LMFDB - Embedded newform 1200.3.bg.i.1057.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$, degree $2$, 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:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1057.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.1057
Dual form			1200.3.bg.i.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{3} +(3.67423 + 3.67423i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(1.22474 - 1.22474i) q^{3} +(3.67423 + 3.67423i) q^{7} -3.00000i q^{9} -6.00000 q^{11} +(-6.12372 + 6.12372i) q^{13} +(-22.0454 - 22.0454i) q^{17} +25.0000i q^{19} +9.00000 q^{21} +(-7.34847 + 7.34847i) q^{23} +(-3.67423 - 3.67423i) q^{27} -42.0000i q^{29} -49.0000 q^{31} +(-7.34847 + 7.34847i) q^{33} +(-4.89898 - 4.89898i) q^{37} +15.0000i q^{39} -60.0000 q^{41} +(-1.22474 + 1.22474i) q^{43} +(51.4393 + 51.4393i) q^{47} -22.0000i q^{49} -54.0000 q^{51} +(14.6969 - 14.6969i) q^{53} +(30.6186 + 30.6186i) q^{57} -78.0000i q^{59} -13.0000 q^{61} +(11.0227 - 11.0227i) q^{63} +(-52.6640 - 52.6640i) q^{67} +18.0000i q^{69} +60.0000 q^{71} +(-63.6867 + 63.6867i) q^{73} +(-22.0454 - 22.0454i) q^{77} -106.000i q^{79} -9.00000 q^{81} +(-80.8332 + 80.8332i) q^{83} +(-51.4393 - 51.4393i) q^{87} +60.0000i q^{89} -45.0000 q^{91} +(-60.0125 + 60.0125i) q^{93} +(121.250 + 121.250i) q^{97} +18.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{11} + 36 q^{21} - 196 q^{31} - 240 q^{41} - 216 q^{51} - 52 q^{61} + 240 q^{71} - 36 q^{81} - 180 q^{91}+O(q^{100}) $ 4 * q - 24 * q^11 + 36 * q^21 - 196 * q^31 - 240 * q^41 - 216 * q^51 - 52 * q^61 + 240 * q^71 - 36 * q^81 - 180 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.67423	+	3.67423i	0.524891	+	0.524891i	0.919044	−	0.394154i	$-0.128962\pi$
$7$	3.67423	+	3.67423i	0.524891	+	0.524891i	−0.394154	+	0.919044i	$0.628962\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	−6.00000			−0.545455			−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000			−0.545455			−0.272727	+	0.962091i	$0.587926\pi$
$12$		0			0
$13$	−6.12372	+	6.12372i	−0.471056	+	0.471056i	−0.902256	−	0.431200i	$-0.858090\pi$
$13$	−6.12372	+	6.12372i	−0.471056	+	0.471056i	0.431200	+	0.902256i	$0.358090\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−22.0454	−	22.0454i	−1.29679	−	1.29679i	−0.930502	−	0.366287i	$-0.880629\pi$
$17$	−22.0454	−	22.0454i	−1.29679	−	1.29679i	−0.366287	−	0.930502i	$-0.619371\pi$
$18$		0			0
$19$			25.0000i			1.31579i	0.753110	+	0.657895i	$0.228553\pi$
$19$			25.0000i			1.31579i	−0.753110	+	0.657895i	$0.771447\pi$
$20$		0			0
$21$	9.00000			0.428571
$22$		0			0
$23$	−7.34847	+	7.34847i	−0.319499	+	0.319499i	−0.848575	−	0.529076i	$-0.822539\pi$
$23$	−7.34847	+	7.34847i	−0.319499	+	0.319499i	0.529076	+	0.848575i	$0.322539\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.136083	−	0.136083i
$28$		0			0
$29$		−	42.0000i		−	1.44828i	−0.689655	−	0.724138i	$-0.742238\pi$
$29$		−	42.0000i		−	1.44828i	0.689655	−	0.724138i	$-0.257762\pi$
$30$		0			0
$31$	−49.0000			−1.58065			−0.790323	−	0.612691i	$-0.790087\pi$
$31$	−49.0000			−1.58065			−0.790323	+	0.612691i	$0.790087\pi$
$32$		0			0
$33$	−7.34847	+	7.34847i	−0.222681	+	0.222681i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.89898	−	4.89898i	−0.132405	−	0.132405i	0.637798	−	0.770203i	$-0.279845\pi$
$37$	−4.89898	−	4.89898i	−0.132405	−	0.132405i	−0.770203	+	0.637798i	$0.779845\pi$
$38$		0			0
$39$			15.0000i			0.384615i
$40$		0			0
$41$	−60.0000			−1.46341			−0.731707	−	0.681619i	$-0.761276\pi$
$41$	−60.0000			−1.46341			−0.731707	+	0.681619i	$0.761276\pi$
$42$		0			0
$43$	−1.22474	+	1.22474i	−0.0284824	+	0.0284824i	−0.721205	−	0.692722i	$-0.756411\pi$
$43$	−1.22474	+	1.22474i	−0.0284824	+	0.0284824i	0.692722	+	0.721205i	$0.256411\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	51.4393	+	51.4393i	1.09445	+	1.09445i	0.995047	+	0.0994059i	$0.0316942\pi$
$47$	51.4393	+	51.4393i	1.09445	+	1.09445i	0.0994059	+	0.995047i	$0.468306\pi$
$48$		0			0
$49$		−	22.0000i		−	0.448980i
$50$		0			0
$51$	−54.0000			−1.05882
$52$		0			0
$53$	14.6969	−	14.6969i	0.277301	−	0.277301i	−0.554730	−	0.832031i	$-0.687178\pi$
$53$	14.6969	−	14.6969i	0.277301	−	0.277301i	0.832031	+	0.554730i	$0.187178\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	30.6186	+	30.6186i	0.537169	+	0.537169i
$58$		0			0
$59$		−	78.0000i		−	1.32203i	−0.750371	−	0.661017i	$-0.770125\pi$
$59$		−	78.0000i		−	1.32203i	0.750371	−	0.661017i	$-0.229875\pi$
$60$		0			0
$61$	−13.0000			−0.213115			−0.106557	−	0.994307i	$-0.533983\pi$
$61$	−13.0000			−0.213115			−0.106557	+	0.994307i	$0.533983\pi$
$62$		0			0
$63$	11.0227	−	11.0227i	0.174964	−	0.174964i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−52.6640	−	52.6640i	−0.786030	−	0.786030i	0.194811	−	0.980841i	$-0.437591\pi$
$67$	−52.6640	−	52.6640i	−0.786030	−	0.786030i	−0.980841	+	0.194811i	$0.937591\pi$
$68$		0			0
$69$			18.0000i			0.260870i
$70$		0			0
$71$	60.0000			0.845070			0.422535	−	0.906347i	$-0.361140\pi$
$71$	60.0000			0.845070			0.422535	+	0.906347i	$0.361140\pi$
$72$		0			0
$73$	−63.6867	+	63.6867i	−0.872421	+	0.872421i	−0.992736	−	0.120315i	$-0.961610\pi$
$73$	−63.6867	+	63.6867i	−0.872421	+	0.872421i	0.120315	+	0.992736i	$0.461610\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22.0454	−	22.0454i	−0.286304	−	0.286304i
$78$		0			0
$79$		−	106.000i		−	1.34177i	−0.741560	−	0.670886i	$-0.765914\pi$
$79$		−	106.000i		−	1.34177i	0.741560	−	0.670886i	$-0.234086\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−80.8332	+	80.8332i	−0.973894	+	0.973894i	−0.999668	−	0.0257743i	$-0.991795\pi$
$83$	−80.8332	+	80.8332i	−0.973894	+	0.973894i	0.0257743	+	0.999668i	$0.491795\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−51.4393	−	51.4393i	−0.591256	−	0.591256i
$88$		0			0
$89$			60.0000i			0.674157i	0.941476	+	0.337079i	$0.109439\pi$
$89$			60.0000i			0.674157i	−0.941476	+	0.337079i	$0.890561\pi$
$90$		0			0
$91$	−45.0000			−0.494505
$92$		0			0
$93$	−60.0125	+	60.0125i	−0.645296	+	0.645296i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	121.250	+	121.250i	1.25000	+	1.25000i	0.955720	+	0.294277i	$0.0950789\pi$
$97$	121.250	+	121.250i	1.25000	+	1.25000i	0.294277	+	0.955720i	$0.404921\pi$
$98$		0			0
$99$			18.0000i			0.181818i
$100$		0			0
$101$	−192.000			−1.90099			−0.950495	−	0.310740i	$-0.899423\pi$
$101$	−192.000			−1.90099			−0.950495	+	0.310740i	$0.899423\pi$
$102$		0			0
$103$	−78.3837	+	78.3837i	−0.761007	+	0.761007i	−0.976504	−	0.215498i	$-0.930863\pi$
$103$	−78.3837	+	78.3837i	−0.761007	+	0.761007i	0.215498	+	0.976504i	$0.430863\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	29.3939	+	29.3939i	0.274709	+	0.274709i	0.830993	−	0.556283i	$-0.187773\pi$
$107$	29.3939	+	29.3939i	0.274709	+	0.274709i	−0.556283	+	0.830993i	$0.687773\pi$
$108$		0			0
$109$			1.00000i			0.00917431i	0.999989	+	0.00458716i	$0.00146014\pi$
$109$			1.00000i			0.00917431i	−0.999989	+	0.00458716i	$0.998540\pi$
$110$		0			0
$111$	−12.0000			−0.108108
$112$		0			0
$113$	58.7878	−	58.7878i	0.520246	−	0.520246i	−0.397400	−	0.917646i	$-0.630087\pi$
$113$	58.7878	−	58.7878i	0.520246	−	0.520246i	0.917646	+	0.397400i	$0.130087\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	18.3712	+	18.3712i	0.157019	+	0.157019i
$118$		0			0
$119$		−	162.000i		−	1.36134i
$120$		0			0
$121$	−85.0000			−0.702479
$122$		0			0
$123$	−73.4847	+	73.4847i	−0.597437	+	0.597437i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−14.6969	−	14.6969i	−0.115724	−	0.115724i	0.646873	−	0.762597i	$-0.276076\pi$
$127$	−14.6969	−	14.6969i	−0.115724	−	0.115724i	−0.762597	+	0.646873i	$0.776076\pi$
$128$		0			0
$129$			3.00000i			0.0232558i
$130$		0			0
$131$	−12.0000			−0.0916031			−0.0458015	−	0.998951i	$-0.514584\pi$
$131$	−12.0000			−0.0916031			−0.0458015	+	0.998951i	$0.514584\pi$
$132$		0			0
$133$	−91.8559	+	91.8559i	−0.690646	+	0.690646i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	80.8332	+	80.8332i	0.590023	+	0.590023i	0.937638	−	0.347614i	$-0.113008\pi$
$137$	80.8332	+	80.8332i	0.590023	+	0.590023i	−0.347614	+	0.937638i	$0.613008\pi$
$138$		0			0
$139$			58.0000i			0.417266i	0.977994	+	0.208633i	$0.0669014\pi$
$139$			58.0000i			0.417266i	−0.977994	+	0.208633i	$0.933099\pi$
$140$		0			0
$141$	126.000			0.893617
$142$		0			0
$143$	36.7423	−	36.7423i	0.256939	−	0.256939i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−26.9444	−	26.9444i	−0.183295	−	0.183295i
$148$		0			0
$149$		−	234.000i		−	1.57047i	−0.619198	−	0.785235i	$-0.712542\pi$
$149$		−	234.000i		−	1.57047i	0.619198	−	0.785235i	$-0.287458\pi$
$150$		0			0
$151$	−85.0000			−0.562914			−0.281457	−	0.959574i	$-0.590818\pi$
$151$	−85.0000			−0.562914			−0.281457	+	0.959574i	$0.590818\pi$
$152$		0			0
$153$	−66.1362	+	66.1362i	−0.432263	+	0.432263i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−74.7094	−	74.7094i	−0.475856	−	0.475856i	0.427947	−	0.903804i	$-0.359237\pi$
$157$	−74.7094	−	74.7094i	−0.475856	−	0.475856i	−0.903804	+	0.427947i	$0.859237\pi$
$158$		0			0
$159$		−	36.0000i		−	0.226415i
$160$		0			0
$161$	−54.0000			−0.335404
$162$		0			0
$163$	−79.6084	+	79.6084i	−0.488395	+	0.488395i	−0.907800	−	0.419404i	$-0.862239\pi$
$163$	−79.6084	+	79.6084i	−0.488395	+	0.488395i	0.419404	+	0.907800i	$0.362239\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	176.363	+	176.363i	1.05607	+	1.05607i	0.998332	+	0.0577355i	$0.0183880\pi$
$167$	176.363	+	176.363i	1.05607	+	1.05607i	0.0577355	+	0.998332i	$0.481612\pi$
$168$		0			0
$169$			94.0000i			0.556213i
$170$		0			0
$171$	75.0000			0.438596
$172$		0			0
$173$	−80.8332	+	80.8332i	−0.467244	+	0.467244i	−0.901020	−	0.433777i	$-0.857181\pi$
$173$	−80.8332	+	80.8332i	−0.467244	+	0.467244i	0.433777	+	0.901020i	$0.357181\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−95.5301	−	95.5301i	−0.539718	−	0.539718i
$178$		0			0
$179$			42.0000i			0.234637i	0.993094	+	0.117318i	$0.0374298\pi$
$179$			42.0000i			0.234637i	−0.993094	+	0.117318i	$0.962570\pi$
$180$		0			0
$181$	145.000			0.801105			0.400552	−	0.916274i	$-0.368818\pi$
$181$	145.000			0.801105			0.400552	+	0.916274i	$0.368818\pi$
$182$		0			0
$183$	−15.9217	+	15.9217i	−0.0870037	+	0.0870037i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	132.272	+	132.272i	0.707339	+	0.707339i
$188$		0			0
$189$		−	27.0000i		−	0.142857i
$190$		0			0
$191$	−294.000			−1.53927			−0.769634	−	0.638486i	$-0.779561\pi$
$191$	−294.000			−1.53927			−0.769634	+	0.638486i	$0.779561\pi$
$192$		0			0
$193$	−25.7196	+	25.7196i	−0.133262	+	0.133262i	−0.770592	−	0.637329i	$-0.780039\pi$
$193$	−25.7196	+	25.7196i	−0.133262	+	0.133262i	0.637329	+	0.770592i	$0.280039\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	22.0454	+	22.0454i	0.111906	+	0.111906i	0.760842	−	0.648937i	$-0.224786\pi$
$197$	22.0454	+	22.0454i	0.111906	+	0.111906i	−0.648937	+	0.760842i	$0.724786\pi$
$198$		0			0
$199$			179.000i			0.899497i	0.893155	+	0.449749i	$0.148486\pi$
$199$			179.000i			0.899497i	−0.893155	+	0.449749i	$0.851514\pi$
$200$		0			0
$201$	−129.000			−0.641791
$202$		0			0
$203$	154.318	−	154.318i	0.760186	−	0.760186i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	22.0454	+	22.0454i	0.106500	+	0.106500i
$208$		0			0
$209$		−	150.000i		−	0.717703i
$210$		0			0
$211$	85.0000			0.402844			0.201422	−	0.979505i	$-0.435444\pi$
$211$	85.0000			0.402844			0.201422	+	0.979505i	$0.435444\pi$
$212$		0			0
$213$	73.4847	−	73.4847i	0.344999	−	0.344999i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−180.037	−	180.037i	−0.829666	−	0.829666i
$218$		0			0
$219$			156.000i			0.712329i
$220$		0			0
$221$	270.000			1.22172
$222$		0			0
$223$	104.103	−	104.103i	0.466831	−	0.466831i	−0.434055	−	0.900886i	$-0.642918\pi$
$223$	104.103	−	104.103i	0.466831	−	0.466831i	0.900886	+	0.434055i	$0.142918\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	373.000i		−	1.62882i	−0.580289	−	0.814410i	$-0.697060\pi$
$229$		−	373.000i		−	1.62882i	0.580289	−	0.814410i	$-0.302940\pi$
$230$		0			0
$231$	−54.0000			−0.233766
$232$		0			0
$233$	220.454	−	220.454i	0.946155	−	0.946155i	−0.0524678	−	0.998623i	$-0.516709\pi$
$233$	220.454	−	220.454i	0.946155	−	0.946155i	0.998623	+	0.0524678i	$0.0167087\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−129.823	−	129.823i	−0.547776	−	0.547776i
$238$		0			0
$239$		−	36.0000i		−	0.150628i	−0.997160	−	0.0753138i	$-0.976004\pi$
$239$		−	36.0000i		−	0.150628i	0.997160	−	0.0753138i	$-0.0239959\pi$
$240$		0			0
$241$	−97.0000			−0.402490			−0.201245	−	0.979541i	$-0.564499\pi$
$241$	−97.0000			−0.402490			−0.201245	+	0.979541i	$0.564499\pi$
$242$		0			0
$243$	−11.0227	+	11.0227i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−153.093	−	153.093i	−0.619810	−	0.619810i
$248$		0			0
$249$			198.000i			0.795181i
$250$		0			0
$251$	−288.000			−1.14741			−0.573705	−	0.819062i	$-0.694494\pi$
$251$	−288.000			−1.14741			−0.573705	+	0.819062i	$0.694494\pi$
$252$		0			0
$253$	44.0908	−	44.0908i	0.174272	−	0.174272i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	338.030	+	338.030i	1.31529	+	1.31529i	0.917458	+	0.397832i	$0.130237\pi$
$257$	338.030	+	338.030i	1.31529	+	1.31529i	0.397832	+	0.917458i	$0.369763\pi$
$258$		0			0
$259$		−	36.0000i		−	0.138996i
$260$		0			0
$261$	−126.000			−0.482759
$262$		0			0
$263$	249.848	−	249.848i	0.949992	−	0.949992i	−0.0488156	−	0.998808i	$-0.515545\pi$
$263$	249.848	−	249.848i	0.949992	−	0.949992i	0.998808	+	0.0488156i	$0.0155447\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	73.4847	+	73.4847i	0.275224	+	0.275224i
$268$		0			0
$269$		−	318.000i		−	1.18216i	−0.806614	−	0.591078i	$-0.798703\pi$
$269$		−	318.000i		−	1.18216i	0.806614	−	0.591078i	$-0.201297\pi$
$270$		0			0
$271$	226.000			0.833948			0.416974	−	0.908918i	$-0.363091\pi$
$271$	226.000			0.833948			0.416974	+	0.908918i	$0.363091\pi$
$272$		0			0
$273$	−55.1135	+	55.1135i	−0.201881	+	0.201881i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−380.896	−	380.896i	−1.37507	−	1.37507i	−0.852742	−	0.522333i	$-0.825062\pi$
$277$	−380.896	−	380.896i	−1.37507	−	1.37507i	−0.522333	−	0.852742i	$-0.674938\pi$
$278$		0			0
$279$			147.000i			0.526882i
$280$		0			0
$281$	126.000			0.448399			0.224199	−	0.974543i	$-0.428023\pi$
$281$	126.000			0.448399			0.224199	+	0.974543i	$0.428023\pi$
$282$		0			0
$283$	−395.593	+	395.593i	−1.39785	+	1.39785i	−0.591683	+	0.806171i	$0.701536\pi$
$283$	−395.593	+	395.593i	−1.39785	+	1.39785i	−0.806171	+	0.591683i	$0.798464\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−220.454	−	220.454i	−0.768133	−	0.768133i
$288$		0			0
$289$			683.000i			2.36332i
$290$		0			0
$291$	297.000			1.02062
$292$		0			0
$293$	22.0454	−	22.0454i	0.0752403	−	0.0752403i	−0.668485	−	0.743725i	$-0.733057\pi$
$293$	22.0454	−	22.0454i	0.0752403	−	0.0752403i	0.743725	+	0.668485i	$0.233057\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	22.0454	+	22.0454i	0.0742270	+	0.0742270i
$298$		0			0
$299$		−	90.0000i		−	0.301003i
$300$		0			0
$301$	−9.00000			−0.0299003
$302$		0			0
$303$	−235.151	+	235.151i	−0.776076	+	0.776076i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−86.9569	−	86.9569i	−0.283247	−	0.283247i	0.551155	−	0.834403i	$-0.314187\pi$
$307$	−86.9569	−	86.9569i	−0.283247	−	0.283247i	−0.834403	+	0.551155i	$0.814187\pi$
$308$		0			0
$309$			192.000i			0.621359i
$310$		0			0
$311$	246.000			0.790997			0.395498	−	0.918467i	$-0.370572\pi$
$311$	246.000			0.790997			0.395498	+	0.918467i	$0.370572\pi$
$312$		0			0
$313$	−37.9671	+	37.9671i	−0.121301	+	0.121301i	−0.765151	−	0.643851i	$-0.777336\pi$
$313$	−37.9671	+	37.9671i	−0.121301	+	0.121301i	0.643851	+	0.765151i	$0.277336\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.6969	+	14.6969i	0.0463626	+	0.0463626i	0.729908	−	0.683545i	$-0.239563\pi$
$317$	14.6969	+	14.6969i	0.0463626	+	0.0463626i	−0.683545	+	0.729908i	$0.739563\pi$
$318$		0			0
$319$			252.000i			0.789969i
$320$		0			0
$321$	72.0000			0.224299
$322$		0			0
$323$	551.135	−	551.135i	1.70630	−	1.70630i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.22474	+	1.22474i	0.00374540	+	0.00374540i
$328$		0			0
$329$			378.000i			1.14894i
$330$		0			0
$331$	178.000			0.537764			0.268882	−	0.963173i	$-0.413346\pi$
$331$	178.000			0.537764			0.268882	+	0.963173i	$0.413346\pi$
$332$		0			0
$333$	−14.6969	+	14.6969i	−0.0441350	+	0.0441350i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	417.638	+	417.638i	1.23928	+	1.23928i	0.960295	+	0.278987i	$0.0899987\pi$
$337$	417.638	+	417.638i	1.23928	+	1.23928i	0.278987	+	0.960295i	$0.410001\pi$
$338$		0			0
$339$		−	144.000i		−	0.424779i
$340$		0			0
$341$	294.000			0.862170
$342$		0			0
$343$	260.871	−	260.871i	0.760556	−	0.760556i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−448.257	−	448.257i	−1.29181	−	1.29181i	−0.933669	−	0.358137i	$-0.883412\pi$
$347$	−448.257	−	448.257i	−1.29181	−	1.29181i	−0.358137	−	0.933669i	$-0.616588\pi$
$348$		0			0
$349$			542.000i			1.55301i	0.630112	+	0.776504i	$0.283009\pi$
$349$			542.000i			1.55301i	−0.630112	+	0.776504i	$0.716991\pi$
$350$		0			0
$351$	45.0000			0.128205
$352$		0			0
$353$	242.499	−	242.499i	0.686967	−	0.686967i	−0.274593	−	0.961561i	$-0.588543\pi$
$353$	242.499	−	242.499i	0.686967	−	0.686967i	0.961561	+	0.274593i	$0.0885432\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−198.409	−	198.409i	−0.555767	−	0.555767i
$358$		0			0
$359$		−	408.000i		−	1.13649i	−0.822859	−	0.568245i	$-0.807623\pi$
$359$		−	408.000i		−	1.13649i	0.822859	−	0.568245i	$-0.192377\pi$
$360$		0			0
$361$	−264.000			−0.731302
$362$		0			0
$363$	−104.103	+	104.103i	−0.286786	+	0.286786i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	199.633	+	199.633i	0.543960	+	0.543960i	0.924687	−	0.380727i	$-0.124326\pi$
$367$	199.633	+	199.633i	0.543960	+	0.543960i	−0.380727	+	0.924687i	$0.624326\pi$
$368$		0			0
$369$			180.000i			0.487805i
$370$		0			0
$371$	108.000			0.291105
$372$		0			0
$373$	319.658	−	319.658i	0.856993	−	0.856993i	−0.133990	−	0.990983i	$-0.542779\pi$
$373$	319.658	−	319.658i	0.856993	−	0.856993i	0.990983	+	0.133990i	$0.0427789\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	257.196	+	257.196i	0.682219	+	0.682219i
$378$		0			0
$379$		−	71.0000i		−	0.187335i	−0.995604	−	0.0936675i	$-0.970141\pi$
$379$		−	71.0000i		−	0.187335i	0.995604	−	0.0936675i	$-0.0298591\pi$
$380$		0			0
$381$	−36.0000			−0.0944882
$382$		0			0
$383$	−58.7878	+	58.7878i	−0.153493	+	0.153493i	−0.779676	−	0.626183i	$-0.784616\pi$
$383$	−58.7878	+	58.7878i	−0.153493	+	0.153493i	0.626183	+	0.779676i	$0.284616\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.67423	+	3.67423i	0.00949415	+	0.00949415i
$388$		0			0
$389$		−	288.000i		−	0.740360i	−0.928960	−	0.370180i	$-0.879296\pi$
$389$		−	288.000i		−	0.740360i	0.928960	−	0.370180i	$-0.120704\pi$
$390$		0			0
$391$	324.000			0.828645
$392$		0			0
$393$	−14.6969	+	14.6969i	−0.0373968	+	0.0373968i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	74.7094	+	74.7094i	0.188185	+	0.188185i	0.794911	−	0.606726i	$-0.207517\pi$
$397$	74.7094	+	74.7094i	0.188185	+	0.188185i	−0.606726	+	0.794911i	$0.707517\pi$
$398$		0			0
$399$			225.000i			0.563910i
$400$		0			0
$401$	36.0000			0.0897756			0.0448878	−	0.998992i	$-0.485707\pi$
$401$	36.0000			0.0897756			0.0448878	+	0.998992i	$0.485707\pi$
$402$		0			0
$403$	300.062	−	300.062i	0.744572	−	0.744572i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	29.3939	+	29.3939i	0.0722208	+	0.0722208i
$408$		0			0
$409$		−	13.0000i		−	0.0317848i	−0.999874	−	0.0158924i	$-0.994941\pi$
$409$		−	13.0000i		−	0.0317848i	0.999874	−	0.0158924i	$-0.00505893\pi$
$410$		0			0
$411$	198.000			0.481752
$412$		0			0
$413$	286.590	−	286.590i	0.693923	−	0.693923i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	71.0352	+	71.0352i	0.170348	+	0.170348i
$418$		0			0
$419$		−	144.000i		−	0.343675i	−0.985125	−	0.171838i	$-0.945030\pi$
$419$		−	144.000i		−	0.343675i	0.985125	−	0.171838i	$-0.0549705\pi$
$420$		0			0
$421$	610.000			1.44893			0.724466	−	0.689311i	$-0.242087\pi$
$421$	610.000			1.44893			0.724466	+	0.689311i	$0.242087\pi$
$422$		0			0
$423$	154.318	−	154.318i	0.364818	−	0.364818i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−47.7650	−	47.7650i	−0.111862	−	0.111862i
$428$		0			0
$429$		−	90.0000i		−	0.209790i
$430$		0			0
$431$	402.000			0.932715			0.466357	−	0.884596i	$-0.345566\pi$
$431$	402.000			0.932715			0.466357	+	0.884596i	$0.345566\pi$
$432$		0			0
$433$	292.714	−	292.714i	0.676014	−	0.676014i	−0.283082	−	0.959096i	$-0.591357\pi$
$433$	292.714	−	292.714i	0.676014	−	0.676014i	0.959096	+	0.283082i	$0.0913568\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−183.712	−	183.712i	−0.420393	−	0.420393i
$438$		0			0
$439$			119.000i			0.271071i	0.990773	+	0.135535i	$0.0432754\pi$
$439$			119.000i			0.271071i	−0.990773	+	0.135535i	$0.956725\pi$
$440$		0			0
$441$	−66.0000			−0.149660
$442$		0			0
$443$	191.060	−	191.060i	0.431287	−	0.431287i	−0.457779	−	0.889066i	$-0.651355\pi$
$443$	191.060	−	191.060i	0.431287	−	0.431287i	0.889066	+	0.457779i	$0.151355\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−286.590	−	286.590i	−0.641142	−	0.641142i
$448$		0			0
$449$		−	84.0000i		−	0.187082i	−0.995615	−	0.0935412i	$-0.970181\pi$
$449$		−	84.0000i		−	0.187082i	0.995615	−	0.0935412i	$-0.0298187\pi$
$450$		0			0
$451$	360.000			0.798226
$452$		0			0
$453$	−104.103	+	104.103i	−0.229809	+	0.229809i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−146.969	−	146.969i	−0.321596	−	0.321596i	0.527783	−	0.849379i	$-0.323023\pi$
$457$	−146.969	−	146.969i	−0.321596	−	0.321596i	−0.849379	+	0.527783i	$0.823023\pi$
$458$		0			0
$459$			162.000i			0.352941i
$460$		0			0
$461$	516.000			1.11931			0.559653	−	0.828727i	$-0.310934\pi$
$461$	516.000			1.11931			0.559653	+	0.828727i	$0.310934\pi$
$462$		0			0
$463$	−524.191	+	524.191i	−1.13216	+	1.13216i	−0.142344	+	0.989817i	$0.545464\pi$
$463$	−524.191	+	524.191i	−1.13216	+	1.13216i	−0.989817	+	0.142344i	$0.954536\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	507.044	+	507.044i	1.08575	+	1.08575i	0.995961	+	0.0897872i	$0.0286187\pi$
$467$	507.044	+	507.044i	1.08575	+	1.08575i	0.0897872	+	0.995961i	$0.471381\pi$
$468$		0			0
$469$		−	387.000i		−	0.825160i
$470$		0			0
$471$	−183.000			−0.388535
$472$		0			0
$473$	7.34847	−	7.34847i	0.0155359	−	0.0155359i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−44.0908	−	44.0908i	−0.0924336	−	0.0924336i
$478$		0			0
$479$			102.000i			0.212944i	0.994316	+	0.106472i	$0.0339554\pi$
$479$			102.000i			0.212944i	−0.994316	+	0.106472i	$0.966045\pi$
$480$		0			0
$481$	60.0000			0.124740
$482$		0			0
$483$	−66.1362	+	66.1362i	−0.136928	+	0.136928i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−243.724	−	243.724i	−0.500460	−	0.500460i	0.411121	−	0.911581i	$-0.365138\pi$
$487$	−243.724	−	243.724i	−0.500460	−	0.500460i	−0.911581	+	0.411121i	$0.865138\pi$
$488$		0			0
$489$			195.000i			0.398773i
$490$		0			0
$491$	−444.000			−0.904277			−0.452138	−	0.891948i	$-0.649339\pi$
$491$	−444.000			−0.904277			−0.452138	+	0.891948i	$0.649339\pi$
$492$		0			0
$493$	−925.907	+	925.907i	−1.87811	+	1.87811i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	220.454	+	220.454i	0.443570	+	0.443570i
$498$		0			0
$499$			707.000i			1.41683i	0.705794	+	0.708417i	$0.250590\pi$
$499$			707.000i			1.41683i	−0.705794	+	0.708417i	$0.749410\pi$
$500$		0			0
$501$	432.000			0.862275
$502$		0			0
$503$	−176.363	+	176.363i	−0.350623	+	0.350623i	−0.860341	−	0.509718i	$-0.829750\pi$
$503$	−176.363	+	176.363i	−0.350623	+	0.350623i	0.509718	+	0.860341i	$0.329750\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	115.126	+	115.126i	0.227073	+	0.227073i
$508$		0			0
$509$		−	666.000i		−	1.30845i	−0.756301	−	0.654224i	$-0.772995\pi$
$509$		−	666.000i		−	1.30845i	0.756301	−	0.654224i	$-0.227005\pi$
$510$		0			0
$511$	−468.000			−0.915851
$512$		0			0
$513$	91.8559	−	91.8559i	0.179056	−	0.179056i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−308.636	−	308.636i	−0.596974	−	0.596974i
$518$		0			0
$519$			198.000i			0.381503i
$520$		0			0
$521$	−558.000			−1.07102			−0.535509	−	0.844530i	$-0.679880\pi$
$521$	−558.000			−1.07102			−0.535509	+	0.844530i	$0.679880\pi$
$522$		0			0
$523$	547.461	−	547.461i	1.04677	−	1.04677i	0.0479193	−	0.998851i	$-0.484741\pi$
$523$	547.461	−	547.461i	1.04677	−	1.04677i	0.998851	−	0.0479193i	$-0.0152590\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1080.22	+	1080.22i	2.04976	+	2.04976i
$528$		0			0
$529$			421.000i			0.795841i
$530$		0			0
$531$	−234.000			−0.440678
$532$		0			0
$533$	367.423	−	367.423i	0.689350	−	0.689350i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	51.4393	+	51.4393i	0.0957901	+	0.0957901i
$538$		0			0
$539$			132.000i			0.244898i
$540$		0			0
$541$	803.000			1.48429			0.742144	−	0.670240i	$-0.233809\pi$
$541$	803.000			1.48429			0.742144	+	0.670240i	$0.233809\pi$
$542$		0			0
$543$	177.588	−	177.588i	0.327050	−	0.327050i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	215.555	+	215.555i	0.394068	+	0.394068i	0.876135	−	0.482067i	$-0.160114\pi$
$547$	215.555	+	215.555i	0.394068	+	0.394068i	−0.482067	+	0.876135i	$0.660114\pi$
$548$		0			0
$549$			39.0000i			0.0710383i
$550$		0			0
$551$	1050.00			1.90563
$552$		0			0
$553$	389.469	−	389.469i	0.704284	−	0.704284i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−448.257	−	448.257i	−0.804770	−	0.804770i	0.179067	−	0.983837i	$-0.442692\pi$
$557$	−448.257	−	448.257i	−0.804770	−	0.804770i	−0.983837	+	0.179067i	$0.942692\pi$
$558$		0			0
$559$		−	15.0000i		−	0.0268336i
$560$		0			0
$561$	324.000			0.577540
$562$		0			0
$563$	257.196	−	257.196i	0.456832	−	0.456832i	−0.440782	−	0.897614i	$-0.645299\pi$
$563$	257.196	−	257.196i	0.456832	−	0.456832i	0.897614	+	0.440782i	$0.145299\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−33.0681	−	33.0681i	−0.0583212	−	0.0583212i
$568$		0			0
$569$			582.000i			1.02285i	0.859329	+	0.511424i	$0.170882\pi$
$569$			582.000i			1.02285i	−0.859329	+	0.511424i	$0.829118\pi$
$570$		0			0
$571$	−215.000			−0.376532			−0.188266	−	0.982118i	$-0.560287\pi$
$571$	−215.000			−0.376532			−0.188266	+	0.982118i	$0.560287\pi$
$572$		0			0
$573$	−360.075	+	360.075i	−0.628403	+	0.628403i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−246.174	−	246.174i	−0.426644	−	0.426644i	0.460839	−	0.887484i	$-0.347548\pi$
$577$	−246.174	−	246.174i	−0.426644	−	0.426644i	−0.887484	+	0.460839i	$0.847548\pi$
$578$		0			0
$579$			63.0000i			0.108808i
$580$		0			0
$581$	−594.000			−1.02238
$582$		0			0
$583$	−88.1816	+	88.1816i	−0.151255	+	0.151255i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.6969	−	14.6969i	−0.0250374	−	0.0250374i	0.694477	−	0.719515i	$-0.255636\pi$
$587$	−14.6969	−	14.6969i	−0.0250374	−	0.0250374i	−0.719515	+	0.694477i	$0.755636\pi$
$588$		0			0
$589$		−	1225.00i		−	2.07980i
$590$		0			0
$591$	54.0000			0.0913706
$592$		0			0
$593$	−323.333	+	323.333i	−0.545249	+	0.545249i	−0.925063	−	0.379814i	$-0.875988\pi$
$593$	−323.333	+	323.333i	−0.545249	+	0.545249i	0.379814	+	0.925063i	$0.375988\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	219.229	+	219.229i	0.367218	+	0.367218i
$598$		0			0
$599$			528.000i			0.881469i	0.897637	+	0.440735i	$0.145282\pi$
$599$			528.000i			0.881469i	−0.897637	+	0.440735i	$0.854718\pi$
$600$		0			0
$601$	493.000			0.820300			0.410150	−	0.912018i	$-0.365476\pi$
$601$	493.000			0.820300			0.410150	+	0.912018i	$0.365476\pi$
$602$		0			0
$603$	−157.992	+	157.992i	−0.262010	+	0.262010i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−729.948	−	729.948i	−1.20255	−	1.20255i	−0.973388	−	0.229162i	$-0.926402\pi$
$607$	−729.948	−	729.948i	−1.20255	−	1.20255i	−0.229162	−	0.973388i	$-0.573598\pi$
$608$		0			0
$609$		−	378.000i		−	0.620690i
$610$		0			0
$611$	−630.000			−1.03110
$612$		0			0
$613$	−112.677	+	112.677i	−0.183812	+	0.183812i	−0.793014	−	0.609203i	$-0.791490\pi$
$613$	−112.677	+	112.677i	−0.183812	+	0.183812i	0.609203	+	0.793014i	$0.291490\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−382.120	−	382.120i	−0.619320	−	0.619320i	0.326037	−	0.945357i	$-0.394287\pi$
$617$	−382.120	−	382.120i	−0.619320	−	0.619320i	−0.945357	+	0.326037i	$0.894287\pi$
$618$		0			0
$619$		−	733.000i		−	1.18417i	−0.805876	−	0.592084i	$-0.798305\pi$
$619$		−	733.000i		−	1.18417i	0.805876	−	0.592084i	$-0.201695\pi$
$620$		0			0
$621$	54.0000			0.0869565
$622$		0			0
$623$	−220.454	+	220.454i	−0.353859	+	0.353859i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−183.712	−	183.712i	−0.293001	−	0.293001i
$628$		0			0
$629$			216.000i			0.343402i
$630$		0			0
$631$	−647.000			−1.02536			−0.512678	−	0.858581i	$-0.671347\pi$
$631$	−647.000			−1.02536			−0.512678	+	0.858581i	$0.671347\pi$
$632$		0			0
$633$	104.103	−	104.103i	0.164460	−	0.164460i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	134.722	+	134.722i	0.211494	+	0.211494i
$638$		0			0
$639$		−	180.000i		−	0.281690i
$640$		0			0
$641$	−636.000			−0.992200			−0.496100	−	0.868265i	$-0.665235\pi$
$641$	−636.000			−0.992200			−0.496100	+	0.868265i	$0.665235\pi$
$642$		0			0
$643$	−396.817	+	396.817i	−0.617134	+	0.617134i	−0.944795	−	0.327661i	$-0.893740\pi$
$643$	−396.817	+	396.817i	−0.617134	+	0.617134i	0.327661	+	0.944795i	$0.393740\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.3939	+	29.3939i	0.0454310	+	0.0454310i	0.729457	−	0.684026i	$-0.239773\pi$
$647$	29.3939	+	29.3939i	0.0454310	+	0.0454310i	−0.684026	+	0.729457i	$0.739773\pi$
$648$		0			0
$649$			468.000i			0.721109i
$650$		0			0
$651$	−441.000			−0.677419
$652$		0			0
$653$	521.741	−	521.741i	0.798991	−	0.798991i	−0.183945	−	0.982936i	$-0.558887\pi$
$653$	521.741	−	521.741i	0.798991	−	0.798991i	0.982936	+	0.183945i	$0.0588869\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	191.060	+	191.060i	0.290807	+	0.290807i
$658$		0			0
$659$		−	36.0000i		−	0.0546282i	−0.999627	−	0.0273141i	$-0.991305\pi$
$659$		−	36.0000i		−	0.0546282i	0.999627	−	0.0273141i	$-0.00869543\pi$
$660$		0			0
$661$	−1022.00			−1.54614			−0.773071	−	0.634319i	$-0.781280\pi$
$661$	−1022.00			−1.54614			−0.773071	+	0.634319i	$0.781280\pi$
$662$		0			0
$663$	330.681	−	330.681i	0.498765	−	0.498765i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	308.636	+	308.636i	0.462722	+	0.462722i
$668$		0			0
$669$		−	255.000i		−	0.381166i
$670$		0			0
$671$	78.0000			0.116244
$672$		0			0
$673$	−857.321	+	857.321i	−1.27388	+	1.27388i	−0.329845	+	0.944035i	$0.606997\pi$
$673$	−857.321	+	857.321i	−1.27388	+	1.27388i	−0.944035	+	0.329845i	$0.893003\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−536.438	−	536.438i	−0.792376	−	0.792376i	0.189504	−	0.981880i	$-0.439312\pi$
$677$	−536.438	−	536.438i	−0.792376	−	0.792376i	−0.981880	+	0.189504i	$0.939312\pi$
$678$		0			0
$679$			891.000i			1.31222i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−764.241	+	764.241i	−1.11895	+	1.11895i	−0.127051	+	0.991896i	$0.540551\pi$
$683$	−764.241	+	764.241i	−1.11895	+	1.11895i	−0.991896	+	0.127051i	$0.959449\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−456.830	−	456.830i	−0.664963	−	0.664963i
$688$		0			0
$689$			180.000i			0.261248i
$690$		0			0
$691$	842.000			1.21852			0.609262	−	0.792969i	$-0.291466\pi$
$691$	842.000			1.21852			0.609262	+	0.792969i	$0.291466\pi$
$692$		0			0
$693$	−66.1362	+	66.1362i	−0.0954347	+	0.0954347i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1322.72	+	1322.72i	1.89774	+	1.89774i
$698$		0			0
$699$		−	540.000i		−	0.772532i
$700$		0			0
$701$	−534.000			−0.761769			−0.380884	−	0.924623i	$-0.624380\pi$
$701$	−534.000			−0.761769			−0.380884	+	0.924623i	$0.624380\pi$
$702$		0			0
$703$	122.474	−	122.474i	0.174217	−	0.174217i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−705.453	−	705.453i	−0.997812	−	0.997812i
$708$		0			0
$709$			743.000i			1.04795i	0.851732	+	0.523977i	$0.175552\pi$
$709$			743.000i			1.04795i	−0.851732	+	0.523977i	$0.824448\pi$
$710$		0			0
$711$	−318.000			−0.447257
$712$		0			0
$713$	360.075	−	360.075i	0.505014	−	0.505014i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−44.0908	−	44.0908i	−0.0614935	−	0.0614935i
$718$		0			0
$719$		−	906.000i		−	1.26008i	−0.776561	−	0.630042i	$-0.783038\pi$
$719$		−	906.000i		−	1.26008i	0.776561	−	0.630042i	$-0.216962\pi$
$720$		0			0
$721$	−576.000			−0.798890
$722$		0			0
$723$	−118.800	+	118.800i	−0.164316	+	0.164316i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	74.7094	+	74.7094i	0.102764	+	0.102764i	0.756619	−	0.653855i	$-0.226850\pi$
$727$	74.7094	+	74.7094i	0.102764	+	0.102764i	−0.653855	+	0.756619i	$0.726850\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	54.0000			0.0738714
$732$		0			0
$733$	631.968	−	631.968i	0.862167	−	0.862167i	−0.129423	−	0.991590i	$-0.541312\pi$
$733$	631.968	−	631.968i	0.862167	−	0.862167i	0.991590	+	0.129423i	$0.0413124\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	315.984	+	315.984i	0.428744	+	0.428744i
$738$		0			0
$739$		−	470.000i		−	0.635995i	−0.948092	−	0.317997i	$-0.896990\pi$
$739$		−	470.000i		−	0.635995i	0.948092	−	0.317997i	$-0.103010\pi$
$740$		0			0
$741$	−375.000			−0.506073
$742$		0			0
$743$	−249.848	+	249.848i	−0.336269	+	0.336269i	−0.854961	−	0.518692i	$-0.826419\pi$
$743$	−249.848	+	249.848i	−0.336269	+	0.336269i	0.518692	+	0.854961i	$0.326419\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	242.499	+	242.499i	0.324631	+	0.324631i
$748$		0			0
$749$			216.000i			0.288385i
$750$		0			0
$751$	−1282.00			−1.70706			−0.853529	−	0.521046i	$-0.825542\pi$
$751$	−1282.00			−1.70706			−0.853529	+	0.521046i	$0.825542\pi$
$752$		0			0
$753$	−352.727	+	352.727i	−0.468428	+	0.468428i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−106.553	−	106.553i	−0.140757	−	0.140757i	0.633217	−	0.773974i	$-0.281734\pi$
$757$	−106.553	−	106.553i	−0.140757	−	0.140757i	−0.773974	+	0.633217i	$0.781734\pi$
$758$		0			0
$759$		−	108.000i		−	0.142292i
$760$		0			0
$761$	−768.000			−1.00920			−0.504599	−	0.863354i	$-0.668360\pi$
$761$	−768.000			−1.00920			−0.504599	+	0.863354i	$0.668360\pi$
$762$		0			0
$763$	−3.67423	+	3.67423i	−0.00481551	+	0.00481551i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	477.650	+	477.650i	0.622752	+	0.622752i
$768$		0			0
$769$		−	491.000i		−	0.638492i	−0.947672	−	0.319246i	$-0.896570\pi$
$769$		−	491.000i		−	0.638492i	0.947672	−	0.319246i	$-0.103430\pi$
$770$		0			0
$771$	828.000			1.07393
$772$		0			0
$773$	−793.635	+	793.635i	−1.02669	+	1.02669i	−0.0270605	+	0.999634i	$0.508615\pi$
$773$	−793.635	+	793.635i	−1.02669	+	1.02669i	−0.999634	+	0.0270605i	$0.991385\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−44.0908	−	44.0908i	−0.0567449	−	0.0567449i
$778$		0			0
$779$		−	1500.00i		−	1.92555i
$780$		0			0
$781$	−360.000			−0.460948
$782$		0			0
$783$	−154.318	+	154.318i	−0.197085	+	0.197085i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−378.446	−	378.446i	−0.480872	−	0.480872i	0.424538	−	0.905410i	$-0.360436\pi$
$787$	−378.446	−	378.446i	−0.480872	−	0.480872i	−0.905410	+	0.424538i	$0.860436\pi$
$788$		0			0
$789$		−	612.000i		−	0.775665i
$790$		0			0
$791$	432.000			0.546144
$792$		0			0
$793$	79.6084	−	79.6084i	0.100389	−	0.100389i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	617.271	+	617.271i	0.774494	+	0.774494i	0.978889	−	0.204395i	$-0.0655227\pi$
$797$	617.271	+	617.271i	0.774494	+	0.774494i	−0.204395	+	0.978889i	$0.565523\pi$
$798$		0			0
$799$		−	2268.00i		−	2.83855i
$800$		0			0
$801$	180.000			0.224719
$802$		0			0
$803$	382.120	−	382.120i	0.475866	−	0.475866i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−389.469	−	389.469i	−0.482613	−	0.482613i
$808$		0			0
$809$		−	348.000i		−	0.430161i	−0.976596	−	0.215080i	$-0.930999\pi$
$809$		−	348.000i		−	0.430161i	0.976596	−	0.215080i	$-0.0690013\pi$
$810$		0			0
$811$	−1045.00			−1.28853			−0.644266	−	0.764801i	$-0.722837\pi$
$811$	−1045.00			−1.28853			−0.644266	+	0.764801i	$0.722837\pi$
$812$		0			0
$813$	276.792	−	276.792i	0.340458	−	0.340458i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−30.6186	−	30.6186i	−0.0374769	−	0.0374769i
$818$		0			0
$819$			135.000i			0.164835i
$820$		0			0
$821$	606.000			0.738124			0.369062	−	0.929405i	$-0.379679\pi$
$821$	606.000			0.738124			0.369062	+	0.929405i	$0.379679\pi$
$822$		0			0
$823$	302.512	−	302.512i	0.367572	−	0.367572i	−0.499019	−	0.866591i	$-0.666306\pi$
$823$	302.512	−	302.512i	0.367572	−	0.367572i	0.866591	+	0.499019i	$0.166306\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−396.817	−	396.817i	−0.479827	−	0.479827i	0.425249	−	0.905076i	$-0.360187\pi$
$827$	−396.817	−	396.817i	−0.479827	−	0.479827i	−0.905076	+	0.425249i	$0.860187\pi$
$828$		0			0
$829$		−	406.000i		−	0.489747i	−0.969555	−	0.244873i	$-0.921254\pi$
$829$		−	406.000i		−	0.489747i	0.969555	−	0.244873i	$-0.0787464\pi$
$830$		0			0
$831$	−933.000			−1.12274
$832$		0			0
$833$	−484.999	+	484.999i	−0.582232	+	0.582232i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	180.037	+	180.037i	0.215099	+	0.215099i
$838$		0			0
$839$			462.000i			0.550656i	0.961350	+	0.275328i	$0.0887864\pi$
$839$			462.000i			0.550656i	−0.961350	+	0.275328i	$0.911214\pi$
$840$		0			0
$841$	−923.000			−1.09750
$842$		0			0
$843$	154.318	−	154.318i	0.183058	−	0.183058i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−312.310	−	312.310i	−0.368725	−	0.368725i
$848$		0			0
$849$			969.000i			1.14134i
$850$		0			0
$851$	72.0000			0.0846063
$852$		0			0
$853$	−285.366	+	285.366i	−0.334543	+	0.334543i	−0.854309	−	0.519766i	$-0.826019\pi$
$853$	−285.366	+	285.366i	−0.334543	+	0.334543i	0.519766	+	0.854309i	$0.326019\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−191.060	−	191.060i	−0.222941	−	0.222941i	0.586795	−	0.809736i	$-0.300389\pi$
$857$	−191.060	−	191.060i	−0.222941	−	0.222941i	−0.809736	+	0.586795i	$0.800389\pi$
$858$		0			0
$859$			410.000i			0.477299i	0.971106	+	0.238650i	$0.0767047\pi$
$859$			410.000i			0.477299i	−0.971106	+	0.238650i	$0.923295\pi$
$860$		0			0
$861$	−540.000			−0.627178
$862$		0			0
$863$	−1116.97	+	1116.97i	−1.29428	+	1.29428i	−0.362174	+	0.932111i	$0.617965\pi$
$863$	−1116.97	+	1116.97i	−1.29428	+	1.29428i	−0.932111	+	0.362174i	$0.882035\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	836.501	+	836.501i	0.964822	+	0.964822i
$868$		0			0
$869$			636.000i			0.731876i
$870$		0			0
$871$	645.000			0.740528
$872$		0			0
$873$	363.749	−	363.749i	0.416666	−	0.416666i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−180.037	−	180.037i	−0.205288	−	0.205288i	0.596973	−	0.802261i	$-0.296370\pi$
$877$	−180.037	−	180.037i	−0.205288	−	0.205288i	−0.802261	+	0.596973i	$0.796370\pi$
$878$		0			0
$879$		−	54.0000i		−	0.0614334i
$880$		0			0
$881$	−120.000			−0.136209			−0.0681044	−	0.997678i	$-0.521695\pi$
$881$	−120.000			−0.136209			−0.0681044	+	0.997678i	$0.521695\pi$
$882$		0			0
$883$	−584.203	+	584.203i	−0.661612	+	0.661612i	−0.955760	−	0.294148i	$-0.904964\pi$
$883$	−584.203	+	584.203i	−0.661612	+	0.661612i	0.294148	+	0.955760i	$0.404964\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−198.409	−	198.409i	−0.223685	−	0.223685i	0.586363	−	0.810048i	$-0.300559\pi$
$887$	−198.409	−	198.409i	−0.223685	−	0.223685i	−0.810048	+	0.586363i	$0.800559\pi$
$888$		0			0
$889$		−	108.000i		−	0.121485i
$890$		0			0
$891$	54.0000			0.0606061
$892$		0			0
$893$	−1285.98	+	1285.98i	−1.44007	+	1.44007i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−110.227	−	110.227i	−0.122884	−	0.122884i
$898$		0			0
$899$			2058.00i			2.28921i
$900$		0			0
$901$	−648.000			−0.719201
$902$		0			0
$903$	−11.0227	+	11.0227i	−0.0122068	+	0.0122068i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1033.68	+	1033.68i	1.13967	+	1.13967i	0.988508	+	0.151166i	$0.0483028\pi$
$907$	1033.68	+	1033.68i	1.13967	+	1.13967i	0.151166	+	0.988508i	$0.451697\pi$
$908$		0			0
$909$			576.000i			0.633663i
$910$		0			0
$911$	−1218.00			−1.33699			−0.668496	−	0.743716i	$-0.733062\pi$
$911$	−1218.00			−1.33699			−0.668496	+	0.743716i	$0.733062\pi$
$912$		0			0
$913$	484.999	−	484.999i	0.531215	−	0.531215i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−44.0908	−	44.0908i	−0.0480816	−	0.0480816i
$918$		0			0
$919$			469.000i			0.510337i	0.966897	+	0.255169i	$0.0821310\pi$
$919$			469.000i			0.510337i	−0.966897	+	0.255169i	$0.917869\pi$
$920$		0			0
$921$	−213.000			−0.231270
$922$		0			0
$923$	−367.423	+	367.423i	−0.398075	+	0.398075i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	235.151	+	235.151i	0.253669	+	0.253669i
$928$		0			0
$929$		−	474.000i		−	0.510226i	−0.966911	−	0.255113i	$-0.917887\pi$
$929$		−	474.000i		−	0.510226i	0.966911	−	0.255113i	$-0.0821127\pi$
$930$		0			0
$931$	550.000			0.590763
$932$		0			0
$933$	301.287	−	301.287i	0.322923	−	0.322923i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−709.127	−	709.127i	−0.756806	−	0.756806i	0.218934	−	0.975740i	$-0.429742\pi$
$937$	−709.127	−	709.127i	−0.756806	−	0.756806i	−0.975740	+	0.218934i	$0.929742\pi$
$938$		0			0
$939$			93.0000i			0.0990415i
$940$		0			0
$941$	990.000			1.05207			0.526036	−	0.850462i	$-0.323678\pi$
$941$	990.000			1.05207			0.526036	+	0.850462i	$0.323678\pi$
$942$		0			0
$943$	440.908	−	440.908i	0.467559	−	0.467559i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1036.13	−	1036.13i	−1.09412	−	1.09412i	−0.995084	−	0.0990391i	$-0.968423\pi$
$947$	−1036.13	−	1036.13i	−1.09412	−	1.09412i	−0.0990391	−	0.995084i	$-0.531577\pi$
$948$		0			0
$949$		−	780.000i		−	0.821918i
$950$		0			0
$951$	36.0000			0.0378549
$952$		0			0
$953$	−631.968	+	631.968i	−0.663136	+	0.663136i	−0.956118	−	0.292982i	$-0.905352\pi$
$953$	−631.968	+	631.968i	−0.663136	+	0.663136i	0.292982	+	0.956118i	$0.405352\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	308.636	+	308.636i	0.322503	+	0.322503i
$958$		0			0
$959$			594.000i			0.619395i
$960$		0			0
$961$	1440.00			1.49844
$962$		0			0
$963$	88.1816	−	88.1816i	0.0915697	−	0.0915697i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	563.383	+	563.383i	0.582609	+	0.582609i	0.935619	−	0.353011i	$-0.114842\pi$
$967$	563.383	+	563.383i	0.582609	+	0.582609i	−0.353011	+	0.935619i	$0.614842\pi$
$968$		0			0
$969$		−	1350.00i		−	1.39319i
$970$		0			0
$971$	−1338.00			−1.37796			−0.688980	−	0.724780i	$-0.741941\pi$
$971$	−1338.00			−1.37796			−0.688980	+	0.724780i	$0.741941\pi$
$972$		0			0
$973$	−213.106	+	213.106i	−0.219019	+	0.219019i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−301.287	−	301.287i	−0.308380	−	0.308380i	0.535901	−	0.844281i	$-0.319972\pi$
$977$	−301.287	−	301.287i	−0.308380	−	0.308380i	−0.844281	+	0.535901i	$0.819972\pi$
$978$		0			0
$979$		−	360.000i		−	0.367722i
$980$		0			0
$981$	3.00000			0.00305810
$982$		0			0
$983$	−426.211	+	426.211i	−0.433582	+	0.433582i	−0.889845	−	0.456263i	$-0.849188\pi$
$983$	−426.211	+	426.211i	−0.433582	+	0.433582i	0.456263	+	0.889845i	$0.349188\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	462.954	+	462.954i	0.469051	+	0.469051i
$988$		0			0
$989$		−	18.0000i		−	0.0182002i
$990$		0			0
$991$	−83.0000			−0.0837538			−0.0418769	−	0.999123i	$-0.513334\pi$
$991$	−83.0000			−0.0837538			−0.0418769	+	0.999123i	$0.513334\pi$
$992$		0			0
$993$	218.005	−	218.005i	0.219541	−	0.219541i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	573.181	+	573.181i	0.574905	+	0.574905i	0.933495	−	0.358590i	$-0.116742\pi$
$997$	573.181	+	573.181i	0.574905	+	0.574905i	−0.358590	+	0.933495i	$0.616742\pi$
$998$		0			0
$999$			36.0000i			0.0360360i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.i.1057.2		4
4.3	odd	2		300.3.k.c.157.1	✓	4
5.2	odd	4	inner	1200.3.bg.i.193.1		4
5.3	odd	4	inner	1200.3.bg.i.193.2		4
5.4	even	2	inner	1200.3.bg.i.1057.1		4
12.11	even	2		900.3.l.d.757.1		4
20.3	even	4		300.3.k.c.193.1	yes	4
20.7	even	4		300.3.k.c.193.2	yes	4
20.19	odd	2		300.3.k.c.157.2	yes	4
60.23	odd	4		900.3.l.d.793.1		4
60.47	odd	4		900.3.l.d.793.2		4
60.59	even	2		900.3.l.d.757.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.3.k.c.157.1	✓	4	4.3	odd	2
300.3.k.c.157.2	yes	4	20.19	odd	2
300.3.k.c.193.1	yes	4	20.3	even	4
300.3.k.c.193.2	yes	4	20.7	even	4
900.3.l.d.757.1		4	12.11	even	2
900.3.l.d.757.2		4	60.59	even	2
900.3.l.d.793.1		4	60.23	odd	4
900.3.l.d.793.2		4	60.47	odd	4
1200.3.bg.i.193.1		4	5.2	odd	4	inner
1200.3.bg.i.193.2		4	5.3	odd	4	inner
1200.3.bg.i.1057.1		4	5.4	even	2	inner
1200.3.bg.i.1057.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{3} +(3.67423 + 3.67423i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{3} +(3.67423 + 3.67423i) q^{7} -3.00000i q^{9} -6.00000 q^{11} +(-6.12372 + 6.12372i) q^{13} +(-22.0454 - 22.0454i) q^{17} +25.0000i q^{19} +9.00000 q^{21} +(-7.34847 + 7.34847i) q^{23} +(-3.67423 - 3.67423i) q^{27} -42.0000i q^{29} -49.0000 q^{31} +(-7.34847 + 7.34847i) q^{33} +(-4.89898 - 4.89898i) q^{37} +15.0000i q^{39} -60.0000 q^{41} +(-1.22474 + 1.22474i) q^{43} +(51.4393 + 51.4393i) q^{47} -22.0000i q^{49} -54.0000 q^{51} +(14.6969 - 14.6969i) q^{53} +(30.6186 + 30.6186i) q^{57} -78.0000i q^{59} -13.0000 q^{61} +(11.0227 - 11.0227i) q^{63} +(-52.6640 - 52.6640i) q^{67} +18.0000i q^{69} +60.0000 q^{71} +(-63.6867 + 63.6867i) q^{73} +(-22.0454 - 22.0454i) q^{77} -106.000i q^{79} -9.00000 q^{81} +(-80.8332 + 80.8332i) q^{83} +(-51.4393 - 51.4393i) q^{87} +60.0000i q^{89} -45.0000 q^{91} +(-60.0125 + 60.0125i) q^{93} +(121.250 + 121.250i) q^{97} +18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{11} + 36 q^{21} - 196 q^{31} - 240 q^{41} - 216 q^{51} - 52 q^{61} + 240 q^{71} - 36 q^{81} - 180 q^{91}+O(q^{100}) \) 4 * q - 24 * q^11 + 36 * q^21 - 196 * q^31 - 240 * q^41 - 216 * q^51 - 52 * q^61 + 240 * q^71 - 36 * q^81 - 180 * q^91

Introduction

L-functions

Modular forms

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