LMFDB - Embedded newform 4140.3.d.a.2161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4140.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.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:	$112.806829445$
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} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + \cdots + 1535848276 $ x^16 - 64x^14 - 16x^13 + 2252x^12 + 648x^11 - 30106x^10 + 12360x^9 + 374528x^8 + 196544x^7 + 1261236x^6 - 4237944x^5 + 38013345x^4 - 8913096x^3 + 327235800x^2 + 72566336x + 1535848276
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.3
Root			$-4.53300 - 2.23607i$ of defining polynomial
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.a.2161.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -5.73038i q^{7} +O(q^{10})$	$q-2.23607i q^{5} -5.73038i q^{7} -14.4494i q^{11} -22.5859 q^{13} -15.8960i q^{17} -21.5685i q^{19} +(-14.6376 + 17.7410i) q^{23} -5.00000 q^{25} +23.3223 q^{29} +52.1637 q^{31} -12.8135 q^{35} -29.3528i q^{37} -69.3985 q^{41} -59.0128i q^{43} -35.0771 q^{47} +16.1627 q^{49} -55.7924i q^{53} -32.3099 q^{55} -34.7849 q^{59} -35.4314i q^{61} +50.5035i q^{65} -111.151i q^{67} +94.6728 q^{71} +8.86831 q^{73} -82.8008 q^{77} +86.9680i q^{79} +153.870i q^{83} -35.5444 q^{85} +152.838i q^{89} +129.426i q^{91} -48.2286 q^{95} +14.2228i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 12 q^{13} + 14 q^{23} - 80 q^{25} - 90 q^{29} + 10 q^{31} - 30 q^{35} - 186 q^{41} + 320 q^{47} + 2 q^{49} - 120 q^{55} + 90 q^{59} + 238 q^{71} - 280 q^{73} - 324 q^{77} - 30 q^{85} - 80 q^{95}+O(q^{100}) $ 16 * q - 12 * q^13 + 14 * q^23 - 80 * q^25 - 90 * q^29 + 10 * q^31 - 30 * q^35 - 186 * q^41 + 320 * q^47 + 2 * q^49 - 120 * q^55 + 90 * q^59 + 238 * q^71 - 280 * q^73 - 324 * q^77 - 30 * q^85 - 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$1$	$1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$		−	5.73038i		−	0.818626i	−0.912394	−	0.409313i	$-0.865768\pi$
$7$		−	5.73038i		−	0.818626i	0.912394	−	0.409313i	$-0.134232\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	14.4494i		−	1.31359i	−0.754071	−	0.656793i	$-0.771913\pi$
$11$		−	14.4494i		−	1.31359i	0.754071	−	0.656793i	$-0.228087\pi$
$12$		0			0
$13$	−22.5859			−1.73737			−0.868687	−	0.495361i	$-0.835036\pi$
$13$	−22.5859			−1.73737			−0.868687	+	0.495361i	$0.835036\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	15.8960i		−	0.935056i	−0.883978	−	0.467528i	$-0.845145\pi$
$17$		−	15.8960i		−	0.935056i	0.883978	−	0.467528i	$-0.154855\pi$
$18$		0			0
$19$		−	21.5685i		−	1.13518i	−0.823310	−	0.567592i	$-0.807875\pi$
$19$		−	21.5685i		−	1.13518i	0.823310	−	0.567592i	$-0.192125\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−14.6376	+	17.7410i	−0.636416	+	0.771346i
$23$	−14.6376	+	17.7410i	−0.636416	+	0.771346i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	23.3223			0.804216			0.402108	−	0.915592i	$-0.368278\pi$
$29$	23.3223			0.804216			0.402108	+	0.915592i	$0.368278\pi$
$30$		0			0
$31$	52.1637			1.68270			0.841349	−	0.540492i	$-0.181762\pi$
$31$	52.1637			1.68270			0.841349	+	0.540492i	$0.181762\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−12.8135			−0.366101
$36$		0			0
$37$		−	29.3528i		−	0.793319i	−0.917966	−	0.396660i	$-0.870169\pi$
$37$		−	29.3528i		−	0.793319i	0.917966	−	0.396660i	$-0.129831\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−69.3985			−1.69265			−0.846323	−	0.532670i	$-0.821189\pi$
$41$	−69.3985			−1.69265			−0.846323	+	0.532670i	$0.821189\pi$
$42$		0			0
$43$		−	59.0128i		−	1.37239i	−0.727418	−	0.686195i	$-0.759280\pi$
$43$		−	59.0128i		−	1.37239i	0.727418	−	0.686195i	$-0.240720\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−35.0771			−0.746320			−0.373160	−	0.927767i	$-0.621726\pi$
$47$	−35.0771			−0.746320			−0.373160	+	0.927767i	$0.621726\pi$
$48$		0			0
$49$	16.1627			0.329852
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	55.7924i		−	1.05269i	−0.850272	−	0.526343i	$-0.823563\pi$
$53$		−	55.7924i		−	1.05269i	0.850272	−	0.526343i	$-0.176437\pi$
$54$		0			0
$55$	−32.3099			−0.587453
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−34.7849			−0.589575			−0.294788	−	0.955563i	$-0.595249\pi$
$59$	−34.7849			−0.589575			−0.294788	+	0.955563i	$0.595249\pi$
$60$		0			0
$61$		−	35.4314i		−	0.580842i	−0.956899	−	0.290421i	$-0.906205\pi$
$61$		−	35.4314i		−	0.580842i	0.956899	−	0.290421i	$-0.0937954\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			50.5035i			0.776977i
$66$		0			0
$67$		−	111.151i		−	1.65898i	−0.558525	−	0.829488i	$-0.688632\pi$
$67$		−	111.151i		−	1.65898i	0.558525	−	0.829488i	$-0.311368\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	94.6728			1.33342			0.666710	−	0.745317i	$-0.267702\pi$
$71$	94.6728			1.33342			0.666710	+	0.745317i	$0.267702\pi$
$72$		0			0
$73$	8.86831			0.121484			0.0607418	−	0.998154i	$-0.480653\pi$
$73$	8.86831			0.121484			0.0607418	+	0.998154i	$0.480653\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−82.8008			−1.07534
$78$		0			0
$79$			86.9680i			1.10086i	0.834881	+	0.550431i	$0.185536\pi$
$79$			86.9680i			1.10086i	−0.834881	+	0.550431i	$0.814464\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			153.870i			1.85386i	0.375234	+	0.926930i	$0.377562\pi$
$83$			153.870i			1.85386i	−0.375234	+	0.926930i	$0.622438\pi$
$84$		0			0
$85$	−35.5444			−0.418170
$86$		0			0
$87$		0			0
$88$		0			0
$89$			152.838i			1.71728i	0.512578	+	0.858641i	$0.328690\pi$
$89$			152.838i			1.71728i	−0.512578	+	0.858641i	$0.671310\pi$
$90$		0			0
$91$			129.426i			1.42226i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−48.2286			−0.507670
$96$		0			0
$97$			14.2228i			0.146627i	0.997309	+	0.0733136i	$0.0233574\pi$
$97$			14.2228i			0.146627i	−0.997309	+	0.0733136i	$0.976643\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−133.799			−1.32474			−0.662369	−	0.749178i	$-0.730449\pi$
$101$	−133.799			−1.32474			−0.662369	+	0.749178i	$0.730449\pi$
$102$		0			0
$103$			31.1112i			0.302050i	0.988530	+	0.151025i	$0.0482574\pi$
$103$			31.1112i			0.302050i	−0.988530	+	0.151025i	$0.951743\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	149.721i		−	1.39926i	−0.714505	−	0.699630i	$-0.753348\pi$
$107$		−	149.721i		−	1.39926i	0.714505	−	0.699630i	$-0.246652\pi$
$108$		0			0
$109$			157.255i			1.44270i	0.692568	+	0.721352i	$0.256479\pi$
$109$			157.255i			1.44270i	−0.692568	+	0.721352i	$0.743521\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			34.7000i			0.307080i	0.988142	+	0.153540i	$0.0490674\pi$
$113$			34.7000i			0.307080i	−0.988142	+	0.153540i	$0.950933\pi$
$114$		0			0
$115$	39.6700	+	32.7306i	0.344957	+	0.284614i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−91.0899			−0.765461
$120$		0			0
$121$	−87.7863			−0.725507
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	142.267			1.12021			0.560107	−	0.828420i	$-0.310760\pi$
$127$	142.267			1.12021			0.560107	+	0.828420i	$0.310760\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.26810			−0.0478481			−0.0239240	−	0.999714i	$-0.507616\pi$
$131$	−6.26810			−0.0478481			−0.0239240	+	0.999714i	$0.507616\pi$
$132$		0			0
$133$	−123.596			−0.929291
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	76.6062i		−	0.559170i	−0.960121	−	0.279585i	$-0.909803\pi$
$137$		−	76.6062i		−	0.559170i	0.960121	−	0.279585i	$-0.0901968\pi$
$138$		0			0
$139$	−71.9853			−0.517880			−0.258940	−	0.965893i	$-0.583373\pi$
$139$	−71.9853			−0.517880			−0.258940	+	0.965893i	$0.583373\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			326.353i			2.28219i
$144$		0			0
$145$		−	52.1502i		−	0.359657i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	55.2401i		−	0.370739i	−0.982669	−	0.185370i	$-0.940652\pi$
$149$		−	55.2401i		−	0.370739i	0.982669	−	0.185370i	$-0.0593482\pi$
$150$		0			0
$151$	107.049			0.708934			0.354467	−	0.935069i	$-0.384662\pi$
$151$	107.049			0.708934			0.354467	+	0.935069i	$0.384662\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	116.641i		−	0.752526i
$156$		0			0
$157$		−	184.159i		−	1.17298i	−0.809955	−	0.586492i	$-0.800508\pi$
$157$		−	184.159i		−	1.17298i	0.809955	−	0.586492i	$-0.199492\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	101.662	+	83.8788i	0.631444	+	0.520986i
$162$		0			0
$163$	149.086			0.914640			0.457320	−	0.889302i	$-0.348809\pi$
$163$	149.086			0.914640			0.457320	+	0.889302i	$0.348809\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	42.2896			0.253231			0.126616	−	0.991952i	$-0.459589\pi$
$167$	42.2896			0.253231			0.126616	+	0.991952i	$0.459589\pi$
$168$		0			0
$169$	341.121			2.01847
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−134.739			−0.778838			−0.389419	−	0.921061i	$-0.627324\pi$
$173$	−134.739			−0.778838			−0.389419	+	0.921061i	$0.627324\pi$
$174$		0			0
$175$			28.6519i			0.163725i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−17.6581			−0.0986485			−0.0493242	−	0.998783i	$-0.515707\pi$
$179$	−17.6581			−0.0986485			−0.0493242	+	0.998783i	$0.515707\pi$
$180$		0			0
$181$		−	44.8426i		−	0.247749i	−0.992298	−	0.123875i	$-0.960468\pi$
$181$		−	44.8426i		−	0.247749i	0.992298	−	0.123875i	$-0.0395321\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−65.6349			−0.354783
$186$		0			0
$187$	−229.688			−1.22828
$188$		0			0
$189$		0			0
$190$		0			0
$191$			126.936i			0.664584i	0.943177	+	0.332292i	$0.107822\pi$
$191$			126.936i			0.664584i	−0.943177	+	0.332292i	$0.892178\pi$
$192$		0			0
$193$	−239.282			−1.23980			−0.619901	−	0.784680i	$-0.712827\pi$
$193$	−239.282			−1.23980			−0.619901	+	0.784680i	$0.712827\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	139.910			0.710203			0.355102	−	0.934828i	$-0.384446\pi$
$197$	139.910			0.710203			0.355102	+	0.934828i	$0.384446\pi$
$198$		0			0
$199$		−	106.506i		−	0.535207i	−0.963529	−	0.267603i	$-0.913768\pi$
$199$		−	106.506i		−	0.535207i	0.963529	−	0.267603i	$-0.0862317\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	133.646i		−	0.658352i
$204$		0			0
$205$			155.180i			0.756974i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−311.653			−1.49116
$210$		0			0
$211$	−131.377			−0.622638			−0.311319	−	0.950305i	$-0.600771\pi$
$211$	−131.377			−0.622638			−0.311319	+	0.950305i	$0.600771\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−131.957			−0.613751
$216$		0			0
$217$		−	298.918i		−	1.37750i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			359.024i			1.62454i
$222$		0			0
$223$	−152.786			−0.685140			−0.342570	−	0.939492i	$-0.611297\pi$
$223$	−152.786			−0.685140			−0.342570	+	0.939492i	$0.611297\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	76.9755i		−	0.339099i	−0.985522	−	0.169550i	$-0.945769\pi$
$227$		−	76.9755i		−	0.339099i	0.985522	−	0.169550i	$-0.0542313\pi$
$228$		0			0
$229$			183.105i			0.799586i	0.916605	+	0.399793i	$0.130918\pi$
$229$			183.105i			0.799586i	−0.916605	+	0.399793i	$0.869082\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	98.8182			0.424112			0.212056	−	0.977257i	$-0.431984\pi$
$233$	98.8182			0.424112			0.212056	+	0.977257i	$0.431984\pi$
$234$		0			0
$235$			78.4347i			0.333765i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	354.489			1.48322			0.741610	−	0.670832i	$-0.234063\pi$
$239$	354.489			1.48322			0.741610	+	0.670832i	$0.234063\pi$
$240$		0			0
$241$		−	371.085i		−	1.53977i	−0.638182	−	0.769885i	$-0.720313\pi$
$241$		−	371.085i		−	1.53977i	0.638182	−	0.769885i	$-0.279687\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	36.1410i		−	0.147514i
$246$		0			0
$247$			487.143i			1.97224i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			357.639i			1.42486i	0.701744	+	0.712429i	$0.252405\pi$
$251$			357.639i			1.42486i	−0.701744	+	0.712429i	$0.747595\pi$
$252$		0			0
$253$	256.347	+	211.505i	1.01323	+	0.835987i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−127.799			−0.497272			−0.248636	−	0.968597i	$-0.579982\pi$
$257$	−127.799			−0.497272			−0.248636	+	0.968597i	$0.579982\pi$
$258$		0			0
$259$	−168.203			−0.649432
$260$		0			0
$261$		0			0
$262$		0			0
$263$			271.488i			1.03227i	0.856506	+	0.516137i	$0.172630\pi$
$263$			271.488i			1.03227i	−0.856506	+	0.516137i	$0.827370\pi$
$264$		0			0
$265$	−124.756			−0.470776
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−238.833			−0.887855			−0.443928	−	0.896063i	$-0.646415\pi$
$269$	−238.833			−0.887855			−0.443928	+	0.896063i	$0.646415\pi$
$270$		0			0
$271$	115.745			0.427103			0.213551	−	0.976932i	$-0.431497\pi$
$271$	115.745			0.427103			0.213551	+	0.976932i	$0.431497\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			72.2472i			0.262717i
$276$		0			0
$277$	26.2892			0.0949067			0.0474534	−	0.998873i	$-0.484889\pi$
$277$	26.2892			0.0949067			0.0474534	+	0.998873i	$0.484889\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			170.570i			0.607010i	0.952830	+	0.303505i	$0.0981570\pi$
$281$			170.570i			0.607010i	−0.952830	+	0.303505i	$0.901843\pi$
$282$		0			0
$283$			136.360i			0.481836i	0.970545	+	0.240918i	$0.0774485\pi$
$283$			136.360i			0.481836i	−0.970545	+	0.240918i	$0.922551\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			397.680i			1.38564i
$288$		0			0
$289$	36.3186			0.125670
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	175.814i		−	0.600049i	−0.953931	−	0.300025i	$-0.903005\pi$
$293$		−	175.814i		−	0.600049i	0.953931	−	0.300025i	$-0.0969949\pi$
$294$		0			0
$295$			77.7815i			0.263666i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	330.602	−	400.695i	1.10569	−	1.34012i
$300$		0			0
$301$	−338.166			−1.12347
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−79.2270			−0.259761
$306$		0			0
$307$	−45.6453			−0.148682			−0.0743409	−	0.997233i	$-0.523685\pi$
$307$	−45.6453			−0.148682			−0.0743409	+	0.997233i	$0.523685\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−369.398			−1.18778			−0.593888	−	0.804548i	$-0.702408\pi$
$311$	−369.398			−1.18778			−0.593888	+	0.804548i	$0.702408\pi$
$312$		0			0
$313$			176.338i			0.563380i	0.959505	+	0.281690i	$0.0908950\pi$
$313$			176.338i			0.563380i	−0.959505	+	0.281690i	$0.909105\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	190.147			0.599831			0.299916	−	0.953966i	$-0.403041\pi$
$317$	190.147			0.599831			0.299916	+	0.953966i	$0.403041\pi$
$318$		0			0
$319$		−	336.994i		−	1.05641i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−342.852			−1.06146
$324$		0			0
$325$	112.929			0.347475
$326$		0			0
$327$		0			0
$328$		0			0
$329$			201.005i			0.610957i
$330$		0			0
$331$	153.515			0.463792			0.231896	−	0.972741i	$-0.425507\pi$
$331$	153.515			0.463792			0.231896	+	0.972741i	$0.425507\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−248.542			−0.741916
$336$		0			0
$337$			545.713i			1.61933i	0.586895	+	0.809663i	$0.300350\pi$
$337$			545.713i			1.61933i	−0.586895	+	0.809663i	$0.699650\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	753.736i		−	2.21037i
$342$		0			0
$343$		−	373.407i		−	1.08865i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−536.006			−1.54469			−0.772343	−	0.635206i	$-0.780915\pi$
$347$	−536.006			−1.54469			−0.772343	+	0.635206i	$0.780915\pi$
$348$		0			0
$349$	327.902			0.939546			0.469773	−	0.882787i	$-0.344336\pi$
$349$	327.902			0.939546			0.469773	+	0.882787i	$0.344336\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	136.555			0.386841			0.193420	−	0.981116i	$-0.438042\pi$
$353$	136.555			0.386841			0.193420	+	0.981116i	$0.438042\pi$
$354$		0			0
$355$		−	211.695i		−	0.596323i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			199.326i			0.555226i	0.960693	+	0.277613i	$0.0895432\pi$
$359$			199.326i			0.555226i	−0.960693	+	0.277613i	$0.910457\pi$
$360$		0			0
$361$	−104.200			−0.288644
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	19.8301i		−	0.0543291i
$366$		0			0
$367$			451.248i			1.22956i	0.788700	+	0.614779i	$0.210755\pi$
$367$			451.248i			1.22956i	−0.788700	+	0.614779i	$0.789245\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−319.711			−0.861756
$372$		0			0
$373$			566.051i			1.51756i	0.651345	+	0.758782i	$0.274205\pi$
$373$			566.051i			1.51756i	−0.651345	+	0.758782i	$0.725795\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−526.754			−1.39723
$378$		0			0
$379$		−	548.345i		−	1.44682i	−0.690419	−	0.723410i	$-0.742574\pi$
$379$		−	548.345i		−	1.44682i	0.690419	−	0.723410i	$-0.257426\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	275.822i		−	0.720161i	−0.932921	−	0.360080i	$-0.882749\pi$
$383$		−	275.822i		−	0.720161i	0.932921	−	0.360080i	$-0.117251\pi$
$384$		0			0
$385$			185.148i			0.480904i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	19.2474i		−	0.0494791i	−0.999694	−	0.0247395i	$-0.992124\pi$
$389$		−	19.2474i		−	0.0494791i	0.999694	−	0.0247395i	$-0.00787564\pi$
$390$		0			0
$391$	282.010	+	232.678i	0.721252	+	0.595085i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	194.466			0.492320
$396$		0			0
$397$	345.136			0.869360			0.434680	−	0.900585i	$-0.356861\pi$
$397$	345.136			0.869360			0.434680	+	0.900585i	$0.356861\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			493.627i			1.23099i	0.788141	+	0.615495i	$0.211044\pi$
$401$			493.627i			1.23099i	−0.788141	+	0.615495i	$0.788956\pi$
$402$		0			0
$403$	−1178.16			−2.92348
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−424.132			−1.04209
$408$		0			0
$409$	−435.568			−1.06496			−0.532479	−	0.846443i	$-0.678740\pi$
$409$	−435.568			−1.06496			−0.532479	+	0.846443i	$0.678740\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			199.331i			0.482642i
$414$		0			0
$415$	344.065			0.829071
$416$		0			0
$417$		0			0
$418$		0			0
$419$			655.748i			1.56503i	0.622631	+	0.782515i	$0.286064\pi$
$419$			655.748i			1.56503i	−0.622631	+	0.782515i	$0.713936\pi$
$420$		0			0
$421$		−	136.328i		−	0.323818i	−0.986806	−	0.161909i	$-0.948235\pi$
$421$		−	136.328i		−	0.323818i	0.986806	−	0.161909i	$-0.0517652\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			79.4798i			0.187011i
$426$		0			0
$427$	−203.035			−0.475493
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	246.087i		−	0.570969i	−0.958383	−	0.285484i	$-0.907846\pi$
$431$		−	246.087i		−	0.570969i	0.958383	−	0.285484i	$-0.0921544\pi$
$432$		0			0
$433$		−	582.571i		−	1.34543i	−0.739902	−	0.672715i	$-0.765128\pi$
$433$		−	582.571i		−	1.34543i	0.739902	−	0.672715i	$-0.234872\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	382.646	+	315.710i	0.875620	+	0.722449i
$438$		0			0
$439$	−708.320			−1.61348			−0.806742	−	0.590903i	$-0.798771\pi$
$439$	−708.320			−1.61348			−0.806742	+	0.590903i	$0.798771\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−338.574			−0.764274			−0.382137	−	0.924106i	$-0.624812\pi$
$443$	−338.574			−0.764274			−0.382137	+	0.924106i	$0.624812\pi$
$444$		0			0
$445$	341.756			0.767991
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−276.872			−0.616641			−0.308320	−	0.951283i	$-0.599767\pi$
$449$	−276.872			−0.616641			−0.308320	+	0.951283i	$0.599767\pi$
$450$		0			0
$451$			1002.77i			2.22344i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	289.405			0.636054
$456$		0			0
$457$		−	641.187i		−	1.40304i	−0.712652	−	0.701518i	$-0.752506\pi$
$457$		−	641.187i		−	1.40304i	0.712652	−	0.701518i	$-0.247494\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−204.088			−0.442707			−0.221354	−	0.975194i	$-0.571048\pi$
$461$	−204.088			−0.442707			−0.221354	+	0.975194i	$0.571048\pi$
$462$		0			0
$463$	−596.319			−1.28795			−0.643973	−	0.765048i	$-0.722715\pi$
$463$	−596.319			−1.28795			−0.643973	+	0.765048i	$0.722715\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			144.242i			0.308870i	0.988003	+	0.154435i	$0.0493558\pi$
$467$			144.242i			0.308870i	−0.988003	+	0.154435i	$0.950644\pi$
$468$		0			0
$469$	−636.940			−1.35808
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−852.701			−1.80275
$474$		0			0
$475$			107.843i			0.227037i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	153.818i		−	0.321124i	−0.987026	−	0.160562i	$-0.948669\pi$
$479$		−	153.818i		−	0.321124i	0.987026	−	0.160562i	$-0.0513307\pi$
$480$		0			0
$481$			662.959i			1.37829i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	31.8032			0.0655736
$486$		0			0
$487$	−612.573			−1.25785			−0.628925	−	0.777466i	$-0.716505\pi$
$487$	−612.573			−1.25785			−0.628925	+	0.777466i	$0.716505\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	612.246			1.24694			0.623468	−	0.781849i	$-0.285723\pi$
$491$	612.246			1.24694			0.623468	+	0.781849i	$0.285723\pi$
$492$		0			0
$493$		−	370.730i		−	0.751988i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	542.511i		−	1.09157i
$498$		0			0
$499$	−639.084			−1.28073			−0.640364	−	0.768071i	$-0.721217\pi$
$499$	−639.084			−1.28073			−0.640364	+	0.768071i	$0.721217\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			797.548i			1.58558i	0.609494	+	0.792791i	$0.291373\pi$
$503$			797.548i			1.58558i	−0.609494	+	0.792791i	$0.708627\pi$
$504$		0			0
$505$			299.183i			0.592441i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	380.000			0.746563			0.373281	−	0.927718i	$-0.378233\pi$
$509$	380.000			0.746563			0.373281	+	0.927718i	$0.378233\pi$
$510$		0			0
$511$		−	50.8188i		−	0.0994497i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	69.5667			0.135081
$516$		0			0
$517$			506.844i			0.980356i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			82.2165i			0.157805i	0.996882	+	0.0789026i	$0.0251416\pi$
$521$			82.2165i			0.157805i	−0.996882	+	0.0789026i	$0.974858\pi$
$522$		0			0
$523$			529.301i			1.01205i	0.862519	+	0.506024i	$0.168885\pi$
$523$			529.301i			1.01205i	−0.862519	+	0.506024i	$0.831115\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	829.191i		−	1.57342i
$528$		0			0
$529$	−100.484	−	519.369i	−0.189950	−	0.981794i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1567.43			2.94076
$534$		0			0
$535$	−334.786			−0.625768
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	233.542i		−	0.433288i
$540$		0			0
$541$	1077.12			1.99097			0.995487	−	0.0948980i	$-0.0302525\pi$
$541$	1077.12			1.99097			0.995487	+	0.0948980i	$0.0302525\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	351.632			0.645197
$546$		0			0
$547$	538.450			0.984369			0.492184	−	0.870491i	$-0.336199\pi$
$547$	538.450			0.984369			0.492184	+	0.870491i	$0.336199\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	503.027i		−	0.912934i
$552$		0			0
$553$	498.360			0.901193
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	543.116i		−	0.975074i	−0.873102	−	0.487537i	$-0.837895\pi$
$557$		−	543.116i		−	0.975074i	0.873102	−	0.487537i	$-0.162105\pi$
$558$		0			0
$559$			1332.85i			2.38435i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	263.998i		−	0.468912i	−0.972127	−	0.234456i	$-0.924669\pi$
$563$		−	263.998i		−	0.468912i	0.972127	−	0.234456i	$-0.0753310\pi$
$564$		0			0
$565$	77.5917			0.137330
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	75.5139i		−	0.132713i	−0.997796	−	0.0663567i	$-0.978862\pi$
$569$		−	75.5139i		−	0.132713i	0.997796	−	0.0663567i	$-0.0211375\pi$
$570$		0			0
$571$			252.684i			0.442529i	0.975214	+	0.221264i	$0.0710184\pi$
$571$			252.684i			0.442529i	−0.975214	+	0.221264i	$0.928982\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	73.1878	−	88.7048i	0.127283	−	0.154269i
$576$		0			0
$577$	443.113			0.767961			0.383980	−	0.923341i	$-0.374553\pi$
$577$	443.113			0.767961			0.383980	+	0.923341i	$0.374553\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	881.736			1.51762
$582$		0			0
$583$	−806.168			−1.38279
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−946.434			−1.61232			−0.806162	−	0.591695i	$-0.798459\pi$
$587$	−946.434			−1.61232			−0.806162	+	0.591695i	$0.798459\pi$
$588$		0			0
$589$		−	1125.09i		−	1.91017i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	349.409			0.589222			0.294611	−	0.955617i	$-0.404810\pi$
$593$	349.409			0.589222			0.294611	+	0.955617i	$0.404810\pi$
$594$		0			0
$595$			203.683i			0.342325i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−448.083			−0.748052			−0.374026	−	0.927418i	$-0.622023\pi$
$599$	−448.083			−0.748052			−0.374026	+	0.927418i	$0.622023\pi$
$600$		0			0
$601$	−467.984			−0.778676			−0.389338	−	0.921095i	$-0.627296\pi$
$601$	−467.984			−0.778676			−0.389338	+	0.921095i	$0.627296\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			196.296i			0.324457i
$606$		0			0
$607$	666.697			1.09835			0.549174	−	0.835708i	$-0.314942\pi$
$607$	666.697			1.09835			0.549174	+	0.835708i	$0.314942\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	792.246			1.29664
$612$		0			0
$613$		−	482.025i		−	0.786338i	−0.919466	−	0.393169i	$-0.871379\pi$
$613$		−	482.025i		−	0.786338i	0.919466	−	0.393169i	$-0.128621\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			137.235i			0.222422i	0.993797	+	0.111211i	$0.0354730\pi$
$617$			137.235i			0.222422i	−0.993797	+	0.111211i	$0.964527\pi$
$618$		0			0
$619$		−	470.924i		−	0.760783i	−0.924826	−	0.380391i	$-0.875789\pi$
$619$		−	470.924i		−	0.760783i	0.924826	−	0.380391i	$-0.124211\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	875.820			1.40581
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−466.591			−0.741798
$630$		0			0
$631$		−	791.573i		−	1.25447i	−0.778829	−	0.627237i	$-0.784186\pi$
$631$		−	791.573i		−	1.25447i	0.778829	−	0.627237i	$-0.215814\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	318.119i		−	0.500975i
$636$		0			0
$637$	−365.049			−0.573076
$638$		0			0
$639$		0			0
$640$		0			0
$641$			587.063i			0.915855i	0.888990	+	0.457927i	$0.151408\pi$
$641$			587.063i			0.915855i	−0.888990	+	0.457927i	$0.848592\pi$
$642$		0			0
$643$		−	1071.84i		−	1.66693i	−0.552573	−	0.833464i	$-0.686354\pi$
$643$		−	1071.84i		−	1.66693i	0.552573	−	0.833464i	$-0.313646\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−276.415			−0.427226			−0.213613	−	0.976918i	$-0.568523\pi$
$647$	−276.415			−0.427226			−0.213613	+	0.976918i	$0.568523\pi$
$648$		0			0
$649$			502.623i			0.774457i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−248.340			−0.380307			−0.190153	−	0.981754i	$-0.560899\pi$
$653$	−248.340			−0.380307			−0.190153	+	0.981754i	$0.560899\pi$
$654$		0			0
$655$			14.0159i			0.0213983i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			211.399i			0.320787i	0.987053	+	0.160394i	$0.0512763\pi$
$659$			211.399i			0.320787i	−0.987053	+	0.160394i	$0.948724\pi$
$660$		0			0
$661$			843.142i			1.27556i	0.770221	+	0.637778i	$0.220146\pi$
$661$			843.142i			1.27556i	−0.770221	+	0.637778i	$0.779854\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			276.369i			0.415592i
$666$		0			0
$667$	−341.381	+	413.760i	−0.511816	+	0.620329i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−511.964			−0.762986
$672$		0			0
$673$	1117.23			1.66008			0.830041	−	0.557703i	$-0.188317\pi$
$673$	1117.23			1.66008			0.830041	+	0.557703i	$0.188317\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	503.627i		−	0.743911i	−0.928251	−	0.371955i	$-0.878687\pi$
$677$		−	503.627i		−	0.743911i	0.928251	−	0.371955i	$-0.121313\pi$
$678$		0			0
$679$	81.5022			0.120033
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1117.09			−1.63557			−0.817783	−	0.575526i	$-0.804797\pi$
$683$	−1117.09			−1.63557			−0.817783	+	0.575526i	$0.804797\pi$
$684$		0			0
$685$	−171.297			−0.250068
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1260.12i			1.82891i
$690$		0			0
$691$	−373.797			−0.540951			−0.270476	−	0.962727i	$-0.587181\pi$
$691$	−373.797			−0.540951			−0.270476	+	0.962727i	$0.587181\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			160.964i			0.231603i
$696$		0			0
$697$			1103.16i			1.58272i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	392.530i		−	0.559957i	−0.960006	−	0.279978i	$-0.909673\pi$
$701$		−	392.530i		−	0.559957i	0.960006	−	0.279978i	$-0.0903273\pi$
$702$		0			0
$703$	−633.096			−0.900564
$704$		0			0
$705$		0			0
$706$		0			0
$707$			766.717i			1.08446i
$708$		0			0
$709$		−	1069.06i		−	1.50784i	−0.656964	−	0.753922i	$-0.728160\pi$
$709$		−	1069.06i		−	1.50784i	0.656964	−	0.753922i	$-0.271840\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−763.549	+	925.433i	−1.07090	+	1.29794i
$714$		0			0
$715$	729.748			1.02063
$716$		0			0
$717$		0			0
$718$		0			0
$719$	520.733			0.724247			0.362123	−	0.932130i	$-0.382052\pi$
$719$	520.733			0.724247			0.362123	+	0.932130i	$0.382052\pi$
$720$		0			0
$721$	178.279			0.247266
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−116.611			−0.160843
$726$		0			0
$727$		−	149.011i		−	0.204967i	−0.994735	−	0.102483i	$-0.967321\pi$
$727$		−	149.011i		−	0.204967i	0.994735	−	0.102483i	$-0.0326788\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−938.064			−1.28326
$732$		0			0
$733$			161.168i			0.219875i	0.993939	+	0.109937i	$0.0350650\pi$
$733$			161.168i			0.219875i	−0.993939	+	0.109937i	$0.964935\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1606.07			−2.17921
$738$		0			0
$739$	−722.812			−0.978095			−0.489048	−	0.872257i	$-0.662656\pi$
$739$	−722.812			−0.978095			−0.489048	+	0.872257i	$0.662656\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	585.251i		−	0.787687i	−0.919178	−	0.393843i	$-0.871145\pi$
$743$		−	585.251i		−	0.787687i	0.919178	−	0.393843i	$-0.128855\pi$
$744$		0			0
$745$	−123.521			−0.165800
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−857.958			−1.14547
$750$		0			0
$751$		−	248.757i		−	0.331234i	−0.986190	−	0.165617i	$-0.947038\pi$
$751$		−	248.757i		−	0.331234i	0.986190	−	0.165617i	$-0.0529615\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	239.369i		−	0.317045i
$756$		0			0
$757$		−	147.079i		−	0.194292i	−0.995270	−	0.0971459i	$-0.969029\pi$
$757$		−	147.079i		−	0.194292i	0.995270	−	0.0971459i	$-0.0309714\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	715.969			0.940826			0.470413	−	0.882446i	$-0.344105\pi$
$761$	715.969			0.940826			0.470413	+	0.882446i	$0.344105\pi$
$762$		0			0
$763$	901.130			1.18104
$764$		0			0
$765$		0			0
$766$		0			0
$767$	785.648			1.02431
$768$		0			0
$769$		−	563.688i		−	0.733014i	−0.930415	−	0.366507i	$-0.880554\pi$
$769$		−	563.688i		−	0.733014i	0.930415	−	0.366507i	$-0.119446\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	776.912i		−	1.00506i	−0.864560	−	0.502530i	$-0.832403\pi$
$773$		−	776.912i		−	1.00506i	0.864560	−	0.502530i	$-0.167597\pi$
$774$		0			0
$775$	−260.818			−0.336540
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1496.82i			1.92147i
$780$		0			0
$781$		−	1367.97i		−	1.75156i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−411.791			−0.524575
$786$		0			0
$787$		−	700.182i		−	0.889685i	−0.895609	−	0.444843i	$-0.853260\pi$
$787$		−	700.182i		−	0.889685i	0.895609	−	0.444843i	$-0.146740\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	198.844			0.251384
$792$		0			0
$793$			800.249i			1.00914i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			973.433i			1.22137i	0.791873	+	0.610686i	$0.209106\pi$
$797$			973.433i			1.22137i	−0.791873	+	0.610686i	$0.790894\pi$
$798$		0			0
$799$			557.583i			0.697852i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	128.142i		−	0.159579i
$804$		0			0
$805$	187.559	−	227.324i	0.232992	−	0.282390i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−437.494			−0.540784			−0.270392	−	0.962750i	$-0.587153\pi$
$809$	−437.494			−0.540784			−0.270392	+	0.962750i	$0.587153\pi$
$810$		0			0
$811$	1277.85			1.57564			0.787821	−	0.615904i	$-0.211209\pi$
$811$	1277.85			1.57564			0.787821	+	0.615904i	$0.211209\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	333.367i		−	0.409039i
$816$		0			0
$817$	−1272.82			−1.55792
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−592.870			−0.722131			−0.361066	−	0.932540i	$-0.617587\pi$
$821$	−592.870			−0.722131			−0.361066	+	0.932540i	$0.617587\pi$
$822$		0			0
$823$	−107.071			−0.130099			−0.0650494	−	0.997882i	$-0.520721\pi$
$823$	−107.071			−0.130099			−0.0650494	+	0.997882i	$0.520721\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	700.568i		−	0.847119i	−0.905868	−	0.423560i	$-0.860780\pi$
$827$		−	700.568i		−	0.847119i	0.905868	−	0.423560i	$-0.139220\pi$
$828$		0			0
$829$	−1195.25			−1.44179			−0.720897	−	0.693042i	$-0.756270\pi$
$829$	−1195.25			−1.44179			−0.720897	+	0.693042i	$0.756270\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	256.922i		−	0.308430i
$834$		0			0
$835$		−	94.5624i		−	0.113248i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	173.900i		−	0.207270i	−0.994615	−	0.103635i	$-0.966953\pi$
$839$		−	173.900i		−	0.207270i	0.994615	−	0.103635i	$-0.0330474\pi$
$840$		0			0
$841$	−297.071			−0.353236
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	762.771i		−	0.902687i
$846$		0			0
$847$			503.049i			0.593919i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	520.747	+	429.654i	0.611924	+	0.504881i
$852$		0			0
$853$	1443.06			1.69174			0.845872	−	0.533386i	$-0.179081\pi$
$853$	1443.06			1.69174			0.845872	+	0.533386i	$0.179081\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−368.637			−0.430148			−0.215074	−	0.976598i	$-0.568999\pi$
$857$	−368.637			−0.430148			−0.215074	+	0.976598i	$0.568999\pi$
$858$		0			0
$859$	−7.48252			−0.00871073			−0.00435537	−	0.999991i	$-0.501386\pi$
$859$	−7.48252			−0.00871073			−0.00435537	+	0.999991i	$0.501386\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	304.740			0.353117			0.176558	−	0.984290i	$-0.443504\pi$
$863$	304.740			0.353117			0.176558	+	0.984290i	$0.443504\pi$
$864$		0			0
$865$			301.286i			0.348307i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1256.64			1.44608
$870$		0			0
$871$			2510.45i			2.88226i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	64.0676			0.0732201
$876$		0			0
$877$	−188.312			−0.214723			−0.107361	−	0.994220i	$-0.534240\pi$
$877$	−188.312			−0.214723			−0.107361	+	0.994220i	$0.534240\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	213.922i		−	0.242817i	−0.992603	−	0.121409i	$-0.961259\pi$
$881$		−	213.922i		−	0.242817i	0.992603	−	0.121409i	$-0.0387411\pi$
$882$		0			0
$883$	−727.655			−0.824071			−0.412036	−	0.911168i	$-0.635182\pi$
$883$	−727.655			−0.824071			−0.412036	+	0.911168i	$0.635182\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	926.692			1.04475			0.522374	−	0.852716i	$-0.325046\pi$
$887$	926.692			1.04475			0.522374	+	0.852716i	$0.325046\pi$
$888$		0			0
$889$		−	815.245i		−	0.917036i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			756.560i			0.847211i
$894$		0			0
$895$			39.4847i			0.0441169i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1216.58			1.35325
$900$		0			0
$901$	−886.873			−0.984321
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−100.271			−0.110797
$906$		0			0
$907$		−	509.932i		−	0.562218i	−0.959676	−	0.281109i	$-0.909298\pi$
$907$		−	509.932i		−	0.562218i	0.959676	−	0.281109i	$-0.0907023\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	241.322i		−	0.264898i	−0.991190	−	0.132449i	$-0.957716\pi$
$911$		−	241.322i		−	0.264898i	0.991190	−	0.132449i	$-0.0422841\pi$
$912$		0			0
$913$	2223.34			2.43520
$914$		0			0
$915$		0			0
$916$		0			0
$917$			35.9186i			0.0391697i
$918$		0			0
$919$		−	489.612i		−	0.532766i	−0.963867	−	0.266383i	$-0.914171\pi$
$919$		−	489.612i		−	0.532766i	0.963867	−	0.266383i	$-0.0858286\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2138.27			−2.31665
$924$		0			0
$925$			146.764i			0.158664i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	337.438			0.363227			0.181613	−	0.983370i	$-0.441868\pi$
$929$	337.438			0.363227			0.181613	+	0.983370i	$0.441868\pi$
$930$		0			0
$931$		−	348.606i		−	0.374442i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			513.597i			0.549302i
$936$		0			0
$937$		−	578.923i		−	0.617847i	−0.951087	−	0.308923i	$-0.900031\pi$
$937$		−	578.923i		−	0.617847i	0.951087	−	0.308923i	$-0.0999687\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			685.415i			0.728390i	0.931323	+	0.364195i	$0.118656\pi$
$941$			685.415i			0.728390i	−0.931323	+	0.364195i	$0.881344\pi$
$942$		0			0
$943$	1015.82	−	1231.20i	1.07723	−	1.30562i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	603.328			0.637094			0.318547	−	0.947907i	$-0.396805\pi$
$947$	603.328			0.637094			0.318547	+	0.947907i	$0.396805\pi$
$948$		0			0
$949$	−200.298			−0.211063
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	898.707i		−	0.943030i	−0.881858	−	0.471515i	$-0.843707\pi$
$953$		−	898.707i		−	0.943030i	0.881858	−	0.471515i	$-0.156293\pi$
$954$		0			0
$955$	283.837			0.297211
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−438.983			−0.457751
$960$		0			0
$961$	1760.05			1.83147
$962$		0			0
$963$		0			0
$964$		0			0
$965$			535.050i			0.554456i
$966$		0			0
$967$	62.7154			0.0648556			0.0324278	−	0.999474i	$-0.489676\pi$
$967$	62.7154			0.0648556			0.0324278	+	0.999474i	$0.489676\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1127.41i			1.16108i	0.814232	+	0.580540i	$0.197158\pi$
$971$			1127.41i			1.16108i	−0.814232	+	0.580540i	$0.802842\pi$
$972$		0			0
$973$			412.503i			0.423950i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	709.445i		−	0.726147i	−0.931761	−	0.363073i	$-0.881727\pi$
$977$		−	709.445i		−	0.726147i	0.931761	−	0.363073i	$-0.118273\pi$
$978$		0			0
$979$	2208.42			2.25580
$980$		0			0
$981$		0			0
$982$		0			0
$983$			409.376i			0.416456i	0.978080	+	0.208228i	$0.0667696\pi$
$983$			409.376i			0.416456i	−0.978080	+	0.208228i	$0.933230\pi$
$984$		0			0
$985$		−	312.848i		−	0.317612i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1046.94	+	863.803i	1.05859	+	0.873410i
$990$		0			0
$991$	192.336			0.194083			0.0970414	−	0.995280i	$-0.469062\pi$
$991$	192.336			0.194083			0.0970414	+	0.995280i	$0.469062\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−238.155			−0.239352
$996$		0			0
$997$	1426.03			1.43032			0.715160	−	0.698961i	$-0.246354\pi$
$997$	1426.03			1.43032			0.715160	+	0.698961i	$0.246354\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.a.2161.3		16
3.2	odd	2		460.3.f.a.321.4	yes	16
12.11	even	2		1840.3.k.c.321.14		16
15.2	even	4		2300.3.d.b.1149.24		32
15.8	even	4		2300.3.d.b.1149.9		32
15.14	odd	2		2300.3.f.e.1701.14		16
23.22	odd	2	inner	4140.3.d.a.2161.14		16
69.68	even	2		460.3.f.a.321.3	✓	16
276.275	odd	2		1840.3.k.c.321.13		16
345.68	odd	4		2300.3.d.b.1149.23		32
345.137	odd	4		2300.3.d.b.1149.10		32
345.344	even	2		2300.3.f.e.1701.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.3	✓	16	69.68	even	2
460.3.f.a.321.4	yes	16	3.2	odd	2
1840.3.k.c.321.13		16	276.275	odd	2
1840.3.k.c.321.14		16	12.11	even	2
2300.3.d.b.1149.9		32	15.8	even	4
2300.3.d.b.1149.10		32	345.137	odd	4
2300.3.d.b.1149.23		32	345.68	odd	4
2300.3.d.b.1149.24		32	15.2	even	4
2300.3.f.e.1701.13		16	345.344	even	2
2300.3.f.e.1701.14		16	15.14	odd	2
4140.3.d.a.2161.3		16	1.1	even	1	trivial
4140.3.d.a.2161.14		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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -5.73038i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} -5.73038i q^{7} -14.4494i q^{11} -22.5859 q^{13} -15.8960i q^{17} -21.5685i q^{19} +(-14.6376 + 17.7410i) q^{23} -5.00000 q^{25} +23.3223 q^{29} +52.1637 q^{31} -12.8135 q^{35} -29.3528i q^{37} -69.3985 q^{41} -59.0128i q^{43} -35.0771 q^{47} +16.1627 q^{49} -55.7924i q^{53} -32.3099 q^{55} -34.7849 q^{59} -35.4314i q^{61} +50.5035i q^{65} -111.151i q^{67} +94.6728 q^{71} +8.86831 q^{73} -82.8008 q^{77} +86.9680i q^{79} +153.870i q^{83} -35.5444 q^{85} +152.838i q^{89} +129.426i q^{91} -48.2286 q^{95} +14.2228i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 12 q^{13} + 14 q^{23} - 80 q^{25} - 90 q^{29} + 10 q^{31} - 30 q^{35} - 186 q^{41} + 320 q^{47} + 2 q^{49} - 120 q^{55} + 90 q^{59} + 238 q^{71} - 280 q^{73} - 324 q^{77} - 30 q^{85} - 80 q^{95}+O(q^{100}) \) 16 * q - 12 * q^13 + 14 * q^23 - 80 * q^25 - 90 * q^29 + 10 * q^31 - 30 * q^35 - 186 * q^41 + 320 * q^47 + 2 * q^49 - 120 * q^55 + 90 * q^59 + 238 * q^71 - 280 * q^73 - 324 * q^77 - 30 * q^85 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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