\pi$
$263$	11.3470			0.699688			0.349844	+	0.936808i	$0.386235\pi$
$264$		0			0
$265$		−	1.82021i		−	0.111815i
$266$			7.21360i			0.442294i
$267$		0			0
$268$		−	2.86045i		−	0.174730i
$269$	−18.2555			−1.11306			−0.556529	−	0.830828i	$-0.687867\pi$
$269$	−18.2555			−1.11306			−0.556529	+	0.830828i	$0.687867\pi$
$270$		0			0
$271$		−	8.22822i		−	0.499829i	−0.968268	−	0.249914i	$-0.919598\pi$
$271$		−	8.22822i		−	0.499829i	0.968268	−	0.249914i	$-0.0804025\pi$
$272$			1.07269i			0.0650416i
$273$		0			0
$274$		−	12.9587i		−	0.782864i
$275$	10.3942			0.626792
$276$		0			0
$277$			14.4646i			0.869095i	0.900649	+	0.434548i	$0.143092\pi$
$277$			14.4646i			0.869095i	−0.900649	+	0.434548i	$0.856908\pi$
$278$			11.9531i			0.716899i
$279$		0			0
$280$			3.36135i			0.200879i
$281$			5.90766i			0.352421i	0.984352	+	0.176211i	$0.0563840\pi$
$281$			5.90766i			0.352421i	−0.984352	+	0.176211i	$0.943616\pi$
$282$		0			0
$283$	26.7225			1.58849			0.794243	−	0.607601i	$-0.207868\pi$
$283$	26.7225			1.58849			0.794243	+	0.607601i	$0.207868\pi$
$284$	2.10243			0.124756
$285$		0			0
$286$	−17.2003			−1.01708
$287$		−	8.36705i		−	0.493891i
$288$		0			0
$289$	15.8493			0.932313
$290$	−1.21378			−0.0712755
$291$		0			0
$292$	−0.281139			−0.0164524
$293$	−15.3541			−0.896996			−0.448498	−	0.893784i	$-0.648041\pi$
$293$	−15.3541			−0.896996			−0.448498	+	0.893784i	$0.648041\pi$
$294$		0			0
$295$	−6.56185			−0.382046
$296$			1.73443i			0.100812i
$297$		0			0
$298$			13.5939i			0.787475i
$299$		−	22.4971i		−	1.30104i
$300$		0			0
$301$	5.59161			0.322295
$302$	1.07562			0.0618949
$303$		0			0
$304$	2.07962			0.119274
$305$			7.26299i			0.415878i
$306$		0			0
$307$			27.7608i			1.58439i	0.610265	+	0.792197i	$0.291063\pi$
$307$			27.7608i			1.58439i	−0.610265	+	0.792197i	$0.708937\pi$
$308$	−8.87833			−0.505890
$309$		0			0
$310$			3.06339i			0.173989i
$311$	29.7913			1.68931			0.844656	−	0.535310i	$-0.179805\pi$
$311$	29.7913			1.68931			0.844656	+	0.535310i	$0.179805\pi$
$312$		0			0
$313$		−	18.8166i		−	1.06358i	−0.846877	−	0.531788i	$-0.821520\pi$
$313$		−	18.8166i		−	1.06358i	0.846877	−	0.531788i	$-0.178480\pi$
$314$	3.39354			0.191509
$315$		0			0
$316$			1.49716i			0.0842216i
$317$		−	9.96393i		−	0.559630i	−0.960054	−	0.279815i	$-0.909727\pi$
$317$		−	9.96393i		−	0.559630i	0.960054	−	0.279815i	$-0.0902731\pi$
$318$		0			0
$319$		−	3.20595i		−	0.179499i
$320$	0.969046			0.0541713
$321$		0			0
$322$		−	11.6124i		−	0.647134i
$323$			2.23079i			0.124124i
$324$		0			0
$325$		−	27.2899i		−	1.51377i
$326$	6.09521			0.337582
$327$		0			0
$328$	−2.41214			−0.133188
$329$			37.3147i			2.05723i
$330$		0			0
$331$		−	20.0451i		−	1.10178i	−0.834578	−	0.550890i	$-0.814288\pi$
$331$		−	20.0451i		−	1.10178i	0.834578	−	0.550890i	$-0.185712\pi$
$332$			9.35419i			0.513378i
$333$		0			0
$334$			7.12890i			0.390076i
$335$		−	2.77191i		−	0.151446i
$336$		0			0
$337$		−	27.6452i		−	1.50593i	−0.658062	−	0.752964i	$-0.728624\pi$
$337$		−	27.6452i		−	1.50593i	0.658062	−	0.752964i	$-0.271376\pi$
$338$			32.1594i			1.74924i
$339$		0			0
$340$			1.03949i			0.0563742i
$341$	−8.09134			−0.438171
$342$		0			0
$343$	6.82641			0.368591
$344$		−	1.61201i		−	0.0869138i
$345$		0			0
$346$		−	12.3333i		−	0.663044i
$347$		−	3.58620i		−	0.192517i	−0.995356	−	0.0962586i	$-0.969312\pi$
$347$		−	3.58620i		−	0.192517i	0.995356	−	0.0962586i	$-0.0306876\pi$
$348$		0			0
$349$	7.16650			0.383614			0.191807	−	0.981433i	$-0.438565\pi$
$349$	7.16650			0.383614			0.191807	+	0.981433i	$0.438565\pi$
$350$		−	14.0863i		−	0.752944i
$351$		0			0
$352$			2.55954i			0.136424i
$353$		−	2.94264i		−	0.156621i	−0.996929	−	0.0783105i	$-0.975047\pi$
$353$		−	2.94264i		−	0.156621i	0.996929	−	0.0783105i	$-0.0249525\pi$
$354$		0			0
$355$	2.03735			0.108131
$356$			3.81081i			0.201973i
$357$		0			0
$358$	−20.1718			−1.06611
$359$		−	14.9608i		−	0.789601i	−0.918767	−	0.394801i	$-0.870814\pi$
$359$		−	14.9608i		−	0.789601i	0.918767	−	0.394801i	$-0.129186\pi$
$360$		0			0
$361$	−14.6752			−0.772379
$362$			2.88222i			0.151486i
$363$		0			0
$364$			23.3100i			1.22178i
$365$	−0.272437			−0.0142600
$366$		0			0
$367$	−20.2011			−1.05449			−0.527243	−	0.849714i	$-0.676774\pi$
$367$	−20.2011			−1.05449			−0.527243	+	0.849714i	$0.676774\pi$
$368$	−3.34775			−0.174513
$369$		0			0
$370$			1.68074i			0.0873776i
$371$		−	6.51547i		−	0.338267i
$372$		0			0
$373$			5.38424i			0.278785i	0.990237	+	0.139393i	$0.0445150\pi$
$373$			5.38424i			0.278785i	−0.990237	+	0.139393i	$0.955485\pi$
$374$	−2.74560			−0.141972
$375$		0			0
$376$	10.7575			0.554776
$377$	−8.41722			−0.433509
$378$		0			0
$379$	−8.94431			−0.459438			−0.229719	−	0.973257i	$-0.573781\pi$
$379$	−8.94431			−0.459438			−0.229719	+	0.973257i	$0.573781\pi$
$380$	2.01524			0.103380
$381$		0			0
$382$		−	11.6916i		−	0.598197i
$383$	7.05417			0.360451			0.180226	−	0.983625i	$-0.442317\pi$
$383$	7.05417			0.360451			0.180226	+	0.983625i	$0.442317\pi$
$384$		0			0
$385$	−8.60351			−0.438475
$386$	−0.0831582			−0.00423264
$387$		0			0
$388$		−	13.0886i		−	0.664475i
$389$		−	2.40158i		−	0.121765i	−0.998145	−	0.0608824i	$-0.980609\pi$
$389$		−	2.40158i		−	0.121765i	0.998145	−	0.0608824i	$-0.0193915\pi$
$390$		0			0
$391$		−	3.59111i		−	0.181610i
$392$			5.03201i			0.254155i
$393$		0			0
$394$	−9.84566			−0.496017
$395$			1.45081i			0.0729983i
$396$		0			0
$397$		−	37.6516i		−	1.88968i	−0.327531	−	0.944841i	$-0.606216\pi$
$397$		−	37.6516i		−	1.88968i	0.327531	−	0.944841i	$-0.393784\pi$
$398$			12.3477i			0.618934i
$399$		0			0
$400$	−4.06095			−0.203048
$401$			23.1534i			1.15622i	0.815958	+	0.578112i	$0.196210\pi$
$401$			23.1534i			1.15622i	−0.815958	+	0.578112i	$0.803790\pi$
$402$		0			0
$403$			21.2438i			1.05823i
$404$			10.3809i			0.516469i
$405$		0			0
$406$	−4.34474			−0.215626
$407$	−4.43935			−0.220050
$408$		0			0
$409$			12.6033i			0.623193i	0.950215	+	0.311596i	$0.100864\pi$
$409$			12.6033i			0.623193i	−0.950215	+	0.311596i	$0.899136\pi$
$410$	−2.33748			−0.115440
$411$		0			0
$412$		−	3.40046i		−	0.167528i
$413$	−23.4883			−1.15578
$414$		0			0
$415$			9.06464i			0.444966i
$416$	6.72007			0.329479
$417$		0			0
$418$			5.32286i			0.260350i
$419$		−	21.0493i		−	1.02833i	−0.857692	−	0.514164i	$-0.828102\pi$
$419$		−	21.0493i		−	1.02833i	0.857692	−	0.514164i	$-0.171898\pi$
$420$		0			0
$421$			22.6340i			1.10311i	0.834137	+	0.551557i	$0.185966\pi$
$421$			22.6340i			1.10311i	−0.834137	+	0.551557i	$0.814034\pi$
$422$		−	22.7720i		−	1.10852i
$423$		0			0
$424$	−1.87835			−0.0912208
$425$		−	4.35616i		−	0.211305i
$426$		0			0
$427$			25.9980i			1.25813i
$428$	−13.0004			−0.628399
$429$		0			0
$430$		−	1.56211i		−	0.0753317i
$431$	−15.1291			−0.728745			−0.364373	−	0.931253i	$-0.618717\pi$
$431$	−15.1291			−0.728745			−0.364373	+	0.931253i	$0.618717\pi$
$432$		0			0
$433$	6.37944			0.306576			0.153288	−	0.988182i	$-0.451014\pi$
$433$	6.37944			0.306576			0.153288	+	0.988182i	$0.451014\pi$
$434$			10.9655i			0.526360i
$435$		0			0
$436$	15.8716			0.760110
$437$	−6.96203			−0.333039
$438$		0			0
$439$			18.1574i			0.866606i	0.901248	+	0.433303i	$0.142652\pi$
$439$			18.1574i			0.866606i	−0.901248	+	0.433303i	$0.857348\pi$
$440$			2.48031i			0.118244i
$441$		0			0
$442$			7.20858i			0.342877i
$443$			15.7071i			0.746269i	0.927777	+	0.373135i	$0.121717\pi$
$443$			15.7071i			0.746269i	−0.927777	+	0.373135i	$0.878283\pi$
$444$		0			0
$445$			3.69285i			0.175058i
$446$	−6.65524	−	13.3682i	−0.315135	−	0.633001i
$447$		0			0
$448$	3.46872			0.163882
$449$	24.7835			1.16961			0.584803	−	0.811175i	$-0.301172\pi$
$449$	24.7835			1.16961			0.584803	+	0.811175i	$0.301172\pi$
$450$		0			0
$451$		−	6.17398i		−	0.290722i
$452$	0.213882			0.0100602
$453$		0			0
$454$	−16.1068			−0.755930
$455$			22.5885i			1.05897i
$456$		0			0
$457$		−	4.04644i		−	0.189284i	−0.995511	−	0.0946422i	$-0.969829\pi$
$457$		−	4.04644i		−	0.189284i	0.995511	−	0.0946422i	$-0.0301707\pi$
$458$	−23.1904			−1.08361
$459$		0			0
$460$	−3.24412			−0.151258
$461$			28.4921i			1.32701i	0.748171	+	0.663506i	$0.230932\pi$
$461$			28.4921i			1.32701i	−0.748171	+	0.663506i	$0.769068\pi$
$462$		0			0
$463$	−5.30080			−0.246349			−0.123174	−	0.992385i	$-0.539307\pi$
$463$	−5.30080			−0.246349			−0.123174	+	0.992385i	$0.539307\pi$
$464$			1.25255i			0.0581481i
$465$		0			0
$466$		−	19.4967i		−	0.903169i
$467$	3.24020			0.149939			0.0749693	−	0.997186i	$-0.476114\pi$
$467$	3.24020			0.149939			0.0749693	+	0.997186i	$0.476114\pi$
$468$		0			0
$469$		−	9.92211i		−	0.458160i
$470$	10.4245			0.480847
$471$		0			0
$472$			6.77146i			0.311682i
$473$	4.12601			0.189714
$474$		0			0
$475$	−8.44521			−0.387493
$476$			3.72087i			0.170546i
$477$		0			0
$478$	7.31789			0.334712
$479$			12.7509i			0.582605i	0.956631	+	0.291303i	$0.0940886\pi$
$479$			12.7509i			0.582605i	−0.956631	+	0.291303i	$0.905911\pi$
$480$		0			0
$481$			11.6555i			0.531445i
$482$		−	4.68387i		−	0.213345i
$483$		0			0
$484$	4.44875			0.202216
$485$		−	12.6835i		−	0.575928i
$486$		0			0
$487$	14.2673			0.646512			0.323256	−	0.946312i	$-0.395223\pi$
$487$	14.2673			0.646512			0.323256	+	0.946312i	$0.395223\pi$
$488$	7.49500			0.339283
$489$		0			0
$490$			4.87625i			0.220286i
$491$	0.174465			0.00787348			0.00393674	−	0.999992i	$-0.498747\pi$
$491$	0.174465			0.00787348			0.00393674	+	0.999992i	$0.498747\pi$
$492$		0			0
$493$	−1.34360			−0.0605128
$494$	13.9752			0.628773
$495$		0			0
$496$	3.16125			0.141944
$497$	7.29272			0.327123
$498$		0			0
$499$	12.8629			0.575824			0.287912	−	0.957657i	$-0.407039\pi$
$499$	12.8629			0.575824			0.287912	+	0.957657i	$0.407039\pi$
$500$	−8.78047			−0.392675
$501$		0			0
$502$	−12.7990			−0.571247
$503$	−20.2862			−0.904517			−0.452258	−	0.891887i	$-0.649382\pi$
$503$	−20.2862			−0.904517			−0.452258	+	0.891887i	$0.649382\pi$
$504$		0			0
$505$			10.0596i			0.447645i
$506$		−	8.56870i		−	0.380925i
$507$		0			0
$508$	2.76173			0.122532
$509$			4.44407i			0.196980i	0.995138	+	0.0984900i	$0.0314013\pi$
$509$			4.44407i			0.196980i	−0.995138	+	0.0984900i	$0.968599\pi$
$510$		0			0
$511$	−0.975193			−0.0431400
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	2.15971			0.0952609
$515$		−	3.29520i		−	0.145204i
$516$		0			0
$517$			27.5343i			1.21096i
$518$			6.01625i			0.264339i
$519$		0			0
$520$	6.51206			0.285573
$521$	−5.99851			−0.262800			−0.131400	−	0.991329i	$-0.541947\pi$
$521$	−5.99851			−0.262800			−0.131400	+	0.991329i	$0.541947\pi$
$522$		0			0
$523$			31.9703i			1.39796i	0.715140	+	0.698981i	$0.246363\pi$
$523$			31.9703i			1.39796i	−0.715140	+	0.698981i	$0.753637\pi$
$524$			11.6965i			0.510964i
$525$		0			0
$526$		−	11.3470i		−	0.494754i
$527$			3.39105i			0.147716i
$528$		0			0
$529$	−11.7926			−0.512721
$530$	−1.82021			−0.0790648
$531$		0			0
$532$	7.21360			0.312749
$533$	−16.2098			−0.702124
$534$		0			0
$535$	−12.5980			−0.544659
$536$	−2.86045			−0.123553
$537$		0			0
$538$			18.2555i			0.787051i
$539$	−12.8796			−0.554765
$540$		0			0
$541$		−	18.2573i		−	0.784944i	−0.919764	−	0.392472i	$-0.871620\pi$
$541$		−	18.2573i		−	0.784944i	0.919764	−	0.392472i	$-0.128380\pi$
$542$	−8.22822			−0.353432
$543$		0			0
$544$	1.07269			0.0459914
$545$	15.3803			0.658818
$546$		0			0
$547$	−4.64075			−0.198424			−0.0992120	−	0.995066i	$-0.531632\pi$
$547$	−4.64075			−0.198424			−0.0992120	+	0.995066i	$0.531632\pi$
$548$	−12.9587			−0.553569
$549$		0			0
$550$		−	10.3942i		−	0.443209i
$551$			2.60482i			0.110969i
$552$		0			0
$553$			5.19321i			0.220838i
$554$	14.4646			0.614543
$555$		0			0
$556$	11.9531			0.506924
$557$	−2.08638			−0.0884026			−0.0442013	−	0.999023i	$-0.514074\pi$
$557$	−2.08638			−0.0884026			−0.0442013	+	0.999023i	$0.514074\pi$
$558$		0			0
$559$		−	10.8328i		−	0.458180i
$560$	3.36135			0.142043
$561$		0			0
$562$	5.90766			0.249200
$563$	−31.9607			−1.34698			−0.673492	−	0.739195i	$-0.735206\pi$
$563$	−31.9607			−1.34698			−0.673492	+	0.739195i	$0.735206\pi$
$564$		0			0
$565$	0.207261			0.00871955
$566$		−	26.7225i		−	1.12323i
$567$		0			0
$568$		−	2.10243i		−	0.0882158i
$569$	36.4343			1.52741			0.763703	−	0.645568i	$-0.223379\pi$
$569$	36.4343			1.52741			0.763703	+	0.645568i	$0.223379\pi$
$570$		0			0
$571$			19.6602i			0.822752i	0.911466	+	0.411376i	$0.134952\pi$
$571$			19.6602i			0.822752i	−0.911466	+	0.411376i	$0.865048\pi$
$572$			17.2003i			0.719181i
$573$		0			0
$574$	−8.36705			−0.349234
$575$	13.5950			0.566952
$576$		0			0
$577$	−13.5502			−0.564104			−0.282052	−	0.959399i	$-0.591015\pi$
$577$	−13.5502			−0.564104			−0.282052	+	0.959399i	$0.591015\pi$
$578$		−	15.8493i		−	0.659245i
$579$		0			0
$580$			1.21378i			0.0503994i
$581$			32.4471i			1.34613i
$582$		0			0
$583$		−	4.80772i		−	0.199115i
$584$			0.281139i			0.0116336i
$585$		0			0
$586$			15.3541i			0.634272i
$587$	25.8057			1.06511			0.532557	−	0.846394i	$-0.321231\pi$
$587$	25.8057			1.06511			0.532557	+	0.846394i	$0.321231\pi$
$588$		0			0
$589$	6.57418			0.270884
$590$			6.56185i			0.270147i
$591$		0			0
$592$	1.73443			0.0712847
$593$	−28.3711			−1.16506			−0.582531	−	0.812809i	$-0.697937\pi$
$593$	−28.3711			−1.16506			−0.582531	+	0.812809i	$0.697937\pi$
$594$		0			0
$595$			3.60570i			0.147819i
$596$	13.5939			0.556829
$597$		0			0
$598$	−22.4971			−0.919975
$599$			16.6569i			0.680585i	0.940320	+	0.340292i	$0.110526\pi$
$599$			16.6569i			0.680585i	−0.940320	+	0.340292i	$0.889474\pi$
$600$		0			0
$601$		−	1.39838i		−	0.0570413i	−0.999593	−	0.0285206i	$-0.990920\pi$
$601$		−	1.39838i		−	0.0570413i	0.999593	−	0.0285206i	$-0.00907963\pi$
$602$		−	5.59161i		−	0.227897i
$603$		0			0
$604$		−	1.07562i		−	0.0437663i
$605$	4.31104			0.175269
$606$		0			0
$607$		−	18.1391i		−	0.736242i	−0.929778	−	0.368121i	$-0.880001\pi$
$607$		−	18.1391i		−	0.736242i	0.929778	−	0.368121i	$-0.119999\pi$
$608$		−	2.07962i		−	0.0843395i
$609$		0			0
$610$	7.26299			0.294070
$611$	72.2912			2.92459
$612$		0			0
$613$		−	19.4232i		−	0.784496i	−0.919860	−	0.392248i	$-0.871698\pi$
$613$		−	19.4232i		−	0.784496i	0.919860	−	0.392248i	$-0.128302\pi$
$614$	27.7608			1.12034
$615$		0			0
$616$			8.87833i			0.357718i
$617$		−	30.8809i		−	1.24322i	−0.783328	−	0.621609i	$-0.786479\pi$
$617$		−	30.8809i		−	1.24322i	0.783328	−	0.621609i	$-0.213521\pi$
$618$		0			0
$619$		−	6.61552i		−	0.265900i	−0.991123	−	0.132950i	$-0.957555\pi$
$619$		−	6.61552i		−	0.265900i	0.991123	−	0.132950i	$-0.0424450\pi$
$620$	3.06339			0.123029
$621$		0			0
$622$		−	29.7913i		−	1.19452i
$623$			13.2186i			0.529594i
$624$		0			0
$625$	11.7961			0.471843
$626$	−18.8166			−0.752062
$627$		0			0
$628$		−	3.39354i		−	0.135417i
$629$			1.86051i			0.0741835i
$630$		0			0
$631$			10.5797i			0.421173i	0.977575	+	0.210587i	$0.0675374\pi$
$631$			10.5797i			0.421173i	−0.977575	+	0.210587i	$0.932463\pi$
$632$	1.49716			0.0595536
$633$		0			0
$634$	−9.96393			−0.395718
$635$	2.67624			0.106203
$636$		0			0
$637$			33.8155i			1.33982i
$638$	−3.20595			−0.126925
$639$		0			0
$640$		−	0.969046i		−	0.0383049i
$641$	26.2818			1.03807			0.519034	−	0.854754i	$-0.326292\pi$
$641$	26.2818			1.03807			0.519034	+	0.854754i	$0.326292\pi$
$642$		0			0
$643$	46.0453			1.81585			0.907925	−	0.419133i	$-0.137666\pi$
$643$	46.0453			1.81585			0.907925	+	0.419133i	$0.137666\pi$
$644$	−11.6124			−0.457592
$645$		0			0
$646$	2.23079			0.0877693
$647$		−	9.16705i		−	0.360394i	−0.983631	−	0.180197i	$-0.942326\pi$
$647$		−	9.16705i		−	0.360394i	0.983631	−	0.180197i	$-0.0576735\pi$
$648$		0			0
$649$	−17.3318			−0.680334
$650$	−27.2899			−1.07040
$651$		0			0
$652$		−	6.09521i		−	0.238707i
$653$	−39.2613			−1.53641			−0.768207	−	0.640202i	$-0.778851\pi$
$653$	−39.2613			−1.53641			−0.768207	+	0.640202i	$0.778851\pi$
$654$		0			0
$655$			11.3344i			0.442873i
$656$			2.41214i			0.0941784i
$657$		0			0
$658$	37.3147			1.45468
$659$		−	5.39054i		−	0.209986i	−0.994473	−	0.104993i	$-0.966518\pi$
$659$		−	5.39054i		−	0.209986i	0.994473	−	0.104993i	$-0.0334820\pi$
$660$		0			0
$661$		−	14.9885i		−	0.582984i	−0.956573	−	0.291492i	$-0.905848\pi$
$661$		−	14.9885i		−	0.582984i	0.956573	−	0.291492i	$-0.0941517\pi$
$662$	−20.0451			−0.779076
$663$		0			0
$664$	9.35419			0.363013
$665$	6.99031			0.271073
$666$		0			0
$667$		−	4.19322i		−	0.162362i
$668$	7.12890			0.275825
$669$		0			0
$670$	−2.77191			−0.107088
$671$			19.1838i			0.740581i
$672$		0			0
$673$	45.6649			1.76025			0.880126	−	0.474740i	$-0.157458\pi$
$673$	45.6649			1.76025			0.880126	+	0.474740i	$0.157458\pi$
$674$	−27.6452			−1.06485
$675$		0			0
$676$	32.1594			1.23690
$677$		−	9.85412i		−	0.378725i	−0.981907	−	0.189362i	$-0.939358\pi$
$677$		−	9.85412i		−	0.378725i	0.981907	−	0.189362i	$-0.0606421\pi$
$678$		0			0
$679$		−	45.4008i		−	1.74232i
$680$	1.03949			0.0398626
$681$		0			0
$682$			8.09134i			0.309834i
$683$		−	17.8209i		−	0.681896i	−0.940082	−	0.340948i	$-0.889252\pi$
$683$		−	17.8209i		−	0.681896i	0.940082	−	0.340948i	$-0.110748\pi$
$684$		0			0
$685$	−12.5576			−0.479801
$686$		−	6.82641i		−	0.260633i
$687$		0			0
$688$	−1.61201			−0.0614573
$689$	−12.6227			−0.480885
$690$		0			0
$691$		−	3.19590i		−	0.121578i	−0.998151	−	0.0607890i	$-0.980638\pi$
$691$		−	3.19590i		−	0.121578i	0.998151	−	0.0607890i	$-0.0193617\pi$
$692$	−12.3333			−0.468843
$693$		0			0
$694$	−3.58620			−0.136130
$695$	11.5831			0.439372
$696$		0			0
$697$	−2.58749			−0.0980083
$698$		−	7.16650i		−	0.271256i
$699$		0			0
$700$	−14.0863			−0.532412
$701$		−	0.307901i		−	0.0116293i	−0.999983	−	0.00581463i	$-0.998149\pi$
$701$		−	0.307901i		−	0.0116293i	0.999983	−	0.00581463i	$-0.00185086\pi$
$702$		0			0
$703$	3.60695			0.136039
$704$	2.55954			0.0964664
$705$		0			0
$706$	−2.94264			−0.110748
$707$			36.0084i			1.35424i
$708$		0			0
$709$		−	22.9088i		−	0.860358i	−0.902744	−	0.430179i	$-0.858450\pi$
$709$		−	22.9088i		−	0.860358i	0.902744	−	0.430179i	$-0.141550\pi$
$710$		−	2.03735i		−	0.0764602i
$711$		0			0
$712$	3.81081			0.142816
$713$	−10.5831			−0.396339
$714$		0			0
$715$			16.6679i			0.623344i
$716$			20.1718i			0.753855i
$717$		0			0
$718$	−14.9608			−0.558332
$719$		−	16.3360i		−	0.609229i	−0.952476	−	0.304614i	$-0.901472\pi$
$719$		−	16.3360i		−	0.609229i	0.952476	−	0.304614i	$-0.0985276\pi$
$720$		0			0
$721$		−	11.7952i		−	0.439277i
$722$			14.6752i			0.546154i
$723$		0			0
$724$	2.88222			0.107117
$725$		−	5.08654i		−	0.188909i
$726$		0			0
$727$	−14.4292			−0.535148			−0.267574	−	0.963537i	$-0.586222\pi$
$727$	−14.4292			−0.535148			−0.267574	+	0.963537i	$0.586222\pi$
$728$	23.3100			0.863928
$729$		0			0
$730$			0.272437i			0.0100833i
$731$		−	1.72919i		−	0.0639565i
$732$		0			0
$733$	20.9730			0.774656			0.387328	−	0.921942i	$-0.373398\pi$
$733$	20.9730			0.774656			0.387328	+	0.921942i	$0.373398\pi$
$734$			20.2011i			0.745634i
$735$		0			0
$736$			3.34775i			0.123400i
$737$		−	7.32145i		−	0.269689i
$738$		0			0
$739$			24.4015i			0.897625i	0.893626	+	0.448812i	$0.148153\pi$
$739$			24.4015i			0.897625i	−0.893626	+	0.448812i	$0.851847\pi$
$740$	1.68074			0.0617853
$741$		0			0
$742$	−6.51547			−0.239191
$743$			19.0115i			0.697465i	0.937222	+	0.348732i	$0.113388\pi$
$743$			19.0115i			0.697465i	−0.937222	+	0.348732i	$0.886612\pi$
$744$		0			0
$745$	13.1731			0.482627
$746$	5.38424			0.197131
$747$		0			0
$748$			2.74560i			0.100389i
$749$	−45.0948			−1.64773
$750$		0			0
$751$	6.71157			0.244909			0.122454	−	0.992474i	$-0.460923\pi$
$751$	6.71157			0.244909			0.122454	+	0.992474i	$0.460923\pi$
$752$		−	10.7575i		−	0.392286i
$753$		0			0
$754$			8.41722i			0.306537i
$755$		−	1.04232i		−	0.0379340i
$756$		0			0
$757$		−	50.8520i		−	1.84825i	−0.382093	−	0.924124i	$-0.624797\pi$
$757$		−	50.8520i		−	1.84825i	0.382093	−	0.924124i	$-0.375203\pi$
$758$			8.94431i			0.324872i
$759$		0			0
$760$		−	2.01524i		−	0.0731005i
$761$	−11.4685			−0.415732			−0.207866	−	0.978157i	$-0.566652\pi$
$761$	−11.4685			−0.415732			−0.207866	+	0.978157i	$0.566652\pi$
$762$		0			0
$763$	55.0540			1.99309
$764$	−11.6916			−0.422989
$765$		0			0
$766$		−	7.05417i		−	0.254877i
$767$			45.5047i			1.64308i
$768$		0			0
$769$	26.5809			0.958531			0.479265	−	0.877670i	$-0.340903\pi$
$769$	26.5809			0.958531			0.479265	+	0.877670i	$0.340903\pi$
$770$			8.60351i			0.310049i
$771$		0			0
$772$			0.0831582i			0.00299293i
$773$	1.89399			0.0681223			0.0340611	−	0.999420i	$-0.489156\pi$
$773$	1.89399			0.0681223			0.0340611	+	0.999420i	$0.489156\pi$
$774$		0			0
$775$	−12.8377			−0.461143
$776$	−13.0886			−0.469855
$777$		0			0
$778$	−2.40158			−0.0861007
$779$			5.01633i			0.179729i
$780$		0			0
$781$	5.38125			0.192556
$782$	−3.59111			−0.128418
$783$		0			0
$784$	5.03201			0.179715
$785$		−	3.28850i		−	0.117371i
$786$		0			0
$787$			20.3861i			0.726685i	0.931656	+	0.363342i	$0.118364\pi$
$787$			20.3861i			0.726685i	−0.931656	+	0.363342i	$0.881636\pi$
$788$			9.84566i			0.350737i
$789$		0			0
$790$	1.45081			0.0516176
$791$	0.741896			0.0263788
$792$		0			0
$793$	50.3669			1.78858
$794$	−37.6516			−1.33621
$795$		0			0
$796$	12.3477			0.437652
$797$			26.3038i			0.931729i	0.884856	+	0.465864i	$0.154256\pi$
$797$			26.3038i			0.931729i	−0.884856	+	0.465864i	$0.845744\pi$
$798$		0			0
$799$	11.5395			0.408238
$800$			4.06095i			0.143576i
$801$		0			0
$802$	23.1534			0.817573
$803$	−0.719588			−0.0253937
$804$		0			0
$805$	−11.2529			−0.396614
$806$	21.2438			0.748281
$807$		0			0
$808$	10.3809			0.365199
$809$	−36.4983			−1.28321			−0.641607	−	0.767034i	$-0.721732\pi$
$809$	−36.4983			−1.28321			−0.641607	+	0.767034i	$0.721732\pi$
$810$		0			0
$811$		−	0.974702i		−	0.0342264i	−0.999854	−	0.0171132i	$-0.994552\pi$
$811$		−	0.974702i		−	0.0342264i	0.999854	−	0.0171132i	$-0.00544757\pi$
$812$			4.34474i			0.152471i
$813$		0			0
$814$			4.43935i			0.155599i
$815$		−	5.90654i		−	0.206897i
$816$		0			0
$817$	−3.35236			−0.117284
$818$	12.6033			0.440664
$819$		0			0
$820$			2.33748i			0.0816283i
$821$			21.5793i			0.753122i	0.926392	+	0.376561i	$0.122894\pi$
$821$			21.5793i			0.753122i	−0.926392	+	0.376561i	$0.877106\pi$
$822$		0			0
$823$		−	20.7826i		−	0.724435i	−0.932094	−	0.362217i	$-0.882020\pi$
$823$		−	20.7826i		−	0.724435i	0.932094	−	0.362217i	$-0.117980\pi$
$824$	−3.40046			−0.118460
$825$		0			0
$826$			23.4883i			0.817262i
$827$	−21.6578			−0.753116			−0.376558	−	0.926393i	$-0.622893\pi$
$827$	−21.6578			−0.753116			−0.376558	+	0.926393i	$0.622893\pi$
$828$		0			0
$829$			7.23302i			0.251213i	0.992080	+	0.125607i	$0.0400877\pi$
$829$			7.23302i			0.251213i	−0.992080	+	0.125607i	$0.959912\pi$
$830$	9.06464			0.314638
$831$		0			0
$832$		−	6.72007i		−	0.232977i
$833$			5.39780i			0.187023i
$834$		0			0
$835$	6.90823			0.239069
$836$	5.32286			0.184095
$837$		0			0
$838$	−21.0493			−0.727137
$839$	−47.1529			−1.62790			−0.813949	−	0.580937i	$-0.802686\pi$
$839$	−47.1529			−1.62790			−0.813949	+	0.580937i	$0.802686\pi$
$840$		0			0
$841$	27.4311			0.945901
$842$	22.6340			0.780019
$843$		0			0
$844$	−22.7720			−0.783845
$845$	31.1639			1.07207
$846$		0			0
$847$	15.4314			0.530231
$848$			1.87835i			0.0645029i
$849$		0			0
$850$	−4.35616			−0.149415
$851$	−5.80644			−0.199042
$852$		0			0
$853$			26.9928i			0.924215i	0.886824	+	0.462107i	$0.152907\pi$
$853$			26.9928i			0.924215i	−0.886824	+	0.462107i	$0.847093\pi$
$854$	25.9980			0.889634
$855$		0			0
$856$			13.0004i			0.444345i
$857$		−	7.48806i		−	0.255787i	−0.991788	−	0.127894i	$-0.959178\pi$
$857$		−	7.48806i		−	0.255787i	0.991788	−	0.127894i	$-0.0408216\pi$
$858$		0			0
$859$			20.7038i			0.706405i	0.935547	+	0.353202i	$0.114907\pi$
$859$			20.7038i			0.706405i	−0.935547	+	0.353202i	$0.885093\pi$
$860$	−1.56211			−0.0532676
$861$		0			0
$862$			15.1291i			0.515301i
$863$	−21.1713			−0.720680			−0.360340	−	0.932821i	$-0.617339\pi$
$863$	−21.1713			−0.720680			−0.360340	+	0.932821i	$0.617339\pi$
$864$		0			0
$865$	−11.9516			−0.406365
$866$		−	6.37944i		−	0.216782i
$867$		0			0
$868$	10.9655			0.372192
$869$			3.83203i			0.129993i
$870$		0			0
$871$	−19.2225			−0.651328
$872$		−	15.8716i		−	0.537479i
$873$		0			0
$874$			6.96203i			0.235494i
$875$	−30.4570			−1.02963
$876$		0			0
$877$			43.7673i			1.47792i	0.673750	+	0.738959i	$0.264682\pi$
$877$			43.7673i			1.47792i	−0.673750	+	0.738959i	$0.735318\pi$
$878$	18.1574			0.612783
$879$		0			0
$880$	2.48031			0.0836113
$881$		−	51.3943i		−	1.73152i	−0.500460	−	0.865759i	$-0.666836\pi$
$881$		−	51.3943i		−	1.73152i	0.500460	−	0.865759i	$-0.333164\pi$
$882$		0			0
$883$			27.0427i			0.910059i	0.890476	+	0.455029i	$0.150371\pi$
$883$			27.0427i			0.910059i	−0.890476	+	0.455029i	$0.849629\pi$
$884$	7.20858			0.242451
$885$		0			0
$886$	15.7071			0.527692
$887$		−	9.49108i		−	0.318679i	−0.987224	−	0.159340i	$-0.949063\pi$
$887$		−	9.49108i		−	0.318679i	0.987224	−	0.159340i	$-0.0509365\pi$
$888$		0			0
$889$	9.57966			0.321291
$890$	3.69285			0.123785
$891$		0			0
$892$	−13.3682	+	6.65524i	−0.447599	+	0.222834i
$893$		−	22.3715i		−	0.748632i
$894$		0			0
$895$			19.5474i			0.653397i
$896$		−	3.46872i		−	0.115882i
$897$		0			0
$898$		−	24.7835i		−	0.827036i
$899$			3.95962i			0.132061i
$900$		0			0
$901$	−2.01490			−0.0671259
$902$	−6.17398			−0.205571
$903$		0			0
$904$		−	0.213882i		−	0.00711361i
$905$	2.79300			0.0928425
$906$		0			0
$907$	38.9365			1.29286			0.646432	−	0.762972i	$-0.276260\pi$
$907$	38.9365			1.29286			0.646432	+	0.762972i	$0.276260\pi$
$908$			16.1068i			0.534523i
$909$		0			0
$910$	22.5885			0.748801
$911$		−	30.5496i		−	1.01215i	−0.862488	−	0.506077i	$-0.831095\pi$
$911$		−	30.5496i		−	1.01215i	0.862488	−	0.506077i	$-0.168905\pi$
$912$		0			0
$913$			23.9424i			0.792379i
$914$	−4.04644			−0.133844
$915$		0			0
$916$			23.1904i			0.766231i
$917$			40.5718i			1.33980i
$918$		0			0
$919$			32.6393i			1.07667i	0.842731	+	0.538335i	$0.180946\pi$
$919$			32.6393i			1.07667i	−0.842731	+	0.538335i	$0.819054\pi$
$920$			3.24412i			0.106955i
$921$		0			0
$922$	28.4921			0.938339
$923$		−	14.1285i		−	0.465044i
$924$		0			0
$925$	−7.04344			−0.231587
$926$			5.30080i			0.174195i
$927$		0			0
$928$	1.25255			0.0411169
$929$			13.5638i			0.445014i	0.974931	+	0.222507i	$0.0714241\pi$
$929$			13.5638i			0.445014i	−0.974931	+	0.222507i	$0.928576\pi$
$930$		0			0
$931$	10.4646			0.342965
$932$	−19.4967			−0.638637
$933$		0			0
$934$		−	3.24020i		−	0.106023i
$935$			2.66062i			0.0870115i
$936$		0			0
$937$			35.2124i			1.15034i	0.818034	+	0.575170i	$0.195064\pi$
$937$			35.2124i			1.15034i	−0.818034	+	0.575170i	$0.804936\pi$
$938$	−9.92211			−0.323968
$939$		0			0
$940$		−	10.4245i		−	0.340010i
$941$		−	10.8191i		−	0.352691i	−0.984328	−	0.176346i	$-0.943572\pi$
$941$		−	10.8191i		−	0.352691i	0.984328	−	0.176346i	$-0.0564277\pi$
$942$		0			0
$943$		−	8.07525i		−	0.262966i
$944$	6.77146			0.220392
$945$		0			0
$946$		−	4.12601i		−	0.134148i
$947$			10.6045i			0.344599i	0.985045	+	0.172300i	$0.0551198\pi$
$947$			10.6045i			0.344599i	−0.985045	+	0.172300i	$0.944880\pi$
$948$		0			0
$949$			1.88928i			0.0613285i
$950$			8.44521i			0.273999i
$951$		0			0
$952$	3.72087			0.120594
$953$	−12.3593			−0.400356			−0.200178	−	0.979760i	$-0.564152\pi$
$953$	−12.3593			−0.400356			−0.200178	+	0.979760i	$0.564152\pi$
$954$		0			0
$955$	−11.3297			−0.366622
$956$		−	7.31789i		−	0.236677i
$957$		0			0
$958$	12.7509			0.411964
$959$	−44.9501			−1.45151
$960$		0			0
$961$	−21.0065			−0.677630
$962$	11.6555			0.375788
$963$		0			0
$964$	−4.68387			−0.150857
$965$			0.0805841i			0.00259409i
$966$		0			0
$967$		−	34.2823i		−	1.10244i	−0.834359	−	0.551222i	$-0.814162\pi$
$967$		−	34.2823i		−	1.10244i	0.834359	−	0.551222i	$-0.185838\pi$
$968$		−	4.44875i		−	0.142988i
$969$		0			0
$970$	−12.6835			−0.407243
$971$	−23.9188			−0.767590			−0.383795	−	0.923418i	$-0.625383\pi$
$971$	−23.9188			−0.767590			−0.383795	+	0.923418i	$0.625383\pi$
$972$		0			0
$973$	41.4619			1.32921
$974$		−	14.2673i		−	0.457153i
$975$		0			0
$976$		−	7.49500i		−	0.239909i
$977$	12.6179			0.403683			0.201842	−	0.979418i	$-0.435307\pi$
$977$	12.6179			0.403683			0.201842	+	0.979418i	$0.435307\pi$
$978$		0			0
$979$			9.75394i			0.311737i
$980$	4.87625			0.155766
$981$		0			0
$982$		−	0.174465i		−	0.00556739i
$983$	52.6913			1.68059			0.840295	−	0.542129i	$-0.182382\pi$
$983$	52.6913			1.68059			0.840295	+	0.542129i	$0.182382\pi$
$984$		0			0
$985$			9.54090i			0.303998i
$986$			1.34360i			0.0427890i
$987$		0			0
$988$		−	13.9752i		−	0.444609i
$989$	5.39661			0.171602
$990$		0			0
$991$			4.80050i			0.152493i	0.997089	+	0.0762465i	$0.0242936\pi$
$991$			4.80050i			0.152493i	−0.997089	+	0.0762465i	$0.975706\pi$
$992$		−	3.16125i		−	0.100370i
$993$		0			0
$994$		−	7.29272i		−	0.231311i
$995$	11.9655			0.379331
$996$		0			0
$997$	−40.0125			−1.26721			−0.633604	−	0.773657i	$-0.718425\pi$
$997$	−40.0125			−1.26721			−0.633604	+	0.773657i	$0.718425\pi$
$998$		−	12.8629i		−	0.407169i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.d.a.4013.3	✓	72
3.2	odd	2	inner	4014.2.d.a.4013.68	yes	72
223.222	odd	2	inner	4014.2.d.a.4013.67	yes	72
669.668	even	2	inner	4014.2.d.a.4013.4	yes	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.d.a.4013.3	✓	72	1.1	even	1	trivial
4014.2.d.a.4013.4	yes	72	669.668	even	2	inner
4014.2.d.a.4013.67	yes	72	223.222	odd	2	inner
4014.2.d.a.4013.68	yes	72	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.969046 q^{5} -3.46872 q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.969046 q^{5} -3.46872 q^{7} +1.00000i q^{8} +0.969046i q^{10} -2.55954 q^{11} +6.72007i q^{13} +3.46872i q^{14} +1.00000 q^{16} +1.07269i q^{17} +2.07962 q^{19} +0.969046 q^{20} +2.55954i q^{22} -3.34775 q^{23} -4.06095 q^{25} +6.72007 q^{26} +3.46872 q^{28} +1.25255i q^{29} +3.16125 q^{31} -1.00000i q^{32} +1.07269 q^{34} +3.36135 q^{35} +1.73443 q^{37} -2.07962i q^{38} -0.969046i q^{40} +2.41214i q^{41} -1.61201 q^{43} +2.55954 q^{44} +3.34775i q^{46} -10.7575i q^{47} +5.03201 q^{49} +4.06095i q^{50} -6.72007i q^{52} +1.87835i q^{53} +2.48031 q^{55} -3.46872i q^{56} +1.25255 q^{58} +6.77146 q^{59} -7.49500i q^{61} -3.16125i q^{62} -1.00000 q^{64} -6.51206i q^{65} +2.86045i q^{67} -1.07269i q^{68} -3.36135i q^{70} -2.10243 q^{71} +0.281139 q^{73} -1.73443i q^{74} -2.07962 q^{76} +8.87833 q^{77} -1.49716i q^{79} -0.969046 q^{80} +2.41214 q^{82} -9.35419i q^{83} -1.03949i q^{85} +1.61201i q^{86} -2.55954i q^{88} -3.81081i q^{89} -23.3100i q^{91} +3.34775 q^{92} -10.7575 q^{94} -2.01524 q^{95} +13.0886i q^{97} -5.03201i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) 72 * q - 72 * q^4 + 16 * q^7	\( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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