LMFDB - Embedded newform 4000.2.d.c.2001.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(2001,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4000.2001");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.d (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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 1000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2001.11
Character	$\chi$	$=$	4000.2001
Dual form			4000.2.d.c.2001.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.207209i q^{3} +2.73181 q^{7} +2.95706 q^{9} +O(q^{10})$	$q-0.207209i q^{3} +2.73181 q^{7} +2.95706 q^{9} +0.871377i q^{11} +1.98325i q^{13} -2.93520 q^{17} -0.905406i q^{19} -0.566054i q^{21} +6.50896 q^{23} -1.23435i q^{27} +7.71622i q^{29} +4.14455 q^{31} +0.180557 q^{33} -0.436790i q^{37} +0.410946 q^{39} +5.84841 q^{41} -3.55191i q^{43} -4.69617 q^{47} +0.462766 q^{49} +0.608198i q^{51} +9.68555i q^{53} -0.187608 q^{57} -12.8197i q^{59} +12.7406i q^{61} +8.07813 q^{63} +10.2605i q^{67} -1.34871i q^{69} +12.9184 q^{71} -9.76540 q^{73} +2.38043i q^{77} -3.02177 q^{79} +8.61542 q^{81} -9.88164i q^{83} +1.59887 q^{87} -8.04615 q^{89} +5.41785i q^{91} -0.858786i q^{93} +12.1937 q^{97} +2.57672i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 24 q^{9}+O(q^{10}) $ 40 * q - 24 * q^9	$ 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) $ 40 * q - 24 * q^9 - 48 * q^31 - 8 * q^39 + 44 * q^41 + 12 * q^49 - 96 * q^71 - 96 * q^79 - 56 * q^81 - 44 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 0.207209i	− 0.119632i	−0.998209	−	0.0598160i	$-0.980949\pi$
$3$	− 0.207209i	− 0.119632i	0.998209	−	0.0598160i	$-0.0190514\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.73181	1.03253	0.516263	−	0.856430i	$-0.327323\pi$
$7$	2.73181	1.03253	0.516263	+	0.856430i	$0.327323\pi$
$8$	0	0
$9$	2.95706	0.985688
$10$	0	0
$11$	0.871377i	0.262730i	0.991334	+	0.131365i	$0.0419360\pi$
$11$	0.871377i	0.262730i	−0.991334	+	0.131365i	$0.958064\pi$
$12$	0	0
$13$	1.98325i	0.550054i	0.961437	+	0.275027i	$0.0886867\pi$
$13$	1.98325i	0.550054i	−0.961437	+	0.275027i	$0.911313\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.93520	−0.711890	−0.355945	−	0.934507i	$-0.615841\pi$
$17$	−2.93520	−0.711890	−0.355945	+	0.934507i	$0.615841\pi$
$18$	0	0
$19$	− 0.905406i	− 0.207714i	−0.994592	−	0.103857i	$-0.966882\pi$
$19$	− 0.905406i	− 0.207714i	0.994592	−	0.103857i	$-0.0331185\pi$
$20$	0	0
$21$	− 0.566054i	− 0.123523i
$22$	0	0
$23$	6.50896	1.35721	0.678606	−	0.734502i	$-0.262584\pi$
$23$	6.50896	1.35721	0.678606	+	0.734502i	$0.262584\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.23435i	− 0.237552i
$28$	0	0
$29$	7.71622i	1.43287i	0.697656	+	0.716433i	$0.254227\pi$
$29$	7.71622i	1.43287i	−0.697656	+	0.716433i	$0.745773\pi$
$30$	0	0
$31$	4.14455	0.744383	0.372191	−	0.928156i	$-0.378606\pi$
$31$	4.14455	0.744383	0.372191	+	0.928156i	$0.378606\pi$
$32$	0	0
$33$	0.180557	0.0314309
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.436790i	− 0.0718078i	−0.999355	−	0.0359039i	$-0.988569\pi$
$37$	− 0.436790i	− 0.0718078i	0.999355	−	0.0359039i	$-0.0114310\pi$
$38$	0	0
$39$	0.410946	0.0658040
$40$	0	0
$41$	5.84841	0.913368	0.456684	−	0.889629i	$-0.349037\pi$
$41$	5.84841	0.913368	0.456684	+	0.889629i	$0.349037\pi$
$42$	0	0
$43$	− 3.55191i	− 0.541661i	−0.962627	−	0.270830i	$-0.912702\pi$
$43$	− 3.55191i	− 0.541661i	0.962627	−	0.270830i	$-0.0872983\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.69617	−0.685006	−0.342503	−	0.939517i	$-0.611275\pi$
$47$	−4.69617	−0.685006	−0.342503	+	0.939517i	$0.611275\pi$
$48$	0	0
$49$	0.462766	0.0661095
$50$	0	0
$51$	0.608198i	0.0851648i
$52$	0	0
$53$	9.68555i	1.33041i	0.746660	+	0.665206i	$0.231656\pi$
$53$	9.68555i	1.33041i	−0.746660	+	0.665206i	$0.768344\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.187608	−0.0248493
$58$	0	0
$59$	− 12.8197i	− 1.66899i	−0.551016	−	0.834495i	$-0.685760\pi$
$59$	− 12.8197i	− 1.66899i	0.551016	−	0.834495i	$-0.314240\pi$
$60$	0	0
$61$	12.7406i	1.63126i	0.578573	+	0.815630i	$0.303610\pi$
$61$	12.7406i	1.63126i	−0.578573	+	0.815630i	$0.696390\pi$
$62$	0	0
$63$	8.07813	1.01775
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.2605i	1.25352i	0.779212	+	0.626760i	$0.215619\pi$
$67$	10.2605i	1.25352i	−0.779212	+	0.626760i	$0.784381\pi$
$68$	0	0
$69$	− 1.34871i	− 0.162366i
$70$	0	0
$71$	12.9184	1.53314	0.766569	−	0.642162i	$-0.221962\pi$
$71$	12.9184	1.53314	0.766569	+	0.642162i	$0.221962\pi$
$72$	0	0
$73$	−9.76540	−1.14295	−0.571477	−	0.820618i	$-0.693629\pi$
$73$	−9.76540	−1.14295	−0.571477	+	0.820618i	$0.693629\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.38043i	0.271276i
$78$	0	0
$79$	−3.02177	−0.339976	−0.169988	−	0.985446i	$-0.554373\pi$
$79$	−3.02177	−0.339976	−0.169988	+	0.985446i	$0.554373\pi$
$80$	0	0
$81$	8.61542	0.957269
$82$	0	0
$83$	− 9.88164i	− 1.08465i	−0.840169	−	0.542325i	$-0.817544\pi$
$83$	− 9.88164i	− 1.08465i	0.840169	−	0.542325i	$-0.182456\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.59887	0.171417
$88$	0	0
$89$	−8.04615	−0.852891	−0.426445	−	0.904513i	$-0.640234\pi$
$89$	−8.04615	−0.852891	−0.426445	+	0.904513i	$0.640234\pi$
$90$	0	0
$91$	5.41785i	0.567945i
$92$	0	0
$93$	− 0.858786i	− 0.0890519i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.1937	1.23808	0.619041	−	0.785359i	$-0.287522\pi$
$97$	12.1937	1.23808	0.619041	+	0.785359i	$0.287522\pi$
$98$	0	0
$99$	2.57672i	0.258970i
$100$	0	0
$101$	− 5.51432i	− 0.548696i	−0.961631	−	0.274348i	$-0.911538\pi$
$101$	− 5.51432i	− 0.548696i	0.961631	−	0.274348i	$-0.0884619\pi$
$102$	0	0
$103$	−4.98581	−0.491266	−0.245633	−	0.969363i	$-0.578996\pi$
$103$	−4.98581	−0.491266	−0.245633	+	0.969363i	$0.578996\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.4436i	− 1.20297i	−0.798883	−	0.601486i	$-0.794575\pi$
$107$	− 12.4436i	− 1.20297i	0.798883	−	0.601486i	$-0.205425\pi$
$108$	0	0
$109$	4.64042i	0.444472i	0.974993	+	0.222236i	$0.0713355\pi$
$109$	4.64042i	0.444472i	−0.974993	+	0.222236i	$0.928664\pi$
$110$	0	0
$111$	−0.0905066	−0.00859051
$112$	0	0
$113$	−17.0652	−1.60536	−0.802680	−	0.596410i	$-0.796593\pi$
$113$	−17.0652	−1.60536	−0.802680	+	0.596410i	$0.796593\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.86459i	0.542181i
$118$	0	0
$119$	−8.01840	−0.735045
$120$	0	0
$121$	10.2407	0.930973
$122$	0	0
$123$	− 1.21184i	− 0.109268i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.92450	−0.348243	−0.174121	−	0.984724i	$-0.555709\pi$
$127$	−3.92450	−0.348243	−0.174121	+	0.984724i	$0.555709\pi$
$128$	0	0
$129$	−0.735986	−0.0647999
$130$	0	0
$131$	16.8728i	1.47419i	0.675792	+	0.737093i	$0.263802\pi$
$131$	16.8728i	1.47419i	−0.675792	+	0.737093i	$0.736198\pi$
$132$	0	0
$133$	− 2.47339i	− 0.214470i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.81761	−0.582468	−0.291234	−	0.956652i	$-0.594066\pi$
$137$	−6.81761	−0.582468	−0.291234	+	0.956652i	$0.594066\pi$
$138$	0	0
$139$	13.4616i	1.14180i	0.821021	+	0.570898i	$0.193405\pi$
$139$	13.4616i	1.14180i	−0.821021	+	0.570898i	$0.806595\pi$
$140$	0	0
$141$	0.973086i	0.0819486i
$142$	0	0
$143$	−1.72816	−0.144516
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 0.0958892i	− 0.00790880i
$148$	0	0
$149$	− 19.6182i	− 1.60719i	−0.595179	−	0.803593i	$-0.702919\pi$
$149$	− 19.6182i	− 1.60719i	0.595179	−	0.803593i	$-0.297081\pi$
$150$	0	0
$151$	21.8652	1.77937	0.889684	−	0.456577i	$-0.150925\pi$
$151$	21.8652	1.77937	0.889684	+	0.456577i	$0.150925\pi$
$152$	0	0
$153$	−8.67957	−0.701702
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.34144i	− 0.426293i	−0.977020	−	0.213147i	$-0.931629\pi$
$157$	− 5.34144i	− 0.426293i	0.977020	−	0.213147i	$-0.0683712\pi$
$158$	0	0
$159$	2.00693	0.159160
$160$	0	0
$161$	17.7812	1.40136
$162$	0	0
$163$	− 17.4676i	− 1.36817i	−0.729401	−	0.684086i	$-0.760201\pi$
$163$	− 17.4676i	− 1.36817i	0.729401	−	0.684086i	$-0.239799\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.5956	−0.897293	−0.448647	−	0.893709i	$-0.648094\pi$
$167$	−11.5956	−0.897293	−0.448647	+	0.893709i	$0.648094\pi$
$168$	0	0
$169$	9.06673	0.697441
$170$	0	0
$171$	− 2.67734i	− 0.204742i
$172$	0	0
$173$	9.07537i	0.689988i	0.938605	+	0.344994i	$0.112119\pi$
$173$	9.07537i	0.689988i	−0.938605	+	0.344994i	$0.887881\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.65636	−0.199664
$178$	0	0
$179$	10.0236i	0.749200i	0.927186	+	0.374600i	$0.122220\pi$
$179$	10.0236i	0.749200i	−0.927186	+	0.374600i	$0.877780\pi$
$180$	0	0
$181$	14.2992i	1.06285i	0.847106	+	0.531424i	$0.178343\pi$
$181$	14.2992i	1.06285i	−0.847106	+	0.531424i	$0.821657\pi$
$182$	0	0
$183$	2.63995	0.195151
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.55767i	− 0.187035i
$188$	0	0
$189$	− 3.37202i	− 0.245278i
$190$	0	0
$191$	13.4414	0.972588	0.486294	−	0.873795i	$-0.338348\pi$
$191$	13.4414	0.972588	0.486294	+	0.873795i	$0.338348\pi$
$192$	0	0
$193$	20.0219	1.44121	0.720606	−	0.693345i	$-0.243864\pi$
$193$	20.0219	1.44121	0.720606	+	0.693345i	$0.243864\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.4963i	− 0.961568i	−0.876839	−	0.480784i	$-0.840352\pi$
$197$	− 13.4963i	− 0.961568i	0.876839	−	0.480784i	$-0.159648\pi$
$198$	0	0
$199$	14.1065	0.999981	0.499990	−	0.866031i	$-0.333337\pi$
$199$	14.1065	0.999981	0.499990	+	0.866031i	$0.333337\pi$
$200$	0	0
$201$	2.12607	0.149961
$202$	0	0
$203$	21.0792i	1.47947i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	19.2474	1.33779
$208$	0	0
$209$	0.788950	0.0545728
$210$	0	0
$211$	− 13.1355i	− 0.904288i	−0.891945	−	0.452144i	$-0.850659\pi$
$211$	− 13.1355i	− 0.904288i	0.891945	−	0.452144i	$-0.149341\pi$
$212$	0	0
$213$	− 2.67681i	− 0.183412i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.3221	0.768594
$218$	0	0
$219$	2.02347i	0.136734i
$220$	0	0
$221$	− 5.82122i	− 0.391578i
$222$	0	0
$223$	−0.718852	−0.0481379	−0.0240689	−	0.999710i	$-0.507662\pi$
$223$	−0.718852	−0.0481379	−0.0240689	+	0.999710i	$0.507662\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0713i	0.801198i	0.916254	+	0.400599i	$0.131198\pi$
$227$	12.0713i	0.801198i	−0.916254	+	0.400599i	$0.868802\pi$
$228$	0	0
$229$	− 10.3408i	− 0.683342i	−0.939820	−	0.341671i	$-0.889007\pi$
$229$	− 10.3408i	− 0.683342i	0.939820	−	0.341671i	$-0.110993\pi$
$230$	0	0
$231$	0.493246	0.0324532
$232$	0	0
$233$	4.03296	0.264208	0.132104	−	0.991236i	$-0.457827\pi$
$233$	4.03296	0.264208	0.132104	+	0.991236i	$0.457827\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.626136i	0.0406719i
$238$	0	0
$239$	11.7267	0.758540	0.379270	−	0.925286i	$-0.376175\pi$
$239$	11.7267	0.758540	0.379270	+	0.925286i	$0.376175\pi$
$240$	0	0
$241$	12.7697	0.822569	0.411284	−	0.911507i	$-0.365080\pi$
$241$	12.7697	0.822569	0.411284	+	0.911507i	$0.365080\pi$
$242$	0	0
$243$	− 5.48825i	− 0.352072i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.79564	0.114254
$248$	0	0
$249$	−2.04756	−0.129759
$250$	0	0
$251$	22.8026i	1.43929i	0.694344	+	0.719644i	$0.255695\pi$
$251$	22.8026i	1.43929i	−0.694344	+	0.719644i	$0.744305\pi$
$252$	0	0
$253$	5.67176i	0.356581i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.6835	−1.41496	−0.707478	−	0.706736i	$-0.750167\pi$
$257$	−22.6835	−1.41496	−0.707478	+	0.706736i	$0.750167\pi$
$258$	0	0
$259$	− 1.19323i	− 0.0741435i
$260$	0	0
$261$	22.8174i	1.41236i
$262$	0	0
$263$	15.0066	0.925346	0.462673	−	0.886529i	$-0.346890\pi$
$263$	15.0066	0.925346	0.462673	+	0.886529i	$0.346890\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.66723i	0.102033i
$268$	0	0
$269$	15.3940i	0.938591i	0.883041	+	0.469295i	$0.155492\pi$
$269$	15.3940i	0.938591i	−0.883041	+	0.469295i	$0.844508\pi$
$270$	0	0
$271$	15.1636	0.921121	0.460561	−	0.887628i	$-0.347648\pi$
$271$	15.1636	0.921121	0.460561	+	0.887628i	$0.347648\pi$
$272$	0	0
$273$	1.12262	0.0679443
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.80905i	0.589369i	0.955595	+	0.294684i	$0.0952145\pi$
$277$	9.80905i	0.589369i	−0.955595	+	0.294684i	$0.904785\pi$
$278$	0	0
$279$	12.2557	0.733729
$280$	0	0
$281$	3.41960	0.203996	0.101998	−	0.994785i	$-0.467476\pi$
$281$	3.41960	0.203996	0.101998	+	0.994785i	$0.467476\pi$
$282$	0	0
$283$	24.5466i	1.45914i	0.683904	+	0.729572i	$0.260281\pi$
$283$	24.5466i	1.45914i	−0.683904	+	0.729572i	$0.739719\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.9767	0.943076
$288$	0	0
$289$	−8.38461	−0.493212
$290$	0	0
$291$	− 2.52664i	− 0.148114i
$292$	0	0
$293$	− 22.9522i	− 1.34088i	−0.741962	−	0.670442i	$-0.766104\pi$
$293$	− 22.9522i	− 1.34088i	0.741962	−	0.670442i	$-0.233896\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.07559	0.0624120
$298$	0	0
$299$	12.9089i	0.746540i
$300$	0	0
$301$	− 9.70312i	− 0.559279i
$302$	0	0
$303$	−1.14261	−0.0656415
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.31183i	0.131943i	0.997822	+	0.0659714i	$0.0210146\pi$
$307$	2.31183i	0.131943i	−0.997822	+	0.0659714i	$0.978985\pi$
$308$	0	0
$309$	1.03310i	0.0587711i
$310$	0	0
$311$	−0.398156	−0.0225774	−0.0112887	−	0.999936i	$-0.503593\pi$
$311$	−0.398156	−0.0225774	−0.0112887	+	0.999936i	$0.503593\pi$
$312$	0	0
$313$	27.7896	1.57076	0.785380	−	0.619013i	$-0.212467\pi$
$313$	27.7896	1.57076	0.785380	+	0.619013i	$0.212467\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.7194i	0.939056i	0.882918	+	0.469528i	$0.155576\pi$
$317$	16.7194i	0.939056i	−0.882918	+	0.469528i	$0.844424\pi$
$318$	0	0
$319$	−6.72374	−0.376457
$320$	0	0
$321$	−2.57843	−0.143914
$322$	0	0
$323$	2.65755i	0.147870i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.961535	0.0531730
$328$	0	0
$329$	−12.8290	−0.707286
$330$	0	0
$331$	− 12.7840i	− 0.702673i	−0.936249	−	0.351337i	$-0.885727\pi$
$331$	− 12.7840i	− 0.702673i	0.936249	−	0.351337i	$-0.114273\pi$
$332$	0	0
$333$	− 1.29162i	− 0.0707801i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.73648	0.258012	0.129006	−	0.991644i	$-0.458821\pi$
$337$	4.73648	0.258012	0.129006	+	0.991644i	$0.458821\pi$
$338$	0	0
$339$	3.53606i	0.192052i
$340$	0	0
$341$	3.61147i	0.195572i
$342$	0	0
$343$	−17.8585	−0.964266
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 25.7080i	− 1.38008i	−0.723773	−	0.690038i	$-0.757594\pi$
$347$	− 25.7080i	− 1.38008i	0.723773	−	0.690038i	$-0.242406\pi$
$348$	0	0
$349$	7.18541i	0.384626i	0.981334	+	0.192313i	$0.0615989\pi$
$349$	7.18541i	0.384626i	−0.981334	+	0.192313i	$0.938401\pi$
$350$	0	0
$351$	2.44803	0.130666
$352$	0	0
$353$	5.97661	0.318103	0.159052	−	0.987270i	$-0.449156\pi$
$353$	5.97661	0.318103	0.159052	+	0.987270i	$0.449156\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.66148i	0.0879349i
$358$	0	0
$359$	−30.7769	−1.62434	−0.812171	−	0.583420i	$-0.801714\pi$
$359$	−30.7769	−1.62434	−0.812171	+	0.583420i	$0.801714\pi$
$360$	0	0
$361$	18.1802	0.956855
$362$	0	0
$363$	− 2.12196i	− 0.111374i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−22.3956	−1.16904	−0.584520	−	0.811379i	$-0.698717\pi$
$367$	−22.3956	−1.16904	−0.584520	+	0.811379i	$0.698717\pi$
$368$	0	0
$369$	17.2941	0.900296
$370$	0	0
$371$	26.4590i	1.37368i
$372$	0	0
$373$	16.1839i	0.837973i	0.907992	+	0.418986i	$0.137615\pi$
$373$	16.1839i	0.837973i	−0.907992	+	0.418986i	$0.862385\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15.3032	−0.788153
$378$	0	0
$379$	− 10.9692i	− 0.563449i	−0.959495	−	0.281724i	$-0.909094\pi$
$379$	− 10.9692i	− 0.563449i	0.959495	−	0.281724i	$-0.0909063\pi$
$380$	0	0
$381$	0.813190i	0.0416610i
$382$	0	0
$383$	−18.3816	−0.939256	−0.469628	−	0.882864i	$-0.655612\pi$
$383$	−18.3816	−0.939256	−0.469628	+	0.882864i	$0.655612\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.5032i	− 0.533909i
$388$	0	0
$389$	− 30.2480i	− 1.53364i	−0.641865	−	0.766818i	$-0.721839\pi$
$389$	− 30.2480i	− 1.53364i	0.641865	−	0.766818i	$-0.278161\pi$
$390$	0	0
$391$	−19.1051	−0.966186
$392$	0	0
$393$	3.49619	0.176360
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 39.1270i	− 1.96373i	−0.189582	−	0.981865i	$-0.560713\pi$
$397$	− 39.1270i	− 1.96373i	0.189582	−	0.981865i	$-0.439287\pi$
$398$	0	0
$399$	−0.512508	−0.0256575
$400$	0	0
$401$	−10.2607	−0.512397	−0.256198	−	0.966624i	$-0.582470\pi$
$401$	−10.2607	−0.512397	−0.256198	+	0.966624i	$0.582470\pi$
$402$	0	0
$403$	8.21966i	0.409451i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.380609	0.0188661
$408$	0	0
$409$	10.4383	0.516140	0.258070	−	0.966126i	$-0.416913\pi$
$409$	10.4383	0.516140	0.258070	+	0.966126i	$0.416913\pi$
$410$	0	0
$411$	1.41267i	0.0696818i
$412$	0	0
$413$	− 35.0211i	− 1.72327i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.78935	0.136595
$418$	0	0
$419$	− 36.6607i	− 1.79099i	−0.445067	−	0.895497i	$-0.646820\pi$
$419$	− 36.6607i	− 1.79099i	0.445067	−	0.895497i	$-0.353180\pi$
$420$	0	0
$421$	− 10.4070i	− 0.507207i	−0.967308	−	0.253603i	$-0.918384\pi$
$421$	− 10.4070i	− 0.507207i	0.967308	−	0.253603i	$-0.0816158\pi$
$422$	0	0
$423$	−13.8869	−0.675202
$424$	0	0
$425$	0	0
$426$	0	0
$427$	34.8047i	1.68432i
$428$	0	0
$429$	0.358089i	0.0172887i
$430$	0	0
$431$	10.3866	0.500306	0.250153	−	0.968206i	$-0.419519\pi$
$431$	10.3866	0.500306	0.250153	+	0.968206i	$0.419519\pi$
$432$	0	0
$433$	7.01402	0.337072	0.168536	−	0.985695i	$-0.446096\pi$
$433$	7.01402	0.337072	0.168536	+	0.985695i	$0.446096\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.89325i	− 0.281912i
$438$	0	0
$439$	3.40264	0.162399	0.0811995	−	0.996698i	$-0.474125\pi$
$439$	3.40264	0.162399	0.0811995	+	0.996698i	$0.474125\pi$
$440$	0	0
$441$	1.36843	0.0651633
$442$	0	0
$443$	14.1103i	0.670399i	0.942147	+	0.335199i	$0.108804\pi$
$443$	14.1103i	0.670399i	−0.942147	+	0.335199i	$0.891196\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4.06506	−0.192271
$448$	0	0
$449$	−18.9957	−0.896463	−0.448231	−	0.893918i	$-0.647946\pi$
$449$	−18.9957	−0.896463	−0.448231	+	0.893918i	$0.647946\pi$
$450$	0	0
$451$	5.09617i	0.239969i
$452$	0	0
$453$	− 4.53066i	− 0.212869i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.3050	1.32405	0.662027	−	0.749480i	$-0.269697\pi$
$457$	28.3050	1.32405	0.662027	+	0.749480i	$0.269697\pi$
$458$	0	0
$459$	3.62308i	0.169111i
$460$	0	0
$461$	− 11.2576i	− 0.524320i	−0.965024	−	0.262160i	$-0.915565\pi$
$461$	− 11.2576i	− 0.524320i	0.965024	−	0.262160i	$-0.0844348\pi$
$462$	0	0
$463$	1.11665	0.0518950	0.0259475	−	0.999663i	$-0.491740\pi$
$463$	1.11665	0.0518950	0.0259475	+	0.999663i	$0.491740\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.34496i	− 0.0622374i	−0.999516	−	0.0311187i	$-0.990093\pi$
$467$	− 1.34496i	− 0.0622374i	0.999516	−	0.0311187i	$-0.00990699\pi$
$468$	0	0
$469$	28.0297i	1.29429i
$470$	0	0
$471$	−1.10679	−0.0509983
$472$	0	0
$473$	3.09505	0.142311
$474$	0	0
$475$	0	0
$476$	0	0
$477$	28.6408i	1.31137i
$478$	0	0
$479$	−26.2181	−1.19794	−0.598968	−	0.800773i	$-0.704422\pi$
$479$	−26.2181	−1.19794	−0.598968	+	0.800773i	$0.704422\pi$
$480$	0	0
$481$	0.866263	0.0394982
$482$	0	0
$483$	− 3.68442i	− 0.167647i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.2782	−0.556378	−0.278189	−	0.960526i	$-0.589734\pi$
$487$	−12.2782	−0.556378	−0.278189	+	0.960526i	$0.589734\pi$
$488$	0	0
$489$	−3.61945	−0.163677
$490$	0	0
$491$	− 32.3053i	− 1.45792i	−0.684558	−	0.728959i	$-0.740005\pi$
$491$	− 32.3053i	− 1.45792i	0.684558	−	0.728959i	$-0.259995\pi$
$492$	0	0
$493$	− 22.6486i	− 1.02004i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	35.2907	1.58300
$498$	0	0
$499$	− 6.14059i	− 0.274890i	−0.990509	−	0.137445i	$-0.956111\pi$
$499$	− 6.14059i	− 0.274890i	0.990509	−	0.137445i	$-0.0438891\pi$
$500$	0	0
$501$	2.40270i	0.107345i
$502$	0	0
$503$	−26.1226	−1.16475	−0.582375	−	0.812921i	$-0.697876\pi$
$503$	−26.1226	−1.16475	−0.582375	+	0.812921i	$0.697876\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.87870i	− 0.0834362i
$508$	0	0
$509$	8.02196i	0.355567i	0.984070	+	0.177784i	$0.0568927\pi$
$509$	8.02196i	0.355567i	−0.984070	+	0.177784i	$0.943107\pi$
$510$	0	0
$511$	−26.6772	−1.18013
$512$	0	0
$513$	−1.11759	−0.0493429
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.09213i	− 0.179972i
$518$	0	0
$519$	1.88049	0.0825446
$520$	0	0
$521$	−37.3348	−1.63567	−0.817833	−	0.575456i	$-0.804825\pi$
$521$	−37.3348	−1.63567	−0.817833	+	0.575456i	$0.804825\pi$
$522$	0	0
$523$	41.8753i	1.83108i	0.402230	+	0.915539i	$0.368235\pi$
$523$	41.8753i	1.83108i	−0.402230	+	0.915539i	$0.631765\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.1651	−0.529919
$528$	0	0
$529$	19.3666	0.842025
$530$	0	0
$531$	− 37.9088i	− 1.64510i
$532$	0	0
$533$	11.5988i	0.502401i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.07698	0.0896283
$538$	0	0
$539$	0.403244i	0.0173690i
$540$	0	0
$541$	33.1542i	1.42541i	0.701464	+	0.712705i	$0.252530\pi$
$541$	33.1542i	1.42541i	−0.701464	+	0.712705i	$0.747470\pi$
$542$	0	0
$543$	2.96291	0.127151
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 21.2749i	− 0.909650i	−0.890581	−	0.454825i	$-0.849702\pi$
$547$	− 21.2749i	− 0.909650i	0.890581	−	0.454825i	$-0.150298\pi$
$548$	0	0
$549$	37.6746i	1.60791i
$550$	0	0
$551$	6.98631	0.297627
$552$	0	0
$553$	−8.25489	−0.351034
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.73333i	0.0734437i	0.999326	+	0.0367219i	$0.0116916\pi$
$557$	1.73333i	0.0734437i	−0.999326	+	0.0367219i	$0.988308\pi$
$558$	0	0
$559$	7.04431	0.297943
$560$	0	0
$561$	−0.529970	−0.0223754
$562$	0	0
$563$	− 47.0184i	− 1.98159i	−0.135369	−	0.990795i	$-0.543222\pi$
$563$	− 47.0184i	− 1.98159i	0.135369	−	0.990795i	$-0.456778\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	23.5357	0.988405
$568$	0	0
$569$	−0.301785	−0.0126515	−0.00632573	−	0.999980i	$-0.502014\pi$
$569$	−0.301785	−0.0126515	−0.00632573	+	0.999980i	$0.502014\pi$
$570$	0	0
$571$	30.2200i	1.26467i	0.774697	+	0.632333i	$0.217903\pi$
$571$	30.2200i	1.26467i	−0.774697	+	0.632333i	$0.782097\pi$
$572$	0	0
$573$	− 2.78518i	− 0.116353i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.58089	−0.190705	−0.0953524	−	0.995444i	$-0.530398\pi$
$577$	−4.58089	−0.190705	−0.0953524	+	0.995444i	$0.530398\pi$
$578$	0	0
$579$	− 4.14872i	− 0.172415i
$580$	0	0
$581$	− 26.9947i	− 1.11993i
$582$	0	0
$583$	−8.43976	−0.349539
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.387716i	0.0160028i	0.999968	+	0.00800138i	$0.00254695\pi$
$587$	0.387716i	0.0160028i	−0.999968	+	0.00800138i	$0.997453\pi$
$588$	0	0
$589$	− 3.75250i	− 0.154619i
$590$	0	0
$591$	−2.79654	−0.115034
$592$	0	0
$593$	14.5864	0.598994	0.299497	−	0.954097i	$-0.403181\pi$
$593$	14.5864	0.598994	0.299497	+	0.954097i	$0.403181\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.92298i	− 0.119630i
$598$	0	0
$599$	−5.66881	−0.231621	−0.115811	−	0.993271i	$-0.536947\pi$
$599$	−5.66881	−0.231621	−0.115811	+	0.993271i	$0.536947\pi$
$600$	0	0
$601$	−29.8385	−1.21714	−0.608569	−	0.793501i	$-0.708256\pi$
$601$	−29.8385	−1.21714	−0.608569	+	0.793501i	$0.708256\pi$
$602$	0	0
$603$	30.3410i	1.23558i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19.4752	−0.790474	−0.395237	−	0.918579i	$-0.629338\pi$
$607$	−19.4752	−0.790474	−0.395237	+	0.918579i	$0.629338\pi$
$608$	0	0
$609$	4.36779	0.176992
$610$	0	0
$611$	− 9.31365i	− 0.376790i
$612$	0	0
$613$	− 39.6234i	− 1.60037i	−0.599752	−	0.800186i	$-0.704734\pi$
$613$	− 39.6234i	− 1.60037i	0.599752	−	0.800186i	$-0.295266\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.0519	−1.29036	−0.645180	−	0.764031i	$-0.723218\pi$
$617$	−32.0519	−1.29036	−0.645180	+	0.764031i	$0.723218\pi$
$618$	0	0
$619$	− 17.9597i	− 0.721861i	−0.932593	−	0.360930i	$-0.882459\pi$
$619$	− 17.9597i	− 0.721861i	0.932593	−	0.360930i	$-0.117541\pi$
$620$	0	0
$621$	− 8.03437i	− 0.322408i
$622$	0	0
$623$	−21.9805	−0.880632
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 0.163477i	− 0.00652865i
$628$	0	0
$629$	1.28207i	0.0511193i
$630$	0	0
$631$	−28.0265	−1.11572	−0.557858	−	0.829936i	$-0.688377\pi$
$631$	−28.0265	−1.11572	−0.557858	+	0.829936i	$0.688377\pi$
$632$	0	0
$633$	−2.72180	−0.108182
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.917780i	0.0363638i
$638$	0	0
$639$	38.2007	1.51120
$640$	0	0
$641$	−11.2647	−0.444929	−0.222465	−	0.974941i	$-0.571410\pi$
$641$	−11.2647	−0.444929	−0.222465	+	0.974941i	$0.571410\pi$
$642$	0	0
$643$	− 48.2471i	− 1.90268i	−0.308141	−	0.951341i	$-0.599707\pi$
$643$	− 48.2471i	− 1.90268i	0.308141	−	0.951341i	$-0.400293\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.8798	1.25333	0.626663	−	0.779291i	$-0.284420\pi$
$647$	31.8798	1.25333	0.626663	+	0.779291i	$0.284420\pi$
$648$	0	0
$649$	11.1708	0.438494
$650$	0	0
$651$	− 2.34604i	− 0.0919484i
$652$	0	0
$653$	36.3662i	1.42312i	0.702627	+	0.711559i	$0.252010\pi$
$653$	36.3662i	1.42312i	−0.702627	+	0.711559i	$0.747990\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−28.8769	−1.12660
$658$	0	0
$659$	− 5.48884i	− 0.213815i	−0.994269	−	0.106907i	$-0.965905\pi$
$659$	− 5.48884i	− 0.213815i	0.994269	−	0.106907i	$-0.0340949\pi$
$660$	0	0
$661$	− 29.5221i	− 1.14828i	−0.818759	−	0.574138i	$-0.805337\pi$
$661$	− 29.5221i	− 1.14828i	0.818759	−	0.574138i	$-0.194663\pi$
$662$	0	0
$663$	−1.20621	−0.0468452
$664$	0	0
$665$	0	0
$666$	0	0
$667$	50.2246i	1.94470i
$668$	0	0
$669$	0.148952i	0.00575883i
$670$	0	0
$671$	−11.1018	−0.428581
$672$	0	0
$673$	−41.5796	−1.60278	−0.801389	−	0.598144i	$-0.795905\pi$
$673$	−41.5796	−1.60278	−0.801389	+	0.598144i	$0.795905\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.3724i	1.35947i	0.733457	+	0.679736i	$0.237906\pi$
$677$	35.3724i	1.35947i	−0.733457	+	0.679736i	$0.762094\pi$
$678$	0	0
$679$	33.3108	1.27835
$680$	0	0
$681$	2.50127	0.0958488
$682$	0	0
$683$	22.5077i	0.861233i	0.902535	+	0.430617i	$0.141704\pi$
$683$	22.5077i	0.861233i	−0.902535	+	0.430617i	$0.858296\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.14271	−0.0817496
$688$	0	0
$689$	−19.2088	−0.731798
$690$	0	0
$691$	− 0.855355i	− 0.0325392i	−0.999868	−	0.0162696i	$-0.994821\pi$
$691$	− 0.855355i	− 0.0325392i	0.999868	−	0.0162696i	$-0.00517901\pi$
$692$	0	0
$693$	7.03910i	0.267393i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−17.1662	−0.650218
$698$	0	0
$699$	− 0.835664i	− 0.0316077i
$700$	0	0
$701$	− 16.7658i	− 0.633236i	−0.948553	−	0.316618i	$-0.897453\pi$
$701$	− 16.7658i	− 0.633236i	0.948553	−	0.316618i	$-0.102547\pi$
$702$	0	0
$703$	−0.395472	−0.0149155
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.0641i	− 0.566542i
$708$	0	0
$709$	21.8389i	0.820177i	0.912046	+	0.410088i	$0.134502\pi$
$709$	21.8389i	0.820177i	−0.912046	+	0.410088i	$0.865498\pi$
$710$	0	0
$711$	−8.93557	−0.335110
$712$	0	0
$713$	26.9767	1.01029
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.42988i	− 0.0907456i
$718$	0	0
$719$	−28.2838	−1.05481	−0.527403	−	0.849615i	$-0.676834\pi$
$719$	−28.2838	−1.05481	−0.527403	+	0.849615i	$0.676834\pi$
$720$	0	0
$721$	−13.6203	−0.507245
$722$	0	0
$723$	− 2.64599i	− 0.0984055i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18.7205	−0.694305	−0.347152	−	0.937809i	$-0.612851\pi$
$727$	−18.7205	−0.694305	−0.347152	+	0.937809i	$0.612851\pi$
$728$	0	0
$729$	24.7091	0.915150
$730$	0	0
$731$	10.4256i	0.385603i
$732$	0	0
$733$	3.61353i	0.133469i	0.997771	+	0.0667344i	$0.0212580\pi$
$733$	3.61353i	0.133469i	−0.997771	+	0.0667344i	$0.978742\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.94077	−0.329338
$738$	0	0
$739$	31.3284i	1.15243i	0.817296	+	0.576217i	$0.195472\pi$
$739$	31.3284i	1.15243i	−0.817296	+	0.576217i	$0.804528\pi$
$740$	0	0
$741$	− 0.372073i	− 0.0136684i
$742$	0	0
$743$	44.8723	1.64621	0.823103	−	0.567892i	$-0.192241\pi$
$743$	44.8723	1.64621	0.823103	+	0.567892i	$0.192241\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 29.2206i	− 1.06913i
$748$	0	0
$749$	− 33.9936i	− 1.24210i
$750$	0	0
$751$	−29.7771	−1.08658	−0.543292	−	0.839544i	$-0.682822\pi$
$751$	−29.7771	−1.08658	−0.543292	+	0.839544i	$0.682822\pi$
$752$	0	0
$753$	4.72489	0.172185
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 33.1452i	− 1.20468i	−0.798239	−	0.602341i	$-0.794235\pi$
$757$	− 33.1452i	− 1.20468i	0.798239	−	0.602341i	$-0.205765\pi$
$758$	0	0
$759$	1.17524	0.0426584
$760$	0	0
$761$	−14.1986	−0.514699	−0.257350	−	0.966318i	$-0.582849\pi$
$761$	−14.1986	−0.514699	−0.257350	+	0.966318i	$0.582849\pi$
$762$	0	0
$763$	12.6767i	0.458929i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.4247	0.918034
$768$	0	0
$769$	5.12825	0.184930	0.0924648	−	0.995716i	$-0.470525\pi$
$769$	5.12825	0.184930	0.0924648	+	0.995716i	$0.470525\pi$
$770$	0	0
$771$	4.70021i	0.169274i
$772$	0	0
$773$	1.17666i	0.0423216i	0.999776	+	0.0211608i	$0.00673619\pi$
$773$	1.17666i	0.0423216i	−0.999776	+	0.0211608i	$0.993264\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.247247	−0.00886992
$778$	0	0
$779$	− 5.29518i	− 0.189720i
$780$	0	0
$781$	11.2568i	0.402801i
$782$	0	0
$783$	9.52455	0.340380
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.644823i	0.0229854i	0.999934	+	0.0114927i	$0.00365833\pi$
$787$	0.644823i	0.0229854i	−0.999934	+	0.0114927i	$0.996342\pi$
$788$	0	0
$789$	− 3.10949i	− 0.110701i
$790$	0	0
$791$	−46.6189	−1.65758
$792$	0	0
$793$	−25.2677	−0.897281
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 0.822745i	− 0.0291431i	−0.999894	−	0.0145716i	$-0.995362\pi$
$797$	− 0.822745i	− 0.0291431i	0.999894	−	0.0145716i	$-0.00463844\pi$
$798$	0	0
$799$	13.7842	0.487649
$800$	0	0
$801$	−23.7930	−0.840684
$802$	0	0
$803$	− 8.50934i	− 0.300288i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.18978	0.112285
$808$	0	0
$809$	−11.5085	−0.404619	−0.202309	−	0.979322i	$-0.564845\pi$
$809$	−11.5085	−0.404619	−0.202309	+	0.979322i	$0.564845\pi$
$810$	0	0
$811$	− 51.7936i	− 1.81872i	−0.416011	−	0.909360i	$-0.636572\pi$
$811$	− 51.7936i	− 1.81872i	0.416011	−	0.909360i	$-0.363428\pi$
$812$	0	0
$813$	− 3.14202i	− 0.110195i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.21592	−0.112511
$818$	0	0
$819$	16.0209i	0.559816i
$820$	0	0
$821$	− 1.66017i	− 0.0579403i	−0.999580	−	0.0289702i	$-0.990777\pi$
$821$	− 1.66017i	− 0.0579403i	0.999580	−	0.0289702i	$-0.00922278\pi$
$822$	0	0
$823$	33.9444	1.18323	0.591613	−	0.806222i	$-0.298491\pi$
$823$	33.9444	1.18323	0.591613	+	0.806222i	$0.298491\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.2510i	1.22580i	0.790162	+	0.612898i	$0.209996\pi$
$827$	35.2510i	1.22580i	−0.790162	+	0.612898i	$0.790004\pi$
$828$	0	0
$829$	− 36.8215i	− 1.27886i	−0.768847	−	0.639432i	$-0.779169\pi$
$829$	− 36.8215i	− 1.27886i	0.768847	−	0.639432i	$-0.220831\pi$
$830$	0	0
$831$	2.03252	0.0705073
$832$	0	0
$833$	−1.35831	−0.0470627
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.11584i	− 0.176829i
$838$	0	0
$839$	−40.2678	−1.39020	−0.695100	−	0.718913i	$-0.744640\pi$
$839$	−40.2678	−1.39020	−0.695100	+	0.718913i	$0.744640\pi$
$840$	0	0
$841$	−30.5401	−1.05311
$842$	0	0
$843$	− 0.708570i	− 0.0244044i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	27.9756	0.961253
$848$	0	0
$849$	5.08627	0.174560
$850$	0	0
$851$	− 2.84305i	− 0.0974585i
$852$	0	0
$853$	52.0753i	1.78302i	0.452997	+	0.891512i	$0.350355\pi$
$853$	52.0753i	1.78302i	−0.452997	+	0.891512i	$0.649645\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.7275	1.66450	0.832250	−	0.554400i	$-0.187052\pi$
$857$	48.7275	1.66450	0.832250	+	0.554400i	$0.187052\pi$
$858$	0	0
$859$	34.8415i	1.18878i	0.804178	+	0.594389i	$0.202606\pi$
$859$	34.8415i	1.18878i	−0.804178	+	0.594389i	$0.797394\pi$
$860$	0	0
$861$	− 3.31051i	− 0.112822i
$862$	0	0
$863$	14.8198	0.504473	0.252237	−	0.967666i	$-0.418834\pi$
$863$	14.8198	0.504473	0.252237	+	0.967666i	$0.418834\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.73736i	0.0590039i
$868$	0	0
$869$	− 2.63310i	− 0.0893218i
$870$	0	0
$871$	−20.3491	−0.689504
$872$	0	0
$873$	36.0575	1.22036
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5.24283i	− 0.177038i	−0.996074	−	0.0885189i	$-0.971787\pi$
$877$	− 5.24283i	− 0.177038i	0.996074	−	0.0885189i	$-0.0282134\pi$
$878$	0	0
$879$	−4.75590	−0.160413
$880$	0	0
$881$	−24.2313	−0.816374	−0.408187	−	0.912898i	$-0.633839\pi$
$881$	−24.2313	−0.816374	−0.408187	+	0.912898i	$0.633839\pi$
$882$	0	0
$883$	− 40.7796i	− 1.37234i	−0.727439	−	0.686172i	$-0.759290\pi$
$883$	− 40.7796i	− 1.37234i	0.727439	−	0.686172i	$-0.240710\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.68906	0.157443	0.0787216	−	0.996897i	$-0.474916\pi$
$887$	4.68906	0.157443	0.0787216	+	0.996897i	$0.474916\pi$
$888$	0	0
$889$	−10.7210	−0.359570
$890$	0	0
$891$	7.50729i	0.251504i
$892$	0	0
$893$	4.25194i	0.142286i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.67483	0.0893100
$898$	0	0
$899$	31.9802i	1.06660i
$900$	0	0
$901$	− 28.4290i	− 0.947108i
$902$	0	0
$903$	−2.01057	−0.0669076
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.54602i	− 0.117744i	−0.998266	−	0.0588718i	$-0.981250\pi$
$907$	− 3.54602i	− 0.117744i	0.998266	−	0.0588718i	$-0.0187503\pi$
$908$	0	0
$909$	− 16.3062i	− 0.540843i
$910$	0	0
$911$	−24.9437	−0.826420	−0.413210	−	0.910636i	$-0.635593\pi$
$911$	−24.9437	−0.826420	−0.413210	+	0.910636i	$0.635593\pi$
$912$	0	0
$913$	8.61063	0.284970
$914$	0	0
$915$	0	0
$916$	0	0
$917$	46.0933i	1.52213i
$918$	0	0
$919$	4.92584	0.162488	0.0812442	−	0.996694i	$-0.474111\pi$
$919$	4.92584	0.162488	0.0812442	+	0.996694i	$0.474111\pi$
$920$	0	0
$921$	0.479030	0.0157846
$922$	0	0
$923$	25.6205i	0.843308i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−14.7434	−0.484235
$928$	0	0
$929$	−39.6646	−1.30135	−0.650677	−	0.759354i	$-0.725515\pi$
$929$	−39.6646	−1.30135	−0.650677	+	0.759354i	$0.725515\pi$
$930$	0	0
$931$	− 0.418991i	− 0.0137319i
$932$	0	0
$933$	0.0825014i	0.00270097i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.3308	1.02353	0.511767	−	0.859125i	$-0.328991\pi$
$937$	31.3308	1.02353	0.511767	+	0.859125i	$0.328991\pi$
$938$	0	0
$939$	− 5.75824i	− 0.187913i
$940$	0	0
$941$	− 13.1179i	− 0.427630i	−0.976874	−	0.213815i	$-0.931411\pi$
$941$	− 13.1179i	− 0.427630i	0.976874	−	0.213815i	$-0.0685889\pi$
$942$	0	0
$943$	38.0671	1.23963
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.0477i	− 1.04141i	−0.853737	−	0.520705i	$-0.825669\pi$
$947$	− 32.0477i	− 1.04141i	0.853737	−	0.520705i	$-0.174331\pi$
$948$	0	0
$949$	− 19.3672i	− 0.628686i
$950$	0	0
$951$	3.46441	0.112341
$952$	0	0
$953$	−33.0115	−1.06935	−0.534674	−	0.845059i	$-0.679565\pi$
$953$	−33.0115	−1.06935	−0.534674	+	0.845059i	$0.679565\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.39322i	0.0450363i
$958$	0	0
$959$	−18.6244	−0.601413
$960$	0	0
$961$	−13.8227	−0.445894
$962$	0	0
$963$	− 36.7967i	− 1.18576i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−58.2596	−1.87350	−0.936751	−	0.349995i	$-0.886183\pi$
$967$	−58.2596	−1.87350	−0.936751	+	0.349995i	$0.886183\pi$
$968$	0	0
$969$	0.550666	0.0176900
$970$	0	0
$971$	− 25.2784i	− 0.811223i	−0.914046	−	0.405612i	$-0.867059\pi$
$971$	− 25.2784i	− 0.811223i	0.914046	−	0.405612i	$-0.132941\pi$
$972$	0	0
$973$	36.7744i	1.17893i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.7917	−0.569207	−0.284604	−	0.958645i	$-0.591862\pi$
$977$	−17.7917	−0.569207	−0.284604	+	0.958645i	$0.591862\pi$
$978$	0	0
$979$	− 7.01124i	− 0.224080i
$980$	0	0
$981$	13.7220i	0.438111i
$982$	0	0
$983$	33.3339	1.06319	0.531594	−	0.846999i	$-0.321593\pi$
$983$	33.3339	1.06319	0.531594	+	0.846999i	$0.321593\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.65828i	0.0846140i
$988$	0	0
$989$	− 23.1192i	− 0.735149i
$990$	0	0
$991$	21.9753	0.698069	0.349034	−	0.937110i	$-0.386510\pi$
$991$	21.9753	0.698069	0.349034	+	0.937110i	$0.386510\pi$
$992$	0	0
$993$	−2.64896	−0.0840622
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 42.6450i	− 1.35058i	−0.737553	−	0.675290i	$-0.764019\pi$
$997$	− 42.6450i	− 1.35058i	0.737553	−	0.675290i	$-0.235981\pi$
$998$	0	0
$999$	−0.539154	−0.0170581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.d.c.2001.11		40
4.3	odd	2		1000.2.d.c.501.15	✓	40
5.2	odd	4		4000.2.f.c.3249.12		20
5.3	odd	4		4000.2.f.d.3249.9		20
5.4	even	2	inner	4000.2.d.c.2001.30		40
8.3	odd	2		1000.2.d.c.501.16	yes	40
8.5	even	2	inner	4000.2.d.c.2001.12		40
20.3	even	4		1000.2.f.d.749.4		20
20.7	even	4		1000.2.f.c.749.17		20
20.19	odd	2		1000.2.d.c.501.26	yes	40
40.3	even	4		1000.2.f.c.749.18		20
40.13	odd	4		4000.2.f.c.3249.11		20
40.19	odd	2		1000.2.d.c.501.25	yes	40
40.27	even	4		1000.2.f.d.749.3		20
40.29	even	2	inner	4000.2.d.c.2001.29		40
40.37	odd	4		4000.2.f.d.3249.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.d.c.501.15	✓	40	4.3	odd	2
1000.2.d.c.501.16	yes	40	8.3	odd	2
1000.2.d.c.501.25	yes	40	40.19	odd	2
1000.2.d.c.501.26	yes	40	20.19	odd	2
1000.2.f.c.749.17		20	20.7	even	4
1000.2.f.c.749.18		20	40.3	even	4
1000.2.f.d.749.3		20	40.27	even	4
1000.2.f.d.749.4		20	20.3	even	4
4000.2.d.c.2001.11		40	1.1	even	1	trivial
4000.2.d.c.2001.12		40	8.5	even	2	inner
4000.2.d.c.2001.29		40	40.29	even	2	inner
4000.2.d.c.2001.30		40	5.4	even	2	inner
4000.2.f.c.3249.11		20	40.13	odd	4
4000.2.f.c.3249.12		20	5.2	odd	4
4000.2.f.d.3249.9		20	5.3	odd	4
4000.2.f.d.3249.10		20	40.37	odd	4

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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.207209i q^{3} +2.73181 q^{7} +2.95706 q^{9} +O(q^{10})\)	\(q-0.207209i q^{3} +2.73181 q^{7} +2.95706 q^{9} +0.871377i q^{11} +1.98325i q^{13} -2.93520 q^{17} -0.905406i q^{19} -0.566054i q^{21} +6.50896 q^{23} -1.23435i q^{27} +7.71622i q^{29} +4.14455 q^{31} +0.180557 q^{33} -0.436790i q^{37} +0.410946 q^{39} +5.84841 q^{41} -3.55191i q^{43} -4.69617 q^{47} +0.462766 q^{49} +0.608198i q^{51} +9.68555i q^{53} -0.187608 q^{57} -12.8197i q^{59} +12.7406i q^{61} +8.07813 q^{63} +10.2605i q^{67} -1.34871i q^{69} +12.9184 q^{71} -9.76540 q^{73} +2.38043i q^{77} -3.02177 q^{79} +8.61542 q^{81} -9.88164i q^{83} +1.59887 q^{87} -8.04615 q^{89} +5.41785i q^{91} -0.858786i q^{93} +12.1937 q^{97} +2.57672i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{9}+O(q^{10}) \) 40 * q - 24 * q^9	\( 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) \) 40 * q - 24 * q^9 - 48 * q^31 - 8 * q^39 + 44 * q^41 + 12 * q^49 - 96 * q^71 - 96 * q^79 - 56 * q^81 - 44 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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