LMFDB - Embedded newform 2070.3.c.a.91.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,3,Mod(91,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2070.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2070.c (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:	$56.4034147226$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.3
Root			$3.68124i$ of defining polynomial
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.a.91.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +1.16919i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +1.16919i q^{7} -2.82843 q^{8} +3.16228i q^{10} -10.6148i q^{11} -15.0913 q^{13} -1.65348i q^{14} +4.00000 q^{16} -20.0887i q^{17} +22.5221i q^{19} -4.47214i q^{20} +15.0116i q^{22} +(20.9683 - 9.45142i) q^{23} -5.00000 q^{25} +21.3424 q^{26} +2.33837i q^{28} -32.5993 q^{29} -27.0975 q^{31} -5.65685 q^{32} +28.4097i q^{34} +2.61438 q^{35} -53.0568i q^{37} -31.8510i q^{38} +6.32456i q^{40} -9.43720 q^{41} +36.4382i q^{43} -21.2296i q^{44} +(-29.6537 + 13.3663i) q^{46} +49.1365 q^{47} +47.6330 q^{49} +7.07107 q^{50} -30.1827 q^{52} +104.253i q^{53} -23.7354 q^{55} -3.30696i q^{56} +46.1023 q^{58} -53.5457 q^{59} -23.5166i q^{61} +38.3217 q^{62} +8.00000 q^{64} +33.7453i q^{65} +59.4754i q^{67} -40.1773i q^{68} -3.69729 q^{70} -55.2130 q^{71} -8.77305 q^{73} +75.0337i q^{74} +45.0441i q^{76} +12.4107 q^{77} +57.0848i q^{79} -8.94427i q^{80} +13.3462 q^{82} +55.1788i q^{83} -44.9196 q^{85} -51.5314i q^{86} +30.0232i q^{88} -139.825i q^{89} -17.6446i q^{91} +(41.9366 - 18.9028i) q^{92} -69.4894 q^{94} +50.3608 q^{95} -19.8635i q^{97} -67.3632 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) $ 16 * q + 32 * q^4 + 24 * q^13 + 64 * q^16 - 4 * q^23 - 80 * q^25 - 96 * q^26 + 108 * q^29 - 116 * q^31 - 60 * q^35 + 156 * q^41 - 124 * q^46 + 128 * q^47 - 28 * q^49 + 48 * q^52 + 160 * q^58 - 204 * q^59 - 64 * q^62 + 128 * q^64 - 120 * q^70 - 236 * q^71 - 112 * q^73 + 936 * q^77 - 64 * q^82 + 60 * q^85 - 8 * q^92 - 216 * q^94 + 160 * q^95 - 256 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			1.16919i			0.167026i	0.996507	+	0.0835132i	$0.0266141\pi$
$7$			1.16919i			0.167026i	−0.996507	+	0.0835132i	$0.973386\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$			3.16228i			0.316228i
$11$		−	10.6148i		−	0.964981i	−0.875901	−	0.482490i	$-0.839732\pi$
$11$		−	10.6148i		−	0.964981i	0.875901	−	0.482490i	$-0.160268\pi$
$12$		0			0
$13$	−15.0913			−1.16087			−0.580436	−	0.814306i	$-0.697118\pi$
$13$	−15.0913			−1.16087			−0.580436	+	0.814306i	$0.697118\pi$
$14$		−	1.65348i		−	0.118106i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	20.0887i		−	1.18169i	−0.806787	−	0.590843i	$-0.798795\pi$
$17$		−	20.0887i		−	1.18169i	0.806787	−	0.590843i	$-0.201205\pi$
$18$		0			0
$19$			22.5221i			1.18537i	0.805434	+	0.592686i	$0.201932\pi$
$19$			22.5221i			1.18537i	−0.805434	+	0.592686i	$0.798068\pi$
$20$		−	4.47214i		−	0.223607i
$21$		0			0
$22$			15.0116i			0.682344i
$23$	20.9683	−	9.45142i	0.911666	−	0.410931i
$23$	20.9683	−	9.45142i	0.911666	−	0.410931i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	21.3424			0.820860
$27$		0			0
$28$			2.33837i			0.0835132i
$29$	−32.5993			−1.12411			−0.562056	−	0.827099i	$-0.689990\pi$
$29$	−32.5993			−1.12411			−0.562056	+	0.827099i	$0.689990\pi$
$30$		0			0
$31$	−27.0975			−0.874113			−0.437056	−	0.899434i	$-0.643979\pi$
$31$	−27.0975			−0.874113			−0.437056	+	0.899434i	$0.643979\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$			28.4097i			0.835578i
$35$	2.61438			0.0746965
$36$		0			0
$37$		−	53.0568i		−	1.43397i	−0.697089	−	0.716984i	$-0.745522\pi$
$37$		−	53.0568i		−	1.43397i	0.697089	−	0.716984i	$-0.254478\pi$
$38$		−	31.8510i		−	0.838184i
$39$		0			0
$40$			6.32456i			0.158114i
$41$	−9.43720			−0.230176			−0.115088	−	0.993355i	$-0.536715\pi$
$41$	−9.43720			−0.230176			−0.115088	+	0.993355i	$0.536715\pi$
$42$		0			0
$43$			36.4382i			0.847400i	0.905802	+	0.423700i	$0.139269\pi$
$43$			36.4382i			0.847400i	−0.905802	+	0.423700i	$0.860731\pi$
$44$		−	21.2296i		−	0.482490i
$45$		0			0
$46$	−29.6537	+	13.3663i	−0.644645	+	0.290572i
$47$	49.1365			1.04546			0.522728	−	0.852499i	$-0.324914\pi$
$47$	49.1365			1.04546			0.522728	+	0.852499i	$0.324914\pi$
$48$		0			0
$49$	47.6330			0.972102
$50$	7.07107			0.141421
$51$		0			0
$52$	−30.1827			−0.580436
$53$			104.253i			1.96703i	0.180815	+	0.983517i	$0.442126\pi$
$53$			104.253i			1.96703i	−0.180815	+	0.983517i	$0.557874\pi$
$54$		0			0
$55$	−23.7354			−0.431552
$56$		−	3.30696i		−	0.0590528i
$57$		0			0
$58$	46.1023			0.794868
$59$	−53.5457			−0.907554			−0.453777	−	0.891115i	$-0.649924\pi$
$59$	−53.5457			−0.907554			−0.453777	+	0.891115i	$0.649924\pi$
$60$		0			0
$61$		−	23.5166i		−	0.385518i	−0.981246	−	0.192759i	$-0.938257\pi$
$61$		−	23.5166i		−	0.385518i	0.981246	−	0.192759i	$-0.0617435\pi$
$62$	38.3217			0.618091
$63$		0			0
$64$	8.00000			0.125000
$65$			33.7453i			0.519158i
$66$		0			0
$67$			59.4754i			0.887692i	0.896103	+	0.443846i	$0.146386\pi$
$67$			59.4754i			0.887692i	−0.896103	+	0.443846i	$0.853614\pi$
$68$		−	40.1773i		−	0.590843i
$69$		0			0
$70$	−3.69729			−0.0528184
$71$	−55.2130			−0.777648			−0.388824	−	0.921312i	$-0.627119\pi$
$71$	−55.2130			−0.777648			−0.388824	+	0.921312i	$0.627119\pi$
$72$		0			0
$73$	−8.77305			−0.120179			−0.0600894	−	0.998193i	$-0.519139\pi$
$73$	−8.77305			−0.120179			−0.0600894	+	0.998193i	$0.519139\pi$
$74$			75.0337i			1.01397i
$75$		0			0
$76$			45.0441i			0.592686i
$77$	12.4107			0.161177
$78$		0			0
$79$			57.0848i			0.722592i	0.932451	+	0.361296i	$0.117666\pi$
$79$			57.0848i			0.722592i	−0.932451	+	0.361296i	$0.882334\pi$
$80$		−	8.94427i		−	0.111803i
$81$		0			0
$82$	13.3462			0.162759
$83$			55.1788i			0.664805i	0.943138	+	0.332403i	$0.107859\pi$
$83$			55.1788i			0.664805i	−0.943138	+	0.332403i	$0.892141\pi$
$84$		0			0
$85$	−44.9196			−0.528466
$86$		−	51.5314i		−	0.599203i
$87$		0			0
$88$			30.0232i			0.341172i
$89$		−	139.825i		−	1.57107i	−0.618815	−	0.785536i	$-0.712387\pi$
$89$		−	139.825i		−	1.57107i	0.618815	−	0.785536i	$-0.287613\pi$
$90$		0			0
$91$		−	17.6446i		−	0.193896i
$92$	41.9366	−	18.9028i	0.455833	−	0.205466i
$93$		0			0
$94$	−69.4894			−0.739249
$95$	50.3608			0.530114
$96$		0			0
$97$		−	19.8635i		−	0.204778i	−0.994744	−	0.102389i	$-0.967351\pi$
$97$		−	19.8635i		−	0.204778i	0.994744	−	0.102389i	$-0.0326487\pi$
$98$	−67.3632			−0.687380
$99$		0			0
$100$	−10.0000			−0.100000
$101$	−86.5639			−0.857068			−0.428534	−	0.903526i	$-0.640970\pi$
$101$	−86.5639			−0.857068			−0.428534	+	0.903526i	$0.640970\pi$
$102$		0			0
$103$			144.118i			1.39920i	0.714535	+	0.699600i	$0.246639\pi$
$103$			144.118i			1.39920i	−0.714535	+	0.699600i	$0.753361\pi$
$104$	42.6847			0.410430
$105$		0			0
$106$		−	147.436i		−	1.39090i
$107$			10.4544i			0.0977050i	0.998806	+	0.0488525i	$0.0155564\pi$
$107$			10.4544i			0.0977050i	−0.998806	+	0.0488525i	$0.984444\pi$
$108$		0			0
$109$		−	69.1221i		−	0.634148i	−0.948401	−	0.317074i	$-0.897300\pi$
$109$		−	69.1221i		−	0.634148i	0.948401	−	0.317074i	$-0.102700\pi$
$110$	33.5669			0.305154
$111$		0			0
$112$			4.67674i			0.0417566i
$113$		−	137.264i		−	1.21473i	−0.794423	−	0.607364i	$-0.792227\pi$
$113$		−	137.264i		−	1.21473i	0.794423	−	0.607364i	$-0.207773\pi$
$114$		0			0
$115$	−21.1340	−	46.8866i	−0.183774	−	0.407710i
$116$	−65.1985			−0.562056
$117$		0			0
$118$	75.7250			0.641738
$119$	23.4874			0.197373
$120$		0			0
$121$	8.32629			0.0688123
$122$			33.2575i			0.272602i
$123$		0			0
$124$	−54.1950			−0.437056
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	57.9697			0.456455			0.228227	−	0.973608i	$-0.426707\pi$
$127$	57.9697			0.456455			0.228227	+	0.973608i	$0.426707\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$		−	47.7230i		−	0.367100i
$131$	95.0810			0.725809			0.362905	−	0.931826i	$-0.381785\pi$
$131$	95.0810			0.725809			0.362905	+	0.931826i	$0.381785\pi$
$132$		0			0
$133$	−26.3325			−0.197988
$134$		−	84.1109i		−	0.627693i
$135$		0			0
$136$			56.8193i			0.417789i
$137$			198.711i			1.45045i	0.688514	+	0.725223i	$0.258263\pi$
$137$			198.711i			1.45045i	−0.688514	+	0.725223i	$0.741737\pi$
$138$		0			0
$139$	−0.129480			−0.000931508			−0.000465754	−	1.00000i	$-0.500148\pi$
$139$	−0.129480			−0.000931508			−0.000465754		1.00000i	$0.500148\pi$
$140$	5.22876			0.0373483
$141$		0			0
$142$	78.0830			0.549880
$143$			160.191i			1.12022i
$144$		0			0
$145$			72.8942i			0.502718i
$146$	12.4070			0.0849792
$147$		0			0
$148$		−	106.114i		−	0.716984i
$149$			63.8745i			0.428688i	0.976758	+	0.214344i	$0.0687613\pi$
$149$			63.8745i			0.428688i	−0.976758	+	0.214344i	$0.931239\pi$
$150$		0			0
$151$	−35.2673			−0.233558			−0.116779	−	0.993158i	$-0.537257\pi$
$151$	−35.2673			−0.233558			−0.116779	+	0.993158i	$0.537257\pi$
$152$		−	63.7020i		−	0.419092i
$153$		0			0
$154$	−17.5513			−0.113970
$155$			60.5919i			0.390915i
$156$		0			0
$157$			289.136i			1.84163i	0.390002	+	0.920814i	$0.372474\pi$
$157$			289.136i			1.84163i	−0.390002	+	0.920814i	$0.627526\pi$
$158$		−	80.7300i		−	0.510950i
$159$		0			0
$160$			12.6491i			0.0790569i
$161$	11.0505	+	24.5159i	0.0686364	+	0.152272i
$162$		0			0
$163$	−46.2536			−0.283764			−0.141882	−	0.989884i	$-0.545315\pi$
$163$	−46.2536			−0.283764			−0.141882	+	0.989884i	$0.545315\pi$
$164$	−18.8744			−0.115088
$165$		0			0
$166$		−	78.0346i		−	0.470088i
$167$	90.8735			0.544153			0.272076	−	0.962276i	$-0.412290\pi$
$167$	90.8735			0.544153			0.272076	+	0.962276i	$0.412290\pi$
$168$		0			0
$169$	58.7484			0.347624
$170$	63.5259			0.373682
$171$		0			0
$172$			72.8764i			0.423700i
$173$	90.7893			0.524794			0.262397	−	0.964960i	$-0.415487\pi$
$173$	90.7893			0.524794			0.262397	+	0.964960i	$0.415487\pi$
$174$		0			0
$175$		−	5.84593i		−	0.0334053i
$176$		−	42.4591i		−	0.241245i
$177$		0			0
$178$			197.743i			1.11092i
$179$	−301.636			−1.68511			−0.842557	−	0.538607i	$-0.818951\pi$
$179$	−301.636			−1.68511			−0.842557	+	0.538607i	$0.818951\pi$
$180$		0			0
$181$			248.547i			1.37319i	0.727040	+	0.686595i	$0.240895\pi$
$181$			248.547i			1.37319i	−0.727040	+	0.686595i	$0.759105\pi$
$182$			24.9532i			0.137105i
$183$		0			0
$184$	−59.3074	+	26.7327i	−0.322323	+	0.145286i
$185$	−118.639			−0.641290
$186$		0			0
$187$	−213.237			−1.14030
$188$	98.2729			0.522728
$189$		0			0
$190$	−71.2210			−0.374847
$191$		−	146.382i		−	0.766397i	−0.923666	−	0.383199i	$-0.874822\pi$
$191$		−	146.382i		−	0.766397i	0.923666	−	0.383199i	$-0.125178\pi$
$192$		0			0
$193$	−120.648			−0.625120			−0.312560	−	0.949898i	$-0.601187\pi$
$193$	−120.648			−0.625120			−0.312560	+	0.949898i	$0.601187\pi$
$194$			28.0912i			0.144800i
$195$		0			0
$196$	95.2660			0.486051
$197$	−31.6502			−0.160661			−0.0803305	−	0.996768i	$-0.525598\pi$
$197$	−31.6502			−0.160661			−0.0803305	+	0.996768i	$0.525598\pi$
$198$		0			0
$199$			37.4496i			0.188189i	0.995563	+	0.0940944i	$0.0299955\pi$
$199$			37.4496i			0.188189i	−0.995563	+	0.0940944i	$0.970004\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	122.420			0.606039
$203$		−	38.1146i		−	0.187757i
$204$		0			0
$205$			21.1022i			0.102938i
$206$		−	203.813i		−	0.989384i
$207$		0			0
$208$	−60.3653			−0.290218
$209$	239.067			1.14386
$210$		0			0
$211$	−43.6572			−0.206906			−0.103453	−	0.994634i	$-0.532989\pi$
$211$	−43.6572			−0.206906			−0.103453	+	0.994634i	$0.532989\pi$
$212$			208.506i			0.983517i
$213$		0			0
$214$		−	14.7848i		−	0.0690879i
$215$	81.4783			0.378969
$216$		0			0
$217$		−	31.6820i		−	0.146000i
$218$			97.7535i			0.448410i
$219$		0			0
$220$	−47.4708			−0.215776
$221$			303.165i			1.37179i
$222$		0			0
$223$	348.623			1.56333			0.781666	−	0.623697i	$-0.214370\pi$
$223$	348.623			1.56333			0.781666	+	0.623697i	$0.214370\pi$
$224$		−	6.61391i		−	0.0295264i
$225$		0			0
$226$			194.121i			0.858943i
$227$		−	175.793i		−	0.774421i	−0.921991	−	0.387210i	$-0.873439\pi$
$227$		−	175.793i		−	0.774421i	0.921991	−	0.387210i	$-0.126561\pi$
$228$		0			0
$229$			1.44148i			0.00629466i	0.999995	+	0.00314733i	$0.00100183\pi$
$229$			1.44148i			0.00629466i	−0.999995	+	0.00314733i	$0.998998\pi$
$230$	29.8880	+	66.3077i	0.129948	+	0.288294i
$231$		0			0
$232$	92.2047			0.397434
$233$	113.416			0.486763			0.243381	−	0.969931i	$-0.421743\pi$
$233$	113.416			0.486763			0.243381	+	0.969931i	$0.421743\pi$
$234$		0			0
$235$		−	109.872i		−	0.467542i
$236$	−107.091			−0.453777
$237$		0			0
$238$	−33.2162			−0.139564
$239$	−367.260			−1.53665			−0.768326	−	0.640059i	$-0.778910\pi$
$239$	−367.260			−1.53665			−0.768326	+	0.640059i	$0.778910\pi$
$240$		0			0
$241$			220.365i			0.914378i	0.889370	+	0.457189i	$0.151144\pi$
$241$			220.365i			0.914378i	−0.889370	+	0.457189i	$0.848856\pi$
$242$	−11.7752			−0.0486577
$243$		0			0
$244$		−	47.0332i		−	0.192759i
$245$		−	106.511i		−	0.434737i
$246$		0			0
$247$		−	339.888i		−	1.37606i
$248$	76.6433			0.309046
$249$		0			0
$250$		−	15.8114i		−	0.0632456i
$251$			440.210i			1.75382i	0.480650	+	0.876912i	$0.340401\pi$
$251$			440.210i			1.75382i	−0.480650	+	0.876912i	$0.659599\pi$
$252$		0			0
$253$	−100.325	−	222.574i	−0.396541	−	0.879740i
$254$	−81.9816			−0.322762
$255$		0			0
$256$	16.0000			0.0625000
$257$	−491.519			−1.91252			−0.956262	−	0.292512i	$-0.905509\pi$
$257$	−491.519			−1.91252			−0.956262	+	0.292512i	$0.905509\pi$
$258$		0			0
$259$	62.0333			0.239511
$260$			67.4905i			0.259579i
$261$		0			0
$262$	−134.465			−0.513225
$263$			267.833i			1.01838i	0.860655	+	0.509189i	$0.170054\pi$
$263$			267.833i			1.01838i	−0.860655	+	0.509189i	$0.829946\pi$
$264$		0			0
$265$	233.116			0.879684
$266$	37.2397			0.139999
$267$		0			0
$268$			118.951i			0.443846i
$269$	−95.6985			−0.355756			−0.177878	−	0.984053i	$-0.556923\pi$
$269$	−95.6985			−0.355756			−0.177878	+	0.984053i	$0.556923\pi$
$270$		0			0
$271$	−452.255			−1.66884			−0.834419	−	0.551131i	$-0.814197\pi$
$271$	−452.255			−1.66884			−0.834419	+	0.551131i	$0.814197\pi$
$272$		−	80.3547i		−	0.295422i
$273$		0			0
$274$		−	281.020i		−	1.02562i
$275$			53.0739i			0.192996i
$276$		0			0
$277$	−87.4783			−0.315806			−0.157903	−	0.987455i	$-0.550473\pi$
$277$	−87.4783			−0.315806			−0.157903	+	0.987455i	$0.550473\pi$
$278$	0.183112			0.000658676
$279$		0			0
$280$	−7.39458			−0.0264092
$281$			232.157i			0.826183i	0.910690	+	0.413091i	$0.135551\pi$
$281$			232.157i			0.826183i	−0.910690	+	0.413091i	$0.864449\pi$
$282$		0			0
$283$			239.367i			0.845820i	0.906172	+	0.422910i	$0.138991\pi$
$283$			239.367i			0.845820i	−0.906172	+	0.422910i	$0.861009\pi$
$284$	−110.426			−0.388824
$285$		0			0
$286$		−	226.545i		−	0.792114i
$287$		−	11.0338i		−	0.0384454i
$288$		0			0
$289$	−114.554			−0.396382
$290$		−	103.088i		−	0.355476i
$291$		0			0
$292$	−17.5461			−0.0600894
$293$			581.830i			1.98577i	0.119092	+	0.992883i	$0.462002\pi$
$293$			581.830i			1.98577i	−0.119092	+	0.992883i	$0.537998\pi$
$294$		0			0
$295$			119.732i			0.405871i
$296$			150.067i			0.506984i
$297$		0			0
$298$		−	90.3322i		−	0.303128i
$299$	−316.440	+	142.635i	−1.05833	+	0.477039i
$300$		0			0
$301$	−42.6030			−0.141538
$302$	49.8755			0.165151
$303$		0			0
$304$			90.0882i			0.296343i
$305$	−52.5847			−0.172409
$306$		0			0
$307$	110.134			0.358742			0.179371	−	0.983782i	$-0.442594\pi$
$307$	110.134			0.358742			0.179371	+	0.983782i	$0.442594\pi$
$308$	24.8213			0.0805887
$309$		0			0
$310$		−	85.6898i		−	0.276419i
$311$	298.100			0.958520			0.479260	−	0.877673i	$-0.340905\pi$
$311$	298.100			0.958520			0.479260	+	0.877673i	$0.340905\pi$
$312$		0			0
$313$		−	109.474i		−	0.349758i	−0.984590	−	0.174879i	$-0.944047\pi$
$313$		−	109.474i		−	0.349758i	0.984590	−	0.174879i	$-0.0559534\pi$
$314$		−	408.900i		−	1.30223i
$315$		0			0
$316$			114.170i			0.361296i
$317$	−474.434			−1.49664			−0.748318	−	0.663340i	$-0.769138\pi$
$317$	−474.434			−1.49664			−0.748318	+	0.663340i	$0.769138\pi$
$318$		0			0
$319$			346.034i			1.08475i
$320$		−	17.8885i		−	0.0559017i
$321$		0			0
$322$	−15.6277	−	34.6707i	−0.0485333	−	0.107673i
$323$	452.438			1.40074
$324$		0			0
$325$	75.4567			0.232174
$326$	65.4124			0.200652
$327$		0			0
$328$	26.6924			0.0813794
$329$			57.4496i			0.174619i
$330$		0			0
$331$	255.367			0.771503			0.385751	−	0.922603i	$-0.373942\pi$
$331$	255.367			0.771503			0.385751	+	0.922603i	$0.373942\pi$
$332$			110.358i			0.332403i
$333$		0			0
$334$	−128.515			−0.384774
$335$	132.991			0.396988
$336$		0			0
$337$			458.486i			1.36049i	0.732983	+	0.680247i	$0.238127\pi$
$337$			458.486i			1.36049i	−0.732983	+	0.680247i	$0.761873\pi$
$338$	−83.0828			−0.245807
$339$		0			0
$340$	−89.8392			−0.264233
$341$			287.634i			0.843502i
$342$		0			0
$343$			112.982i			0.329393i
$344$		−	103.063i		−	0.299601i
$345$		0			0
$346$	−128.395			−0.371085
$347$	−510.676			−1.47169			−0.735845	−	0.677151i	$-0.763215\pi$
$347$	−510.676			−1.47169			−0.735845	+	0.677151i	$0.763215\pi$
$348$		0			0
$349$	−591.870			−1.69590			−0.847951	−	0.530075i	$-0.822164\pi$
$349$	−591.870			−1.69590			−0.847951	+	0.530075i	$0.822164\pi$
$350$			8.26739i			0.0236211i
$351$		0			0
$352$			60.0463i			0.170586i
$353$	−168.334			−0.476866			−0.238433	−	0.971159i	$-0.576634\pi$
$353$	−168.334			−0.476866			−0.238433	+	0.971159i	$0.576634\pi$
$354$		0			0
$355$			123.460i			0.347775i
$356$		−	279.651i		−	0.785536i
$357$		0			0
$358$	426.577			1.19156
$359$		−	37.5719i		−	0.104657i	−0.998630	−	0.0523285i	$-0.983336\pi$
$359$		−	37.5719i		−	0.104657i	0.998630	−	0.0523285i	$-0.0166643\pi$
$360$		0			0
$361$	−146.243			−0.405105
$362$		−	351.499i		−	0.970992i
$363$		0			0
$364$		−	35.2891i		−	0.0969482i
$365$			19.6171i			0.0537456i
$366$		0			0
$367$			427.036i			1.16359i	0.813337	+	0.581793i	$0.197649\pi$
$367$			427.036i			1.16359i	−0.813337	+	0.581793i	$0.802351\pi$
$368$	83.8733	−	37.8057i	0.227917	−	0.102733i
$369$		0			0
$370$	167.780			0.453461
$371$	−121.891			−0.328547
$372$		0			0
$373$			369.980i			0.991904i	0.868350	+	0.495952i	$0.165181\pi$
$373$			369.980i			0.991904i	−0.868350	+	0.495952i	$0.834819\pi$
$374$	301.563			0.806317
$375$		0			0
$376$	−138.979			−0.369625
$377$	491.966			1.30495
$378$		0			0
$379$			78.8009i			0.207918i	0.994582	+	0.103959i	$0.0331511\pi$
$379$			78.8009i			0.207918i	−0.994582	+	0.103959i	$0.966849\pi$
$380$	100.722			0.265057
$381$		0			0
$382$			207.015i			0.541925i
$383$		−	254.083i		−	0.663403i	−0.943384	−	0.331702i	$-0.892377\pi$
$383$		−	254.083i		−	0.663403i	0.943384	−	0.331702i	$-0.107623\pi$
$384$		0			0
$385$		−	27.7511i		−	0.0720807i
$386$	170.622			0.442027
$387$		0			0
$388$		−	39.7270i		−	0.102389i
$389$			322.363i			0.828698i	0.910118	+	0.414349i	$0.135991\pi$
$389$			322.363i			0.828698i	−0.910118	+	0.414349i	$0.864009\pi$
$390$		0			0
$391$	−189.866	−	421.226i	−0.485592	−	1.07730i
$392$	−134.726			−0.343690
$393$		0			0
$394$	44.7602			0.113605
$395$	127.645			0.323153
$396$		0			0
$397$	715.832			1.80310			0.901552	−	0.432671i	$-0.142429\pi$
$397$	715.832			1.80310			0.901552	+	0.432671i	$0.142429\pi$
$398$		−	52.9617i		−	0.133070i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$		−	441.435i		−	1.10083i	−0.834890	−	0.550417i	$-0.814469\pi$
$401$		−	441.435i		−	1.10083i	0.834890	−	0.550417i	$-0.185531\pi$
$402$		0			0
$403$	408.937			1.01473
$404$	−173.128			−0.428534
$405$		0			0
$406$			53.9022i			0.132764i
$407$	−563.187			−1.38375
$408$		0			0
$409$	−608.727			−1.48833			−0.744164	−	0.667996i	$-0.767152\pi$
$409$	−608.727			−1.48833			−0.744164	+	0.667996i	$0.767152\pi$
$410$		−	29.8430i		−	0.0727879i
$411$		0			0
$412$			288.235i			0.699600i
$413$		−	62.6048i		−	0.151586i
$414$		0			0
$415$	123.384			0.297310
$416$	85.3695			0.205215
$417$		0			0
$418$	−338.091			−0.808831
$419$			287.454i			0.686047i	0.939327	+	0.343023i	$0.111451\pi$
$419$			287.454i			0.686047i	−0.939327	+	0.343023i	$0.888549\pi$
$420$		0			0
$421$		−	130.848i		−	0.310803i	−0.987851	−	0.155402i	$-0.950333\pi$
$421$		−	130.848i		−	0.310803i	0.987851	−	0.155402i	$-0.0496672\pi$
$422$	61.7406			0.146305
$423$		0			0
$424$		−	294.871i		−	0.695452i
$425$			100.443i			0.236337i
$426$		0			0
$427$	27.4952			0.0643917
$428$			20.9089i			0.0488525i
$429$		0			0
$430$	−115.228			−0.267972
$431$			239.816i			0.556417i	0.960521	+	0.278209i	$0.0897406\pi$
$431$			239.816i			0.556417i	−0.960521	+	0.278209i	$0.910259\pi$
$432$		0			0
$433$		−	305.542i		−	0.705640i	−0.935691	−	0.352820i	$-0.885223\pi$
$433$		−	305.542i		−	0.705640i	0.935691	−	0.352820i	$-0.114777\pi$
$434$			44.8051i			0.103238i
$435$		0			0
$436$		−	138.244i		−	0.317074i
$437$	212.865	+	472.250i	0.487106	+	1.08066i
$438$		0			0
$439$	−579.018			−1.31895			−0.659474	−	0.751727i	$-0.729221\pi$
$439$	−579.018			−1.31895			−0.659474	+	0.751727i	$0.729221\pi$
$440$	67.1338			0.152577
$441$		0			0
$442$		−	428.740i		−	0.969999i
$443$	606.274			1.36856			0.684282	−	0.729217i	$-0.260116\pi$
$443$	606.274			1.36856			0.684282	+	0.729217i	$0.260116\pi$
$444$		0			0
$445$	−312.659			−0.702605
$446$	−493.028			−1.10544
$447$		0			0
$448$			9.35348i			0.0208783i
$449$	202.580			0.451181			0.225590	−	0.974222i	$-0.427569\pi$
$449$	202.580			0.451181			0.225590	+	0.974222i	$0.427569\pi$
$450$		0			0
$451$			100.174i			0.222115i
$452$		−	274.529i		−	0.607364i
$453$		0			0
$454$			248.609i			0.547598i
$455$	−39.4545			−0.0867131
$456$		0			0
$457$			373.760i			0.817857i	0.912567	+	0.408928i	$0.134097\pi$
$457$			373.760i			0.817857i	−0.912567	+	0.408928i	$0.865903\pi$
$458$		−	2.03856i		−	0.00445100i
$459$		0			0
$460$	−42.2681	−	93.7732i	−0.0918871	−	0.203855i
$461$	−709.377			−1.53878			−0.769390	−	0.638780i	$-0.779439\pi$
$461$	−709.377			−1.53878			−0.769390	+	0.638780i	$0.779439\pi$
$462$		0			0
$463$	266.488			0.575569			0.287785	−	0.957695i	$-0.407081\pi$
$463$	266.488			0.575569			0.287785	+	0.957695i	$0.407081\pi$
$464$	−130.397			−0.281028
$465$		0			0
$466$	−160.394			−0.344193
$467$		−	686.135i		−	1.46924i	−0.678479	−	0.734620i	$-0.737361\pi$
$467$		−	686.135i		−	1.46924i	0.678479	−	0.734620i	$-0.262639\pi$
$468$		0			0
$469$	−69.5378			−0.148268
$470$			155.383i			0.330602i
$471$		0			0
$472$	151.450			0.320869
$473$	386.784			0.817725
$474$		0			0
$475$		−	112.610i		−	0.237074i
$476$	46.9747			0.0986864
$477$		0			0
$478$	519.384			1.08658
$479$			176.517i			0.368511i	0.982878	+	0.184256i	$0.0589874\pi$
$479$			176.517i			0.368511i	−0.982878	+	0.184256i	$0.941013\pi$
$480$		0			0
$481$			800.698i			1.66465i
$482$		−	311.643i		−	0.646563i
$483$		0			0
$484$	16.6526			0.0344062
$485$	−44.4161			−0.0915796
$486$		0			0
$487$	524.802			1.07762			0.538811	−	0.842427i	$-0.318874\pi$
$487$	524.802			1.07762			0.538811	+	0.842427i	$0.318874\pi$
$488$			66.5149i			0.136301i
$489$		0			0
$490$			150.629i			0.307406i
$491$	551.047			1.12229			0.561147	−	0.827716i	$-0.310360\pi$
$491$	551.047			1.12229			0.561147	+	0.827716i	$0.310360\pi$
$492$		0			0
$493$			654.876i			1.32835i
$494$			480.674i			0.973024i
$495$		0			0
$496$	−108.390			−0.218528
$497$		−	64.5542i		−	0.129888i
$498$		0			0
$499$	−736.739			−1.47643			−0.738216	−	0.674565i	$-0.764331\pi$
$499$	−736.739			−1.47643			−0.738216	+	0.674565i	$0.764331\pi$
$500$			22.3607i			0.0447214i
$501$		0			0
$502$		−	622.551i		−	1.24014i
$503$		−	699.308i		−	1.39027i	−0.718877	−	0.695137i	$-0.755344\pi$
$503$		−	699.308i		−	1.39027i	0.718877	−	0.695137i	$-0.244656\pi$
$504$		0			0
$505$			193.563i			0.383292i
$506$	141.881	+	314.768i	0.280397	+	0.622070i
$507$		0			0
$508$	115.939			0.228227
$509$	316.428			0.621666			0.310833	−	0.950464i	$-0.399392\pi$
$509$	316.428			0.621666			0.310833	+	0.950464i	$0.399392\pi$
$510$		0			0
$511$		−	10.2573i		−	0.0200730i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	695.112			1.35236
$515$	322.257			0.625741
$516$		0			0
$517$		−	521.573i		−	1.00885i
$518$	−87.7283			−0.169360
$519$		0			0
$520$		−	95.4460i		−	0.183550i
$521$		−	798.875i		−	1.53335i	−0.642035	−	0.766675i	$-0.721910\pi$
$521$		−	798.875i		−	1.53335i	0.642035	−	0.766675i	$-0.278090\pi$
$522$		0			0
$523$		−	819.091i		−	1.56614i	−0.621934	−	0.783070i	$-0.713653\pi$
$523$		−	819.091i		−	1.56614i	0.621934	−	0.783070i	$-0.286347\pi$
$524$	190.162			0.362905
$525$		0			0
$526$		−	378.773i		−	0.720101i
$527$			544.353i			1.03293i
$528$		0			0
$529$	350.341	−	396.361i	0.662271	−	0.749265i
$530$	−329.676			−0.622031
$531$		0			0
$532$	−52.6649			−0.0989942
$533$	142.420			0.267204
$534$		0			0
$535$	23.3768			0.0436950
$536$		−	168.222i		−	0.313847i
$537$		0			0
$538$	135.338			0.251558
$539$		−	505.614i		−	0.938060i
$540$		0			0
$541$	289.616			0.535335			0.267668	−	0.963511i	$-0.413747\pi$
$541$	289.616			0.535335			0.267668	+	0.963511i	$0.413747\pi$
$542$	639.585			1.18005
$543$		0			0
$544$			113.639i			0.208895i
$545$	−154.562			−0.283600
$546$		0			0
$547$	996.117			1.82106			0.910528	−	0.413448i	$-0.135676\pi$
$547$	996.117			1.82106			0.910528	+	0.413448i	$0.135676\pi$
$548$			397.422i			0.725223i
$549$		0			0
$550$		−	75.0579i		−	0.136469i
$551$		−	734.202i		−	1.33249i
$552$		0			0
$553$	−66.7427			−0.120692
$554$	123.713			0.223309
$555$		0			0
$556$	−0.258959			−0.000465754
$557$		−	937.756i		−	1.68358i	−0.539802	−	0.841792i	$-0.681501\pi$
$557$		−	937.756i		−	1.68358i	0.539802	−	0.841792i	$-0.318499\pi$
$558$		0			0
$559$		−	549.901i		−	0.983723i
$560$	10.4575			0.0186741
$561$		0			0
$562$		−	328.320i		−	0.584200i
$563$			785.252i			1.39476i	0.716700	+	0.697382i	$0.245652\pi$
$563$			785.252i			1.39476i	−0.716700	+	0.697382i	$0.754348\pi$
$564$		0			0
$565$	−306.932			−0.543243
$566$		−	338.516i		−	0.598085i
$567$		0			0
$568$	156.166			0.274940
$569$		−	403.851i		−	0.709755i	−0.934913	−	0.354877i	$-0.884523\pi$
$569$		−	403.851i		−	0.709755i	0.934913	−	0.354877i	$-0.115477\pi$
$570$		0			0
$571$		−	335.474i		−	0.587519i	−0.955879	−	0.293760i	$-0.905093\pi$
$571$		−	335.474i		−	0.587519i	0.955879	−	0.293760i	$-0.0949065\pi$
$572$			320.383i			0.560110i
$573$		0			0
$574$			15.6042i			0.0271850i
$575$	−104.842	+	47.2571i	−0.182333	+	0.0821863i
$576$		0			0
$577$	−152.703			−0.264649			−0.132325	−	0.991206i	$-0.542244\pi$
$577$	−152.703			−0.264649			−0.132325	+	0.991206i	$0.542244\pi$
$578$	162.004			0.280284
$579$		0			0
$580$			145.788i			0.251359i
$581$	−64.5143			−0.111040
$582$		0			0
$583$	1106.62			1.89815
$584$	24.8139			0.0424896
$585$		0			0
$586$		−	822.831i		−	1.40415i
$587$	−129.262			−0.220207			−0.110104	−	0.993920i	$-0.535118\pi$
$587$	−129.262			−0.220207			−0.110104	+	0.993920i	$0.535118\pi$
$588$		0			0
$589$		−	610.291i		−	1.03615i
$590$		−	169.326i		−	0.286994i
$591$		0			0
$592$		−	212.227i		−	0.358492i
$593$	879.853			1.48373			0.741866	−	0.670548i	$-0.233941\pi$
$593$	879.853			1.48373			0.741866	+	0.670548i	$0.233941\pi$
$594$		0			0
$595$		−	52.5194i		−	0.0882678i
$596$			127.749i			0.214344i
$597$		0			0
$598$	447.514	−	201.716i	0.748351	−	0.337317i
$599$	709.708			1.18482			0.592411	−	0.805636i	$-0.298176\pi$
$599$	709.708			1.18482			0.592411	+	0.805636i	$0.298176\pi$
$600$		0			0
$601$	238.861			0.397439			0.198719	−	0.980056i	$-0.436322\pi$
$601$	238.861			0.397439			0.198719	+	0.980056i	$0.436322\pi$
$602$	60.2498			0.100083
$603$		0			0
$604$	−70.5346			−0.116779
$605$		−	18.6182i		−	0.0307738i
$606$		0			0
$607$	−1082.53			−1.78341			−0.891703	−	0.452621i	$-0.850489\pi$
$607$	−1082.53			−1.78341			−0.891703	+	0.452621i	$0.850489\pi$
$608$		−	127.404i		−	0.209546i
$609$		0			0
$610$	74.3659			0.121911
$611$	−741.535			−1.21364
$612$		0			0
$613$			54.0902i			0.0882384i	0.999026	+	0.0441192i	$0.0140481\pi$
$613$			54.0902i			0.0882384i	−0.999026	+	0.0441192i	$0.985952\pi$
$614$	−155.753			−0.253669
$615$		0			0
$616$	−35.1026			−0.0569848
$617$			454.342i			0.736373i	0.929752	+	0.368186i	$0.120021\pi$
$617$			454.342i			0.736373i	−0.929752	+	0.368186i	$0.879979\pi$
$618$		0			0
$619$		−	153.676i		−	0.248264i	−0.992266	−	0.124132i	$-0.960385\pi$
$619$		−	153.676i		−	0.248264i	0.992266	−	0.124132i	$-0.0396147\pi$
$620$			121.184i			0.195458i
$621$		0			0
$622$	−421.577			−0.677776
$623$	163.482			0.262411
$624$		0			0
$625$	25.0000			0.0400000
$626$			154.820i			0.247316i
$627$		0			0
$628$			578.271i			0.920814i
$629$	−1065.84			−1.69450
$630$		0			0
$631$		−	1241.15i		−	1.96696i	−0.181027	−	0.983478i	$-0.557942\pi$
$631$		−	1241.15i		−	1.96696i	0.181027	−	0.983478i	$-0.442058\pi$
$632$		−	161.460i		−	0.255475i
$633$		0			0
$634$	670.950			1.05828
$635$		−	129.624i		−	0.204133i
$636$		0			0
$637$	−718.846			−1.12849
$638$		−	489.366i		−	0.767032i
$639$		0			0
$640$			25.2982i			0.0395285i
$641$		−	545.978i		−	0.851759i	−0.904780	−	0.425880i	$-0.859965\pi$
$641$		−	545.978i		−	0.851759i	0.904780	−	0.425880i	$-0.140035\pi$
$642$		0			0
$643$			539.118i			0.838441i	0.907884	+	0.419221i	$0.137697\pi$
$643$			539.118i			0.838441i	−0.907884	+	0.419221i	$0.862303\pi$
$644$	22.1009	+	49.0317i	0.0343182	+	0.0761362i
$645$		0			0
$646$	−639.844			−0.990470
$647$	−652.715			−1.00883			−0.504417	−	0.863460i	$-0.668292\pi$
$647$	−652.715			−1.00883			−0.504417	+	0.863460i	$0.668292\pi$
$648$		0			0
$649$			568.376i			0.875772i
$650$	−106.712			−0.164172
$651$		0			0
$652$	−92.5071			−0.141882
$653$	−374.815			−0.573989			−0.286995	−	0.957932i	$-0.592656\pi$
$653$	−374.815			−0.573989			−0.286995	+	0.957932i	$0.592656\pi$
$654$		0			0
$655$		−	212.608i		−	0.324592i
$656$	−37.7488			−0.0575439
$657$		0			0
$658$		−	81.2460i		−	0.123474i
$659$			440.090i			0.667815i	0.942606	+	0.333907i	$0.108367\pi$
$659$			440.090i			0.667815i	−0.942606	+	0.333907i	$0.891633\pi$
$660$		0			0
$661$		−	72.6725i		−	0.109943i	−0.998488	−	0.0549716i	$-0.982493\pi$
$661$		−	72.6725i		−	0.109943i	0.998488	−	0.0549716i	$-0.0175068\pi$
$662$	−361.144			−0.545535
$663$		0			0
$664$		−	156.069i		−	0.235044i
$665$			58.8812i			0.0885431i
$666$		0			0
$667$	−683.552	+	308.110i	−1.02482	+	0.461933i
$668$	181.747			0.272076
$669$		0			0
$670$	−188.078			−0.280713
$671$	−249.623			−0.372017
$672$		0			0
$673$	−1052.88			−1.56445			−0.782227	−	0.622993i	$-0.785916\pi$
$673$	−1052.88			−1.56445			−0.782227	+	0.622993i	$0.785916\pi$
$674$		−	648.398i		−	0.962014i
$675$		0			0
$676$	117.497			0.173812
$677$			393.587i			0.581369i	0.956819	+	0.290684i	$0.0938830\pi$
$677$			393.587i			0.581369i	−0.956819	+	0.290684i	$0.906117\pi$
$678$		0			0
$679$	23.2241			0.0342034
$680$	127.052			0.186841
$681$		0			0
$682$		−	406.776i		−	0.596446i
$683$	645.240			0.944714			0.472357	−	0.881407i	$-0.343403\pi$
$683$	645.240			0.944714			0.472357	+	0.881407i	$0.343403\pi$
$684$		0			0
$685$	444.331			0.648659
$686$		−	159.781i		−	0.232916i
$687$		0			0
$688$			145.753i			0.211850i
$689$		−	1573.31i		−	2.28347i
$690$		0			0
$691$	548.216			0.793366			0.396683	−	0.917956i	$-0.370161\pi$
$691$	548.216			0.793366			0.396683	+	0.917956i	$0.370161\pi$
$692$	181.579			0.262397
$693$		0			0
$694$	722.205			1.04064
$695$			0.289525i			0.000416583i
$696$		0			0
$697$			189.581i			0.271995i
$698$	837.030			1.19918
$699$		0			0
$700$		−	11.6919i		−	0.0167026i
$701$			92.2064i			0.131536i	0.997835	+	0.0657678i	$0.0209496\pi$
$701$			92.2064i			0.131536i	−0.997835	+	0.0657678i	$0.979050\pi$
$702$		0			0
$703$	1194.95			1.69978
$704$		−	84.9183i		−	0.120623i
$705$		0			0
$706$	238.060			0.337195
$707$		−	101.209i		−	0.143153i
$708$		0			0
$709$		−	922.612i		−	1.30129i	−0.759384	−	0.650643i	$-0.774499\pi$
$709$		−	922.612i		−	1.30129i	0.759384	−	0.650643i	$-0.225501\pi$
$710$		−	174.599i		−	0.245914i
$711$		0			0
$712$			395.486i			0.555458i
$713$	−568.189	+	256.110i	−0.796899	+	0.359201i
$714$		0			0
$715$	358.199			0.500977
$716$	−603.271			−0.842557
$717$		0			0
$718$			53.1347i			0.0740037i
$719$	−606.887			−0.844071			−0.422035	−	0.906579i	$-0.638684\pi$
$719$	−606.887			−0.844071			−0.422035	+	0.906579i	$0.638684\pi$
$720$		0			0
$721$	−168.500			−0.233704
$722$	206.819			0.286452
$723$		0			0
$724$			497.095i			0.686595i
$725$	162.996			0.224823
$726$		0			0
$727$		−	787.544i		−	1.08328i	−0.840611	−	0.541640i	$-0.817804\pi$
$727$		−	787.544i		−	1.08328i	0.840611	−	0.541640i	$-0.182196\pi$
$728$			49.9064i			0.0685527i
$729$		0			0
$730$		−	27.7428i		−	0.0380038i
$731$	731.995			1.00136
$732$		0			0
$733$			415.085i			0.566282i	0.959078	+	0.283141i	$0.0913765\pi$
$733$			415.085i			0.566282i	−0.959078	+	0.283141i	$0.908624\pi$
$734$		−	603.921i		−	0.822780i
$735$		0			0
$736$	−118.615	+	53.4653i	−0.161161	+	0.0726431i
$737$	631.319			0.856606
$738$		0			0
$739$	639.964			0.865987			0.432993	−	0.901397i	$-0.357457\pi$
$739$	639.964			0.865987			0.432993	+	0.901397i	$0.357457\pi$
$740$	−237.277			−0.320645
$741$		0			0
$742$	172.380			0.232318
$743$			743.739i			1.00099i	0.865738	+	0.500497i	$0.166849\pi$
$743$			743.739i			1.00099i	−0.865738	+	0.500497i	$0.833151\pi$
$744$		0			0
$745$	142.828			0.191715
$746$		−	523.231i		−	0.701382i
$747$		0			0
$748$	−426.474			−0.570152
$749$	−12.2232			−0.0163193
$750$		0			0
$751$		−	517.047i		−	0.688478i	−0.938882	−	0.344239i	$-0.888137\pi$
$751$		−	517.047i		−	0.688478i	0.938882	−	0.344239i	$-0.111863\pi$
$752$	196.546			0.261364
$753$		0			0
$754$	−695.746			−0.922740
$755$			78.8601i			0.104450i
$756$		0			0
$757$		−	911.823i		−	1.20452i	−0.798299	−	0.602261i	$-0.794267\pi$
$757$		−	911.823i		−	1.20452i	0.798299	−	0.602261i	$-0.205733\pi$
$758$		−	111.441i		−	0.147020i
$759$		0			0
$760$	−142.442			−0.187424
$761$	−954.073			−1.25371			−0.626855	−	0.779136i	$-0.715658\pi$
$761$	−954.073			−1.25371			−0.626855	+	0.779136i	$0.715658\pi$
$762$		0			0
$763$	80.8166			0.105920
$764$		−	292.764i		−	0.383199i
$765$		0			0
$766$			359.328i			0.469097i
$767$	808.076			1.05355
$768$		0			0
$769$		−	276.277i		−	0.359268i	−0.983734	−	0.179634i	$-0.942509\pi$
$769$		−	276.277i		−	0.359268i	0.983734	−	0.179634i	$-0.0574913\pi$
$770$			39.2459i			0.0509687i
$771$		0			0
$772$	−241.296			−0.312560
$773$			40.8440i			0.0528383i	0.999651	+	0.0264192i	$0.00841046\pi$
$773$			40.8440i			0.0528383i	−0.999651	+	0.0264192i	$0.991590\pi$
$774$		0			0
$775$	135.488			0.174823
$776$			56.1824i			0.0724000i
$777$		0			0
$778$		−	455.891i		−	0.585978i
$779$		−	212.545i		−	0.272843i
$780$		0			0
$781$			586.074i			0.750415i
$782$	268.512	+	595.703i	0.343365	+	0.761768i
$783$		0			0
$784$	190.532			0.243026
$785$	646.527			0.823601
$786$		0			0
$787$		−	1364.69i		−	1.73404i	−0.498274	−	0.867019i	$-0.666033\pi$
$787$		−	1364.69i		−	1.73404i	0.498274	−	0.867019i	$-0.333967\pi$
$788$	−63.3005			−0.0803305
$789$		0			0
$790$	−180.518			−0.228504
$791$	160.487			0.202892
$792$		0			0
$793$			354.897i			0.447537i
$794$	−1012.34			−1.27499
$795$		0			0
$796$			74.8991i			0.0940944i
$797$		−	20.2192i		−	0.0253692i	−0.999920	−	0.0126846i	$-0.995962\pi$
$797$		−	20.2192i		−	0.0253692i	0.999920	−	0.0126846i	$-0.00403774\pi$
$798$		0			0
$799$		−	987.086i		−	1.23540i
$800$	28.2843			0.0353553
$801$		0			0
$802$			624.283i			0.778407i
$803$			93.1240i			0.115970i
$804$		0			0
$805$	54.8191	−	24.7096i	0.0680983	−	0.0306951i
$806$	−578.325			−0.717525
$807$		0			0
$808$	244.840			0.303019
$809$	1032.11			1.27579			0.637894	−	0.770124i	$-0.279806\pi$
$809$	1032.11			1.27579			0.637894	+	0.770124i	$0.279806\pi$
$810$		0			0
$811$	−70.2781			−0.0866561			−0.0433281	−	0.999061i	$-0.513796\pi$
$811$	−70.2781			−0.0866561			−0.0433281	+	0.999061i	$0.513796\pi$
$812$		−	76.2292i		−	0.0938783i
$813$		0			0
$814$	796.467			0.978460
$815$			103.426i			0.126903i
$816$		0			0
$817$	−820.663			−1.00448
$818$	860.869			1.05241
$819$		0			0
$820$			42.2044i			0.0514688i
$821$	−84.7400			−0.103216			−0.0516078	−	0.998667i	$-0.516435\pi$
$821$	−84.7400			−0.103216			−0.0516078	+	0.998667i	$0.516435\pi$
$822$		0			0
$823$	46.0292			0.0559286			0.0279643	−	0.999609i	$-0.491098\pi$
$823$	46.0292			0.0559286			0.0279643	+	0.999609i	$0.491098\pi$
$824$		−	407.626i		−	0.494692i
$825$		0			0
$826$			88.5366i			0.107187i
$827$		−	1283.77i		−	1.55232i	−0.630536	−	0.776160i	$-0.717165\pi$
$827$		−	1283.77i		−	1.55232i	0.630536	−	0.776160i	$-0.282835\pi$
$828$		0			0
$829$	−483.453			−0.583177			−0.291588	−	0.956544i	$-0.594184\pi$
$829$	−483.453			−0.583177			−0.291588	+	0.956544i	$0.594184\pi$
$830$	−174.491			−0.210230
$831$		0			0
$832$	−120.731			−0.145109
$833$		−	956.883i		−	1.14872i
$834$		0			0
$835$		−	203.199i		−	0.243352i
$836$	478.134			0.571930
$837$		0			0
$838$		−	406.521i		−	0.485108i
$839$			1060.74i			1.26429i	0.774852	+	0.632143i	$0.217824\pi$
$839$			1060.74i			1.26429i	−0.774852	+	0.632143i	$0.782176\pi$
$840$		0			0
$841$	221.712			0.263629
$842$			185.047i			0.219771i
$843$		0			0
$844$	−87.3144			−0.103453
$845$		−	131.365i		−	0.155462i
$846$		0			0
$847$			9.73498i			0.0114935i
$848$			417.011i			0.491759i
$849$		0			0
$850$		−	142.048i		−	0.167116i
$851$	−501.463	−	1112.51i	−0.589263	−	1.30730i
$852$		0			0
$853$	1653.13			1.93802			0.969010	−	0.247023i	$-0.0794522\pi$
$853$	1653.13			1.93802			0.969010	+	0.247023i	$0.0794522\pi$
$854$	−38.8841			−0.0455318
$855$		0			0
$856$		−	29.5696i		−	0.0345439i
$857$	−1453.42			−1.69593			−0.847967	−	0.530048i	$-0.822174\pi$
$857$	−1453.42			−1.69593			−0.847967	+	0.530048i	$0.822174\pi$
$858$		0			0
$859$	936.399			1.09010			0.545052	−	0.838402i	$-0.316510\pi$
$859$	936.399			1.09010			0.545052	+	0.838402i	$0.316510\pi$
$860$	162.957			0.189484
$861$		0			0
$862$		−	339.151i		−	0.393446i
$863$	−961.830			−1.11452			−0.557259	−	0.830339i	$-0.688147\pi$
$863$	−961.830			−1.11452			−0.557259	+	0.830339i	$0.688147\pi$
$864$		0			0
$865$		−	203.011i		−	0.234695i
$866$			432.102i			0.498963i
$867$		0			0
$868$		−	63.3640i		−	0.0730000i
$869$	605.943			0.697287
$870$		0			0
$871$		−	897.563i		−	1.03050i
$872$			195.507i			0.224205i
$873$		0			0
$874$	−301.037	−	667.862i	−0.344436	−	0.764144i
$875$	−13.0719			−0.0149393
$876$		0			0
$877$	−1244.55			−1.41910			−0.709549	−	0.704657i	$-0.751101\pi$
$877$	−1244.55			−1.41910			−0.709549	+	0.704657i	$0.751101\pi$
$878$	818.855			0.932637
$879$		0			0
$880$	−94.9415			−0.107888
$881$			961.719i			1.09162i	0.837908	+	0.545811i	$0.183778\pi$
$881$			961.719i			1.09162i	−0.837908	+	0.545811i	$0.816222\pi$
$882$		0			0
$883$	42.4945			0.0481251			0.0240626	−	0.999710i	$-0.492340\pi$
$883$	42.4945			0.0481251			0.0240626	+	0.999710i	$0.492340\pi$
$884$			606.330i			0.685893i
$885$		0			0
$886$	−857.401			−0.967722
$887$	551.962			0.622279			0.311140	−	0.950364i	$-0.399289\pi$
$887$	551.962			0.622279			0.311140	+	0.950364i	$0.399289\pi$
$888$		0			0
$889$			67.7774i			0.0762400i
$890$	442.167			0.496817
$891$		0			0
$892$	697.246			0.781666
$893$			1106.65i			1.23925i
$894$		0			0
$895$			674.478i			0.753606i
$896$		−	13.2278i		−	0.0147632i
$897$		0			0
$898$	−286.492			−0.319033
$899$	883.359			0.982601
$900$		0			0
$901$	2094.30			2.32442
$902$		−	141.667i		−	0.157059i
$903$		0			0
$904$			388.242i			0.429472i
$905$	555.769			0.614109
$906$		0			0
$907$			237.524i			0.261878i	0.991390	+	0.130939i	$0.0417993\pi$
$907$			237.524i			0.261878i	−0.991390	+	0.130939i	$0.958201\pi$
$908$		−	351.587i		−	0.387210i
$909$		0			0
$910$	55.7970			0.0613154
$911$			402.885i			0.442245i	0.975246	+	0.221122i	$0.0709720\pi$
$911$			402.885i			0.442245i	−0.975246	+	0.221122i	$0.929028\pi$
$912$		0			0
$913$	585.712			0.641524
$914$		−	528.577i		−	0.578312i
$915$		0			0
$916$			2.88295i			0.00314733i
$917$			111.167i			0.121229i
$918$		0			0
$919$		−	1338.53i		−	1.45650i	−0.685310	−	0.728251i	$-0.740333\pi$
$919$		−	1338.53i		−	1.45650i	0.685310	−	0.728251i	$-0.259667\pi$
$920$	59.7761	+	132.615i	0.0649740	+	0.144147i
$921$		0			0
$922$	1003.21			1.08808
$923$	833.238			0.902750
$924$		0			0
$925$			265.284i			0.286794i
$926$	−376.872			−0.406989
$927$		0			0
$928$	184.409			0.198717
$929$	−621.771			−0.669291			−0.334646	−	0.942344i	$-0.608617\pi$
$929$	−621.771			−0.669291			−0.334646	+	0.942344i	$0.608617\pi$
$930$		0			0
$931$			1072.79i			1.15230i
$932$	226.831			0.243381
$933$		0			0
$934$			970.341i			1.03891i
$935$			476.812i			0.509960i
$936$		0			0
$937$		−	915.580i		−	0.977140i	−0.872525	−	0.488570i	$-0.837519\pi$
$937$		−	915.580i		−	0.977140i	0.872525	−	0.488570i	$-0.162481\pi$
$938$	98.3412			0.104841
$939$		0			0
$940$		−	219.745i		−	0.233771i
$941$			834.255i			0.886563i	0.896383	+	0.443281i	$0.146186\pi$
$941$			834.255i			0.886563i	−0.896383	+	0.443281i	$0.853814\pi$
$942$		0			0
$943$	−197.882	+	89.1950i	−0.209843	+	0.0945864i
$944$	−214.183			−0.226889
$945$		0			0
$946$	−546.995			−0.578219
$947$	−918.683			−0.970099			−0.485049	−	0.874487i	$-0.661198\pi$
$947$	−918.683			−0.970099			−0.485049	+	0.874487i	$0.661198\pi$
$948$		0			0
$949$	132.397			0.139512
$950$			159.255i			0.167637i
$951$		0			0
$952$	−66.4323			−0.0697818
$953$			789.341i			0.828270i	0.910215	+	0.414135i	$0.135916\pi$
$953$			789.341i			0.828270i	−0.910215	+	0.414135i	$0.864084\pi$
$954$		0			0
$955$	−327.320			−0.342743
$956$	−734.519			−0.768326
$957$		0			0
$958$		−	249.632i		−	0.260577i
$959$	−232.330			−0.242263
$960$		0			0
$961$	−226.725			−0.235927
$962$		−	1132.36i		−	1.17709i
$963$		0			0
$964$			440.730i			0.457189i
$965$			269.778i			0.279562i
$966$		0			0
$967$	875.476			0.905353			0.452677	−	0.891675i	$-0.350469\pi$
$967$	875.476			0.905353			0.452677	+	0.891675i	$0.350469\pi$
$968$	−23.5503			−0.0243288
$969$		0			0
$970$	62.8138			0.0647565
$971$		−	1232.49i		−	1.26930i	−0.772801	−	0.634648i	$-0.781145\pi$
$971$		−	1232.49i		−	1.26930i	0.772801	−	0.634648i	$-0.218855\pi$
$972$		0			0
$973$		−	0.151386i		−	0.000155587i
$974$	−742.182			−0.761994
$975$		0			0
$976$		−	94.0663i		−	0.0963794i
$977$		−	879.183i		−	0.899880i	−0.893059	−	0.449940i	$-0.851445\pi$
$977$		−	879.183i		−	0.899880i	0.893059	−	0.449940i	$-0.148555\pi$
$978$		0			0
$979$	−1484.22			−1.51606
$980$		−	213.021i		−	0.217369i
$981$		0			0
$982$	−779.298			−0.793582
$983$		−	247.090i		−	0.251363i	−0.992071	−	0.125682i	$-0.959888\pi$
$983$		−	247.090i		−	0.251363i	0.992071	−	0.125682i	$-0.0401118\pi$
$984$		0			0
$985$			70.7721i			0.0718498i
$986$		−	926.134i		−	0.939284i
$987$		0			0
$988$		−	679.776i		−	0.688032i
$989$	344.393	+	764.048i	0.348223	+	0.772546i
$990$		0			0
$991$	−1750.79			−1.76669			−0.883347	−	0.468721i	$-0.844715\pi$
$991$	−1750.79			−1.76669			−0.883347	+	0.468721i	$0.844715\pi$
$992$	153.287			0.154523
$993$		0			0
$994$			91.2935i			0.0918445i
$995$	83.7398			0.0841606
$996$		0			0
$997$	−456.156			−0.457529			−0.228765	−	0.973482i	$-0.573469\pi$
$997$	−456.156			−0.457529			−0.228765	+	0.973482i	$0.573469\pi$
$998$	1041.91			1.04399
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.a.91.3		16
3.2	odd	2		230.3.d.a.91.12	yes	16
12.11	even	2		1840.3.k.d.321.12		16
15.2	even	4		1150.3.c.c.1149.18		32
15.8	even	4		1150.3.c.c.1149.15		32
15.14	odd	2		1150.3.d.b.551.5		16
23.22	odd	2	inner	2070.3.c.a.91.6		16
69.68	even	2		230.3.d.a.91.11	✓	16
276.275	odd	2		1840.3.k.d.321.11		16
345.68	odd	4		1150.3.c.c.1149.17		32
345.137	odd	4		1150.3.c.c.1149.16		32
345.344	even	2		1150.3.d.b.551.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.11	✓	16	69.68	even	2
230.3.d.a.91.12	yes	16	3.2	odd	2
1150.3.c.c.1149.15		32	15.8	even	4
1150.3.c.c.1149.16		32	345.137	odd	4
1150.3.c.c.1149.17		32	345.68	odd	4
1150.3.c.c.1149.18		32	15.2	even	4
1150.3.d.b.551.5		16	15.14	odd	2
1150.3.d.b.551.6		16	345.344	even	2
1840.3.k.d.321.11		16	276.275	odd	2
1840.3.k.d.321.12		16	12.11	even	2
2070.3.c.a.91.3		16	1.1	even	1	trivial
2070.3.c.a.91.6		16	23.22	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +1.16919i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +1.16919i q^{7} -2.82843 q^{8} +3.16228i q^{10} -10.6148i q^{11} -15.0913 q^{13} -1.65348i q^{14} +4.00000 q^{16} -20.0887i q^{17} +22.5221i q^{19} -4.47214i q^{20} +15.0116i q^{22} +(20.9683 - 9.45142i) q^{23} -5.00000 q^{25} +21.3424 q^{26} +2.33837i q^{28} -32.5993 q^{29} -27.0975 q^{31} -5.65685 q^{32} +28.4097i q^{34} +2.61438 q^{35} -53.0568i q^{37} -31.8510i q^{38} +6.32456i q^{40} -9.43720 q^{41} +36.4382i q^{43} -21.2296i q^{44} +(-29.6537 + 13.3663i) q^{46} +49.1365 q^{47} +47.6330 q^{49} +7.07107 q^{50} -30.1827 q^{52} +104.253i q^{53} -23.7354 q^{55} -3.30696i q^{56} +46.1023 q^{58} -53.5457 q^{59} -23.5166i q^{61} +38.3217 q^{62} +8.00000 q^{64} +33.7453i q^{65} +59.4754i q^{67} -40.1773i q^{68} -3.69729 q^{70} -55.2130 q^{71} -8.77305 q^{73} +75.0337i q^{74} +45.0441i q^{76} +12.4107 q^{77} +57.0848i q^{79} -8.94427i q^{80} +13.3462 q^{82} +55.1788i q^{83} -44.9196 q^{85} -51.5314i q^{86} +30.0232i q^{88} -139.825i q^{89} -17.6446i q^{91} +(41.9366 - 18.9028i) q^{92} -69.4894 q^{94} +50.3608 q^{95} -19.8635i q^{97} -67.3632 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) \) 16 * q + 32 * q^4 + 24 * q^13 + 64 * q^16 - 4 * q^23 - 80 * q^25 - 96 * q^26 + 108 * q^29 - 116 * q^31 - 60 * q^35 + 156 * q^41 - 124 * q^46 + 128 * q^47 - 28 * q^49 + 48 * q^52 + 160 * q^58 - 204 * q^59 - 64 * q^62 + 128 * q^64 - 120 * q^70 - 236 * q^71 - 112 * q^73 + 936 * q^77 - 64 * q^82 + 60 * q^85 - 8 * q^92 - 216 * q^94 + 160 * q^95 - 256 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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