LMFDB - Embedded newform 7800.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.a (trivial)

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:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.35386$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.35386 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.35386 q^{7} +1.00000 q^{9} -3.81322 q^{11} +1.00000 q^{13} +1.00000 q^{17} +4.24835 q^{19} +3.35386 q^{21} +6.24835 q^{23} -1.00000 q^{27} -6.78899 q^{29} -5.97577 q^{31} +3.81322 q^{33} +4.45936 q^{37} -1.00000 q^{39} -0.621911 q^{41} -1.54064 q^{43} +9.14284 q^{47} +4.24835 q^{49} -1.00000 q^{51} +7.08580 q^{53} -4.24835 q^{57} +0.272582 q^{59} +4.78899 q^{61} -3.35386 q^{63} -1.81322 q^{67} -6.24835 q^{69} -9.16707 q^{71} +3.32962 q^{73} +12.7890 q^{77} -5.54064 q^{79} +1.00000 q^{81} +6.27258 q^{83} +6.78899 q^{87} +3.78899 q^{89} -3.35386 q^{91} +5.97577 q^{93} +14.7077 q^{97} -3.81322 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} - 6 q^{19} + q^{21} - 3 q^{27} - q^{29} - 7 q^{31} + 3 q^{33} + 14 q^{37} - 3 q^{39} - 4 q^{43} - q^{47} - 6 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{57} - 7 q^{59} - 5 q^{61} - q^{63} + 3 q^{67} - 10 q^{71} - 10 q^{73} + 19 q^{77} - 16 q^{79} + 3 q^{81} + 11 q^{83} + q^{87} - 8 q^{89} - q^{91} + 7 q^{93} + 26 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^17 - 6 * q^19 + q^21 - 3 * q^27 - q^29 - 7 * q^31 + 3 * q^33 + 14 * q^37 - 3 * q^39 - 4 * q^43 - q^47 - 6 * q^49 - 3 * q^51 + 5 * q^53 + 6 * q^57 - 7 * q^59 - 5 * q^61 - q^63 + 3 * q^67 - 10 * q^71 - 10 * q^73 + 19 * q^77 - 16 * q^79 + 3 * q^81 + 11 * q^83 + q^87 - 8 * q^89 - q^91 + 7 * q^93 + 26 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.35386	−1.26764	−0.633819	−	0.773481i	$-0.718514\pi$
$7$	−3.35386	−1.26764	−0.633819	+	0.773481i	$0.718514\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.81322	−1.14973	−0.574864	−	0.818249i	$-0.694945\pi$
$11$	−3.81322	−1.14973	−0.574864	+	0.818249i	$0.694945\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	4.24835	0.974638	0.487319	−	0.873224i	$-0.337975\pi$
$19$	4.24835	0.974638	0.487319	+	0.873224i	$0.337975\pi$
$20$	0	0
$21$	3.35386	0.731871
$22$	0	0
$23$	6.24835	1.30287	0.651435	−	0.758704i	$-0.274167\pi$
$23$	6.24835	1.30287	0.651435	+	0.758704i	$0.274167\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.78899	−1.26068	−0.630341	−	0.776318i	$-0.717085\pi$
$29$	−6.78899	−1.26068	−0.630341	+	0.776318i	$0.717085\pi$
$30$	0	0
$31$	−5.97577	−1.07328	−0.536640	−	0.843811i	$-0.680307\pi$
$31$	−5.97577	−1.07328	−0.536640	+	0.843811i	$0.680307\pi$
$32$	0	0
$33$	3.81322	0.663796
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.45936	0.733115	0.366557	−	0.930395i	$-0.380536\pi$
$37$	4.45936	0.733115	0.366557	+	0.930395i	$0.380536\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−0.621911	−0.0971262	−0.0485631	−	0.998820i	$-0.515464\pi$
$41$	−0.621911	−0.0971262	−0.0485631	+	0.998820i	$0.515464\pi$
$42$	0	0
$43$	−1.54064	−0.234945	−0.117472	−	0.993076i	$-0.537479\pi$
$43$	−1.54064	−0.234945	−0.117472	+	0.993076i	$0.537479\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.14284	1.33362	0.666810	−	0.745228i	$-0.267659\pi$
$47$	9.14284	1.33362	0.666810	+	0.745228i	$0.267659\pi$
$48$	0	0
$49$	4.24835	0.606907
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	7.08580	0.973310	0.486655	−	0.873594i	$-0.338217\pi$
$53$	7.08580	0.973310	0.486655	+	0.873594i	$0.338217\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.24835	−0.562708
$58$	0	0
$59$	0.272582	0.0354871	0.0177436	−	0.999843i	$-0.494352\pi$
$59$	0.272582	0.0354871	0.0177436	+	0.999843i	$0.494352\pi$
$60$	0	0
$61$	4.78899	0.613167	0.306583	−	0.951844i	$-0.400814\pi$
$61$	4.78899	0.613167	0.306583	+	0.951844i	$0.400814\pi$
$62$	0	0
$63$	−3.35386	−0.422546
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.81322	−0.221520	−0.110760	−	0.993847i	$-0.535328\pi$
$67$	−1.81322	−0.221520	−0.110760	+	0.993847i	$0.535328\pi$
$68$	0	0
$69$	−6.24835	−0.752213
$70$	0	0
$71$	−9.16707	−1.08793	−0.543966	−	0.839107i	$-0.683078\pi$
$71$	−9.16707	−1.08793	−0.543966	+	0.839107i	$0.683078\pi$
$72$	0	0
$73$	3.32962	0.389703	0.194851	−	0.980833i	$-0.437578\pi$
$73$	3.32962	0.389703	0.194851	+	0.980833i	$0.437578\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.7890	1.45744
$78$	0	0
$79$	−5.54064	−0.623370	−0.311685	−	0.950185i	$-0.600893\pi$
$79$	−5.54064	−0.623370	−0.311685	+	0.950185i	$0.600893\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.27258	0.688505	0.344253	−	0.938877i	$-0.388132\pi$
$83$	6.27258	0.688505	0.344253	+	0.938877i	$0.388132\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.78899	0.727856
$88$	0	0
$89$	3.78899	0.401632	0.200816	−	0.979629i	$-0.435641\pi$
$89$	3.78899	0.401632	0.200816	+	0.979629i	$0.435641\pi$
$90$	0	0
$91$	−3.35386	−0.351580
$92$	0	0
$93$	5.97577	0.619658
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.7077	1.49334	0.746671	−	0.665194i	$-0.231651\pi$
$97$	14.7077	1.49334	0.746671	+	0.665194i	$0.231651\pi$
$98$	0	0
$99$	−3.81322	−0.383243
$100$	0	0
$101$	7.03733	0.700241	0.350120	−	0.936705i	$-0.386141\pi$
$101$	7.03733	0.700241	0.350120	+	0.936705i	$0.386141\pi$
$102$	0	0
$103$	−18.7935	−1.85178	−0.925890	−	0.377794i	$-0.876683\pi$
$103$	−18.7935	−1.85178	−0.925890	+	0.377794i	$0.876683\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.87026	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$107$	−4.87026	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$108$	0	0
$109$	1.32962	0.127355	0.0636774	−	0.997971i	$-0.479717\pi$
$109$	1.32962	0.127355	0.0636774	+	0.997971i	$0.479717\pi$
$110$	0	0
$111$	−4.45936	−0.423264
$112$	0	0
$113$	−5.78899	−0.544582	−0.272291	−	0.962215i	$-0.587781\pi$
$113$	−5.78899	−0.544582	−0.272291	+	0.962215i	$0.587781\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−3.35386	−0.307447
$120$	0	0
$121$	3.54064	0.321876
$122$	0	0
$123$	0.621911	0.0560758
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.16707	0.635975	0.317988	−	0.948095i	$-0.396993\pi$
$127$	7.16707	0.635975	0.317988	+	0.948095i	$0.396993\pi$
$128$	0	0
$129$	1.54064	0.135646
$130$	0	0
$131$	−7.00453	−0.611988	−0.305994	−	0.952033i	$-0.598989\pi$
$131$	−7.00453	−0.611988	−0.305994	+	0.952033i	$0.598989\pi$
$132$	0	0
$133$	−14.2483	−1.23549
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.12974	0.0965202	0.0482601	−	0.998835i	$-0.484632\pi$
$137$	1.12974	0.0965202	0.0482601	+	0.998835i	$0.484632\pi$
$138$	0	0
$139$	−4.16255	−0.353063	−0.176531	−	0.984295i	$-0.556488\pi$
$139$	−4.16255	−0.353063	−0.176531	+	0.984295i	$0.556488\pi$
$140$	0	0
$141$	−9.14284	−0.769966
$142$	0	0
$143$	−3.81322	−0.318877
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.24835	−0.350398
$148$	0	0
$149$	14.2857	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$149$	14.2857	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$150$	0	0
$151$	9.85055	0.801627	0.400813	−	0.916160i	$-0.368728\pi$
$151$	9.85055	0.801627	0.400813	+	0.916160i	$0.368728\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.2483	1.37657	0.688284	−	0.725441i	$-0.258364\pi$
$157$	17.2483	1.37657	0.688284	+	0.725441i	$0.258364\pi$
$158$	0	0
$159$	−7.08580	−0.561941
$160$	0	0
$161$	−20.9561	−1.65157
$162$	0	0
$163$	−13.4922	−1.05679	−0.528394	−	0.848999i	$-0.677206\pi$
$163$	−13.4922	−1.05679	−0.528394	+	0.848999i	$0.677206\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.54064	0.273983	0.136991	−	0.990572i	$-0.456257\pi$
$167$	3.54064	0.273983	0.136991	+	0.990572i	$0.456257\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.24835	0.324879
$172$	0	0
$173$	−2.37809	−0.180803	−0.0904014	−	0.995905i	$-0.528815\pi$
$173$	−2.37809	−0.180803	−0.0904014	+	0.995905i	$0.528815\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.272582	−0.0204885
$178$	0	0
$179$	−10.7935	−0.806745	−0.403372	−	0.915036i	$-0.632162\pi$
$179$	−10.7935	−0.806745	−0.403372	+	0.915036i	$0.632162\pi$
$180$	0	0
$181$	−8.11861	−0.603451	−0.301726	−	0.953395i	$-0.597563\pi$
$181$	−8.11861	−0.603451	−0.301726	+	0.953395i	$0.597563\pi$
$182$	0	0
$183$	−4.78899	−0.354012
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.81322	−0.278850
$188$	0	0
$189$	3.35386	0.243957
$190$	0	0
$191$	−25.5780	−1.85076	−0.925379	−	0.379044i	$-0.876253\pi$
$191$	−25.5780	−1.85076	−0.925379	+	0.379044i	$0.876253\pi$
$192$	0	0
$193$	−19.1671	−1.37968	−0.689838	−	0.723964i	$-0.742318\pi$
$193$	−19.1671	−1.37968	−0.689838	+	0.723964i	$0.742318\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.04394	0.359366	0.179683	−	0.983725i	$-0.442493\pi$
$197$	5.04394	0.359366	0.179683	+	0.983725i	$0.442493\pi$
$198$	0	0
$199$	−2.91873	−0.206903	−0.103451	−	0.994634i	$-0.532989\pi$
$199$	−2.91873	−0.206903	−0.103451	+	0.994634i	$0.532989\pi$
$200$	0	0
$201$	1.81322	0.127895
$202$	0	0
$203$	22.7693	1.59809
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.24835	0.434290
$208$	0	0
$209$	−16.1999	−1.12057
$210$	0	0
$211$	−15.1671	−1.04414	−0.522072	−	0.852901i	$-0.674841\pi$
$211$	−15.1671	−1.04414	−0.522072	+	0.852901i	$0.674841\pi$
$212$	0	0
$213$	9.16707	0.628118
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.0419	1.36053
$218$	0	0
$219$	−3.32962	−0.224995
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	−28.4199	−1.90314	−0.951570	−	0.307431i	$-0.900531\pi$
$223$	−28.4199	−1.90314	−0.951570	+	0.307431i	$0.900531\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.4725	−0.827827	−0.413913	−	0.910316i	$-0.635838\pi$
$227$	−12.4725	−0.827827	−0.413913	+	0.910316i	$0.635838\pi$
$228$	0	0
$229$	−29.1559	−1.92668	−0.963339	−	0.268285i	$-0.913543\pi$
$229$	−29.1559	−1.92668	−0.963339	+	0.268285i	$0.913543\pi$
$230$	0	0
$231$	−12.7890	−0.841453
$232$	0	0
$233$	−2.21101	−0.144848	−0.0724242	−	0.997374i	$-0.523074\pi$
$233$	−2.21101	−0.144848	−0.0724242	+	0.997374i	$0.523074\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.54064	0.359903
$238$	0	0
$239$	−13.8506	−0.895918	−0.447959	−	0.894054i	$-0.647849\pi$
$239$	−13.8506	−0.895918	−0.447959	+	0.894054i	$0.647849\pi$
$240$	0	0
$241$	7.49217	0.482613	0.241307	−	0.970449i	$-0.422424\pi$
$241$	7.49217	0.482613	0.241307	+	0.970449i	$0.422424\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.24835	0.270316
$248$	0	0
$249$	−6.27258	−0.397509
$250$	0	0
$251$	−4.78446	−0.301992	−0.150996	−	0.988534i	$-0.548248\pi$
$251$	−4.78446	−0.301992	−0.150996	+	0.988534i	$0.548248\pi$
$252$	0	0
$253$	−23.8263	−1.49795
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.4199	−0.961870	−0.480935	−	0.876756i	$-0.659703\pi$
$257$	−15.4199	−0.961870	−0.480935	+	0.876756i	$0.659703\pi$
$258$	0	0
$259$	−14.9561	−0.929324
$260$	0	0
$261$	−6.78899	−0.420228
$262$	0	0
$263$	−29.8748	−1.84216	−0.921079	−	0.389375i	$-0.872691\pi$
$263$	−29.8748	−1.84216	−0.921079	+	0.389375i	$0.872691\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.78899	−0.231882
$268$	0	0
$269$	7.16255	0.436708	0.218354	−	0.975870i	$-0.429931\pi$
$269$	7.16255	0.436708	0.218354	+	0.975870i	$0.429931\pi$
$270$	0	0
$271$	16.1847	0.983151	0.491575	−	0.870835i	$-0.336421\pi$
$271$	16.1847	0.983151	0.491575	+	0.870835i	$0.336421\pi$
$272$	0	0
$273$	3.35386	0.202985
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.5386	−1.53446	−0.767232	−	0.641370i	$-0.778366\pi$
$277$	−25.5386	−1.53446	−0.767232	+	0.641370i	$0.778366\pi$
$278$	0	0
$279$	−5.97577	−0.357760
$280$	0	0
$281$	8.65925	0.516567	0.258284	−	0.966069i	$-0.416843\pi$
$281$	8.65925	0.516567	0.258284	+	0.966069i	$0.416843\pi$
$282$	0	0
$283$	15.9121	0.945877	0.472939	−	0.881095i	$-0.343193\pi$
$283$	15.9121	0.945877	0.472939	+	0.881095i	$0.343193\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.08580	0.123121
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−14.7077	−0.862181
$292$	0	0
$293$	−13.2529	−0.774241	−0.387121	−	0.922029i	$-0.626530\pi$
$293$	−13.2529	−0.774241	−0.387121	+	0.922029i	$0.626530\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.81322	0.221265
$298$	0	0
$299$	6.24835	0.361351
$300$	0	0
$301$	5.16707	0.297825
$302$	0	0
$303$	−7.03733	−0.404284
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.12521	−0.121292	−0.0606462	−	0.998159i	$-0.519316\pi$
$307$	−2.12521	−0.121292	−0.0606462	+	0.998159i	$0.519316\pi$
$308$	0	0
$309$	18.7935	1.06913
$310$	0	0
$311$	19.9121	1.12911	0.564556	−	0.825395i	$-0.309047\pi$
$311$	19.9121	1.12911	0.564556	+	0.825395i	$0.309047\pi$
$312$	0	0
$313$	6.50122	0.367471	0.183735	−	0.982976i	$-0.441181\pi$
$313$	6.50122	0.367471	0.183735	+	0.982976i	$0.441181\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.3296	1.31032	0.655161	−	0.755489i	$-0.272601\pi$
$317$	23.3296	1.31032	0.655161	+	0.755489i	$0.272601\pi$
$318$	0	0
$319$	25.8879	1.44944
$320$	0	0
$321$	4.87026	0.271831
$322$	0	0
$323$	4.24835	0.236384
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.32962	−0.0735283
$328$	0	0
$329$	−30.6638	−1.69055
$330$	0	0
$331$	−7.32962	−0.402873	−0.201436	−	0.979502i	$-0.564561\pi$
$331$	−7.32962	−0.402873	−0.201436	+	0.979502i	$0.564561\pi$
$332$	0	0
$333$	4.45936	0.244372
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.03733	0.492295	0.246147	−	0.969232i	$-0.420835\pi$
$337$	9.03733	0.492295	0.246147	+	0.969232i	$0.420835\pi$
$338$	0	0
$339$	5.78899	0.314415
$340$	0	0
$341$	22.7869	1.23398
$342$	0	0
$343$	9.22864	0.498300
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.1559	1.13571	0.567855	−	0.823128i	$-0.307773\pi$
$347$	21.1559	1.13571	0.567855	+	0.823128i	$0.307773\pi$
$348$	0	0
$349$	0.956060	0.0511767	0.0255884	−	0.999673i	$-0.491854\pi$
$349$	0.956060	0.0511767	0.0255884	+	0.999673i	$0.491854\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−29.7869	−1.58540	−0.792699	−	0.609614i	$-0.791325\pi$
$353$	−29.7869	−1.58540	−0.792699	+	0.609614i	$0.791325\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.35386	0.177505
$358$	0	0
$359$	35.2660	1.86127	0.930634	−	0.365953i	$-0.119257\pi$
$359$	35.2660	1.86127	0.930634	+	0.365953i	$0.119257\pi$
$360$	0	0
$361$	−0.951534	−0.0500807
$362$	0	0
$363$	−3.54064	−0.185835
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.8375	−0.722309	−0.361155	−	0.932506i	$-0.617617\pi$
$367$	−13.8375	−0.722309	−0.361155	+	0.932506i	$0.617617\pi$
$368$	0	0
$369$	−0.621911	−0.0323754
$370$	0	0
$371$	−23.7648	−1.23380
$372$	0	0
$373$	18.2812	0.946562	0.473281	−	0.880911i	$-0.343069\pi$
$373$	18.2812	0.946562	0.473281	+	0.880911i	$0.343069\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.78899	−0.349651
$378$	0	0
$379$	15.0086	0.770939	0.385469	−	0.922721i	$-0.374040\pi$
$379$	15.0086	0.770939	0.385469	+	0.922721i	$0.374040\pi$
$380$	0	0
$381$	−7.16707	−0.367180
$382$	0	0
$383$	15.5012	0.792076	0.396038	−	0.918234i	$-0.370385\pi$
$383$	15.5012	0.792076	0.396038	+	0.918234i	$0.370385\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.54064	−0.0783150
$388$	0	0
$389$	−17.0419	−0.864057	−0.432028	−	0.901860i	$-0.642202\pi$
$389$	−17.0419	−0.864057	−0.432028	+	0.901860i	$0.642202\pi$
$390$	0	0
$391$	6.24835	0.315993
$392$	0	0
$393$	7.00453	0.353332
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.2044	1.16460	0.582298	−	0.812975i	$-0.302154\pi$
$397$	23.2044	1.16460	0.582298	+	0.812975i	$0.302154\pi$
$398$	0	0
$399$	14.2483	0.713310
$400$	0	0
$401$	−11.1186	−0.555237	−0.277618	−	0.960691i	$-0.589545\pi$
$401$	−11.1186	−0.555237	−0.277618	+	0.960691i	$0.589545\pi$
$402$	0	0
$403$	−5.97577	−0.297674
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17.0045	−0.842883
$408$	0	0
$409$	3.87479	0.191596	0.0957979	−	0.995401i	$-0.469460\pi$
$409$	3.87479	0.191596	0.0957979	+	0.995401i	$0.469460\pi$
$410$	0	0
$411$	−1.12974	−0.0557260
$412$	0	0
$413$	−0.914200	−0.0449848
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.16255	0.203841
$418$	0	0
$419$	13.8748	0.677828	0.338914	−	0.940817i	$-0.389940\pi$
$419$	13.8748	0.677828	0.338914	+	0.940817i	$0.389940\pi$
$420$	0	0
$421$	−37.5870	−1.83188	−0.915940	−	0.401316i	$-0.868553\pi$
$421$	−37.5870	−1.83188	−0.915940	+	0.401316i	$0.868553\pi$
$422$	0	0
$423$	9.14284	0.444540
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.0616	−0.777274
$428$	0	0
$429$	3.81322	0.184104
$430$	0	0
$431$	6.58250	0.317068	0.158534	−	0.987354i	$-0.449323\pi$
$431$	6.58250	0.317068	0.158534	+	0.987354i	$0.449323\pi$
$432$	0	0
$433$	−20.5452	−0.987338	−0.493669	−	0.869650i	$-0.664344\pi$
$433$	−20.5452	−0.987338	−0.493669	+	0.869650i	$0.664344\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.5452	1.26983
$438$	0	0
$439$	−23.4922	−1.12122	−0.560610	−	0.828080i	$-0.689433\pi$
$439$	−23.4922	−1.12122	−0.560610	+	0.828080i	$0.689433\pi$
$440$	0	0
$441$	4.24835	0.202302
$442$	0	0
$443$	5.04394	0.239645	0.119822	−	0.992795i	$-0.461767\pi$
$443$	5.04394	0.239645	0.119822	+	0.992795i	$0.461767\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−14.2857	−0.675690
$448$	0	0
$449$	18.3341	0.865242	0.432621	−	0.901576i	$-0.357589\pi$
$449$	18.3341	0.865242	0.432621	+	0.901576i	$0.357589\pi$
$450$	0	0
$451$	2.37148	0.111669
$452$	0	0
$453$	−9.85055	−0.462819
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.2089	−1.22600	−0.613001	−	0.790082i	$-0.710038\pi$
$457$	−26.2089	−1.22600	−0.613001	+	0.790082i	$0.710038\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−34.4967	−1.60667	−0.803336	−	0.595526i	$-0.796943\pi$
$461$	−34.4967	−1.60667	−0.803336	+	0.595526i	$0.796943\pi$
$462$	0	0
$463$	−5.43060	−0.252382	−0.126191	−	0.992006i	$-0.540275\pi$
$463$	−5.43060	−0.252382	−0.126191	+	0.992006i	$0.540275\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−35.0419	−1.62154	−0.810772	−	0.585362i	$-0.800952\pi$
$467$	−35.0419	−1.62154	−0.810772	+	0.585362i	$0.800952\pi$
$468$	0	0
$469$	6.08127	0.280807
$470$	0	0
$471$	−17.2483	−0.794762
$472$	0	0
$473$	5.87479	0.270123
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.08580	0.324437
$478$	0	0
$479$	24.0989	1.10111	0.550553	−	0.834800i	$-0.314417\pi$
$479$	24.0989	1.10111	0.550553	+	0.834800i	$0.314417\pi$
$480$	0	0
$481$	4.45936	0.203329
$482$	0	0
$483$	20.9561	0.953534
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.2286	1.41510	0.707552	−	0.706661i	$-0.249799\pi$
$487$	31.2286	1.41510	0.707552	+	0.706661i	$0.249799\pi$
$488$	0	0
$489$	13.4922	0.610137
$490$	0	0
$491$	−13.2529	−0.598094	−0.299047	−	0.954238i	$-0.596669\pi$
$491$	−13.2529	−0.598094	−0.299047	+	0.954238i	$0.596669\pi$
$492$	0	0
$493$	−6.78899	−0.305761
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.7450	1.37910
$498$	0	0
$499$	−40.6329	−1.81898	−0.909490	−	0.415726i	$-0.863528\pi$
$499$	−40.6329	−1.81898	−0.909490	+	0.415726i	$0.863528\pi$
$500$	0	0
$501$	−3.54064	−0.158184
$502$	0	0
$503$	−31.0792	−1.38575	−0.692876	−	0.721056i	$-0.743657\pi$
$503$	−31.0792	−1.38575	−0.692876	+	0.721056i	$0.743657\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−20.4109	−0.904697	−0.452349	−	0.891841i	$-0.649414\pi$
$509$	−20.4109	−0.904697	−0.452349	+	0.891841i	$0.649414\pi$
$510$	0	0
$511$	−11.1671	−0.494002
$512$	0	0
$513$	−4.24835	−0.187569
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−34.8637	−1.53330
$518$	0	0
$519$	2.37809	0.104387
$520$	0	0
$521$	−9.57797	−0.419619	−0.209809	−	0.977742i	$-0.567284\pi$
$521$	−9.57797	−0.419619	−0.209809	+	0.977742i	$0.567284\pi$
$522$	0	0
$523$	−19.1297	−0.836485	−0.418243	−	0.908335i	$-0.637354\pi$
$523$	−19.1297	−0.836485	−0.418243	+	0.908335i	$0.637354\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.97577	−0.260308
$528$	0	0
$529$	16.0419	0.697472
$530$	0	0
$531$	0.272582	0.0118290
$532$	0	0
$533$	−0.621911	−0.0269380
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.7935	0.465774
$538$	0	0
$539$	−16.1999	−0.697778
$540$	0	0
$541$	−26.6572	−1.14608	−0.573041	−	0.819527i	$-0.694236\pi$
$541$	−26.6572	−1.14608	−0.573041	+	0.819527i	$0.694236\pi$
$542$	0	0
$543$	8.11861	0.348403
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.9561	−0.724989	−0.362494	−	0.931986i	$-0.618075\pi$
$547$	−16.9561	−0.724989	−0.362494	+	0.931986i	$0.618075\pi$
$548$	0	0
$549$	4.78899	0.204389
$550$	0	0
$551$	−28.8420	−1.22871
$552$	0	0
$553$	18.5825	0.790208
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.5012	1.25001	0.625003	−	0.780622i	$-0.285098\pi$
$557$	29.5012	1.25001	0.625003	+	0.780622i	$0.285098\pi$
$558$	0	0
$559$	−1.54064	−0.0651620
$560$	0	0
$561$	3.81322	0.160994
$562$	0	0
$563$	41.4043	1.74498	0.872491	−	0.488629i	$-0.162503\pi$
$563$	41.4043	1.74498	0.872491	+	0.488629i	$0.162503\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.35386	−0.140849
$568$	0	0
$569$	−46.4618	−1.94778	−0.973890	−	0.227020i	$-0.927102\pi$
$569$	−46.4618	−1.94778	−0.973890	+	0.227020i	$0.927102\pi$
$570$	0	0
$571$	−1.82632	−0.0764291	−0.0382146	−	0.999270i	$-0.512167\pi$
$571$	−1.82632	−0.0764291	−0.0382146	+	0.999270i	$0.512167\pi$
$572$	0	0
$573$	25.5780	1.06854
$574$	0	0
$575$	0	0
$576$	0	0
$577$	35.7980	1.49029	0.745146	−	0.666901i	$-0.232380\pi$
$577$	35.7980	1.49029	0.745146	+	0.666901i	$0.232380\pi$
$578$	0	0
$579$	19.1671	0.796556
$580$	0	0
$581$	−21.0373	−0.872776
$582$	0	0
$583$	−27.0197	−1.11904
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.9297	1.44170	0.720852	−	0.693089i	$-0.243751\pi$
$587$	34.9297	1.44170	0.720852	+	0.693089i	$0.243751\pi$
$588$	0	0
$589$	−25.3871	−1.04606
$590$	0	0
$591$	−5.04394	−0.207480
$592$	0	0
$593$	−22.4967	−0.923829	−0.461914	−	0.886925i	$-0.652837\pi$
$593$	−22.4967	−0.923829	−0.461914	+	0.886925i	$0.652837\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.91873	0.119455
$598$	0	0
$599$	24.7450	1.01106	0.505528	−	0.862810i	$-0.331298\pi$
$599$	24.7450	1.01106	0.505528	+	0.862810i	$0.331298\pi$
$600$	0	0
$601$	−1.57345	−0.0641822	−0.0320911	−	0.999485i	$-0.510217\pi$
$601$	−1.57345	−0.0641822	−0.0320911	+	0.999485i	$0.510217\pi$
$602$	0	0
$603$	−1.81322	−0.0738400
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−36.9449	−1.49955	−0.749774	−	0.661694i	$-0.769838\pi$
$607$	−36.9449	−1.49955	−0.749774	+	0.661694i	$0.769838\pi$
$608$	0	0
$609$	−22.7693	−0.922658
$610$	0	0
$611$	9.14284	0.369880
$612$	0	0
$613$	17.2529	0.696837	0.348419	−	0.937339i	$-0.386719\pi$
$613$	17.2529	0.696837	0.348419	+	0.937339i	$0.386719\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.9515	0.642185	0.321092	−	0.947048i	$-0.395950\pi$
$617$	15.9515	0.642185	0.321092	+	0.947048i	$0.395950\pi$
$618$	0	0
$619$	11.7122	0.470755	0.235377	−	0.971904i	$-0.424367\pi$
$619$	11.7122	0.470755	0.235377	+	0.971904i	$0.424367\pi$
$620$	0	0
$621$	−6.24835	−0.250738
$622$	0	0
$623$	−12.7077	−0.509124
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.1999	0.646961
$628$	0	0
$629$	4.45936	0.177806
$630$	0	0
$631$	−28.0373	−1.11615	−0.558074	−	0.829791i	$-0.688460\pi$
$631$	−28.0373	−1.11615	−0.558074	+	0.829791i	$0.688460\pi$
$632$	0	0
$633$	15.1671	0.602837
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.24835	0.168326
$638$	0	0
$639$	−9.16707	−0.362644
$640$	0	0
$641$	−35.6683	−1.40881	−0.704407	−	0.709797i	$-0.748787\pi$
$641$	−35.6683	−1.40881	−0.704407	+	0.709797i	$0.748787\pi$
$642$	0	0
$643$	−34.0858	−1.34421	−0.672106	−	0.740455i	$-0.734610\pi$
$643$	−34.0858	−1.34421	−0.672106	+	0.740455i	$0.734610\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.7890	−0.620729	−0.310365	−	0.950618i	$-0.600451\pi$
$647$	−15.7890	−0.620729	−0.310365	+	0.950618i	$0.600451\pi$
$648$	0	0
$649$	−1.03941	−0.0408005
$650$	0	0
$651$	−20.0419	−0.785502
$652$	0	0
$653$	16.9031	0.661468	0.330734	−	0.943724i	$-0.392704\pi$
$653$	16.9031	0.661468	0.330734	+	0.943724i	$0.392704\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.32962	0.129901
$658$	0	0
$659$	16.1252	0.628149	0.314075	−	0.949398i	$-0.398306\pi$
$659$	16.1252	0.628149	0.314075	+	0.949398i	$0.398306\pi$
$660$	0	0
$661$	20.5250	0.798329	0.399165	−	0.916879i	$-0.369300\pi$
$661$	20.5250	0.798329	0.399165	+	0.916879i	$0.369300\pi$
$662$	0	0
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−42.4199	−1.64251
$668$	0	0
$669$	28.4199	1.09878
$670$	0	0
$671$	−18.2614	−0.704975
$672$	0	0
$673$	−39.9561	−1.54019	−0.770096	−	0.637927i	$-0.779792\pi$
$673$	−39.9561	−1.54019	−0.770096	+	0.637927i	$0.779792\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.49670	−0.0959559	−0.0479779	−	0.998848i	$-0.515278\pi$
$677$	−2.49670	−0.0959559	−0.0479779	+	0.998848i	$0.515278\pi$
$678$	0	0
$679$	−49.3275	−1.89302
$680$	0	0
$681$	12.4725	0.477946
$682$	0	0
$683$	30.3099	1.15978	0.579888	−	0.814696i	$-0.303096\pi$
$683$	30.3099	1.15978	0.579888	+	0.814696i	$0.303096\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	29.1559	1.11237
$688$	0	0
$689$	7.08580	0.269947
$690$	0	0
$691$	−50.3563	−1.91564	−0.957822	−	0.287362i	$-0.907222\pi$
$691$	−50.3563	−1.91564	−0.957822	+	0.287362i	$0.907222\pi$
$692$	0	0
$693$	12.7890	0.485813
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.621911	−0.0235566
$698$	0	0
$699$	2.21101	0.0836282
$700$	0	0
$701$	10.3670	0.391555	0.195777	−	0.980648i	$-0.437277\pi$
$701$	10.3670	0.391555	0.195777	+	0.980648i	$0.437277\pi$
$702$	0	0
$703$	18.9449	0.714522
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−23.6022	−0.887652
$708$	0	0
$709$	−7.87479	−0.295744	−0.147872	−	0.989007i	$-0.547242\pi$
$709$	−7.87479	−0.295744	−0.147872	+	0.989007i	$0.547242\pi$
$710$	0	0
$711$	−5.54064	−0.207790
$712$	0	0
$713$	−37.3387	−1.39834
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.8506	0.517258
$718$	0	0
$719$	18.1252	0.675956	0.337978	−	0.941154i	$-0.390257\pi$
$719$	18.1252	0.675956	0.337978	+	0.941154i	$0.390257\pi$
$720$	0	0
$721$	63.0307	2.34739
$722$	0	0
$723$	−7.49217	−0.278637
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−28.5340	−1.05827	−0.529134	−	0.848538i	$-0.677483\pi$
$727$	−28.5340	−1.05827	−0.529134	+	0.848538i	$0.677483\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.54064	−0.0569825
$732$	0	0
$733$	27.4245	1.01295	0.506473	−	0.862256i	$-0.330949\pi$
$733$	27.4245	1.01295	0.506473	+	0.862256i	$0.330949\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.91420	0.254688
$738$	0	0
$739$	2.52093	0.0927339	0.0463670	−	0.998924i	$-0.485236\pi$
$739$	2.52093	0.0927339	0.0463670	+	0.998924i	$0.485236\pi$
$740$	0	0
$741$	−4.24835	−0.156067
$742$	0	0
$743$	−13.9364	−0.511275	−0.255638	−	0.966773i	$-0.582285\pi$
$743$	−13.9364	−0.511275	−0.255638	+	0.966773i	$0.582285\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.27258	0.229502
$748$	0	0
$749$	16.3341	0.596837
$750$	0	0
$751$	−24.2574	−0.885165	−0.442583	−	0.896728i	$-0.645938\pi$
$751$	−24.2574	−0.885165	−0.442583	+	0.896728i	$0.645938\pi$
$752$	0	0
$753$	4.78446	0.174355
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.32510	0.0481615	0.0240807	−	0.999710i	$-0.492334\pi$
$757$	1.32510	0.0481615	0.0240807	+	0.999710i	$0.492334\pi$
$758$	0	0
$759$	23.8263	0.864841
$760$	0	0
$761$	−7.32962	−0.265699	−0.132849	−	0.991136i	$-0.542413\pi$
$761$	−7.32962	−0.265699	−0.132849	+	0.991136i	$0.542413\pi$
$762$	0	0
$763$	−4.45936	−0.161440
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.272582	0.00984235
$768$	0	0
$769$	4.70771	0.169764	0.0848822	−	0.996391i	$-0.472949\pi$
$769$	4.70771	0.169764	0.0848822	+	0.996391i	$0.472949\pi$
$770$	0	0
$771$	15.4199	0.555336
$772$	0	0
$773$	18.7077	0.672870	0.336435	−	0.941707i	$-0.390779\pi$
$773$	18.7077	0.672870	0.336435	+	0.941707i	$0.390779\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14.9561	0.536546
$778$	0	0
$779$	−2.64210	−0.0946629
$780$	0	0
$781$	34.9561	1.25083
$782$	0	0
$783$	6.78899	0.242619
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15.0176	0.535321	0.267660	−	0.963513i	$-0.413750\pi$
$787$	15.0176	0.535321	0.267660	+	0.963513i	$0.413750\pi$
$788$	0	0
$789$	29.8748	1.06357
$790$	0	0
$791$	19.4154	0.690333
$792$	0	0
$793$	4.78899	0.170062
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.2135	1.77865	0.889326	−	0.457274i	$-0.151174\pi$
$797$	50.2135	1.77865	0.889326	+	0.457274i	$0.151174\pi$
$798$	0	0
$799$	9.14284	0.323450
$800$	0	0
$801$	3.78899	0.133877
$802$	0	0
$803$	−12.6966	−0.448053
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.16255	−0.252134
$808$	0	0
$809$	−13.7143	−0.482170	−0.241085	−	0.970504i	$-0.577503\pi$
$809$	−13.7143	−0.482170	−0.241085	+	0.970504i	$0.577503\pi$
$810$	0	0
$811$	30.3473	1.06564	0.532818	−	0.846230i	$-0.321133\pi$
$811$	30.3473	1.06564	0.532818	+	0.846230i	$0.321133\pi$
$812$	0	0
$813$	−16.1847	−0.567622
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.54516	−0.228986
$818$	0	0
$819$	−3.35386	−0.117193
$820$	0	0
$821$	−0.287762	−0.0100430	−0.00502148	−	0.999987i	$-0.501598\pi$
$821$	−0.287762	−0.0100430	−0.00502148	+	0.999987i	$0.501598\pi$
$822$	0	0
$823$	−41.1075	−1.43292	−0.716458	−	0.697630i	$-0.754238\pi$
$823$	−41.1075	−1.43292	−0.716458	+	0.697630i	$0.754238\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.0176	−1.28723	−0.643615	−	0.765350i	$-0.722566\pi$
$827$	−37.0176	−1.28723	−0.643615	+	0.765350i	$0.722566\pi$
$828$	0	0
$829$	−50.7011	−1.76092	−0.880461	−	0.474118i	$-0.842767\pi$
$829$	−50.7011	−1.76092	−0.880461	+	0.474118i	$0.842767\pi$
$830$	0	0
$831$	25.5386	0.885923
$832$	0	0
$833$	4.24835	0.147197
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.97577	0.206553
$838$	0	0
$839$	44.4199	1.53355	0.766773	−	0.641918i	$-0.221861\pi$
$839$	44.4199	1.53355	0.766773	+	0.641918i	$0.221861\pi$
$840$	0	0
$841$	17.0903	0.589322
$842$	0	0
$843$	−8.65925	−0.298240
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11.8748	−0.408022
$848$	0	0
$849$	−15.9121	−0.546103
$850$	0	0
$851$	27.8637	0.955154
$852$	0	0
$853$	−14.4088	−0.493349	−0.246674	−	0.969098i	$-0.579338\pi$
$853$	−14.4088	−0.493349	−0.246674	+	0.969098i	$0.579338\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−45.1559	−1.54250	−0.771249	−	0.636534i	$-0.780368\pi$
$857$	−45.1559	−1.54250	−0.771249	+	0.636534i	$0.780368\pi$
$858$	0	0
$859$	−7.54969	−0.257592	−0.128796	−	0.991671i	$-0.541111\pi$
$859$	−7.54969	−0.257592	−0.128796	+	0.991671i	$0.541111\pi$
$860$	0	0
$861$	−2.08580	−0.0710839
$862$	0	0
$863$	28.8551	0.982238	0.491119	−	0.871092i	$-0.336588\pi$
$863$	28.8551	0.982238	0.491119	+	0.871092i	$0.336588\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	21.1277	0.716707
$870$	0	0
$871$	−1.81322	−0.0614386
$872$	0	0
$873$	14.7077	0.497781
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25.7011	0.867865	0.433932	−	0.900945i	$-0.357126\pi$
$877$	25.7011	0.867865	0.433932	+	0.900945i	$0.357126\pi$
$878$	0	0
$879$	13.2529	0.447008
$880$	0	0
$881$	4.32962	0.145869	0.0729343	−	0.997337i	$-0.476764\pi$
$881$	4.32962	0.145869	0.0729343	+	0.997337i	$0.476764\pi$
$882$	0	0
$883$	−53.3760	−1.79625	−0.898123	−	0.439745i	$-0.855069\pi$
$883$	−53.3760	−1.79625	−0.898123	+	0.439745i	$0.855069\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.19988	−0.275325	−0.137663	−	0.990479i	$-0.543959\pi$
$887$	−8.19988	−0.275325	−0.137663	+	0.990479i	$0.543959\pi$
$888$	0	0
$889$	−24.0373	−0.806186
$890$	0	0
$891$	−3.81322	−0.127748
$892$	0	0
$893$	38.8420	1.29980
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.24835	−0.208626
$898$	0	0
$899$	40.5694	1.35307
$900$	0	0
$901$	7.08580	0.236062
$902$	0	0
$903$	−5.16707	−0.171949
$904$	0	0
$905$	0	0
$906$	0	0
$907$	22.4088	0.744073	0.372036	−	0.928218i	$-0.378660\pi$
$907$	22.4088	0.744073	0.372036	+	0.928218i	$0.378660\pi$
$908$	0	0
$909$	7.03733	0.233414
$910$	0	0
$911$	−2.57345	−0.0852620	−0.0426310	−	0.999091i	$-0.513574\pi$
$911$	−2.57345	−0.0852620	−0.0426310	+	0.999091i	$0.513574\pi$
$912$	0	0
$913$	−23.9187	−0.791594
$914$	0	0
$915$	0	0
$916$	0	0
$917$	23.4922	0.775780
$918$	0	0
$919$	22.7824	0.751521	0.375761	−	0.926717i	$-0.377381\pi$
$919$	22.7824	0.751521	0.375761	+	0.926717i	$0.377381\pi$
$920$	0	0
$921$	2.12521	0.0700282
$922$	0	0
$923$	−9.16707	−0.301738
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−18.7935	−0.617260
$928$	0	0
$929$	−14.5936	−0.478801	−0.239401	−	0.970921i	$-0.576951\pi$
$929$	−14.5936	−0.478801	−0.239401	+	0.970921i	$0.576951\pi$
$930$	0	0
$931$	18.0485	0.591515
$932$	0	0
$933$	−19.9121	−0.651894
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.11408	−0.167070	−0.0835349	−	0.996505i	$-0.526621\pi$
$937$	−5.11408	−0.167070	−0.0835349	+	0.996505i	$0.526621\pi$
$938$	0	0
$939$	−6.50122	−0.212159
$940$	0	0
$941$	−44.5734	−1.45305	−0.726526	−	0.687139i	$-0.758867\pi$
$941$	−44.5734	−1.45305	−0.726526	+	0.687139i	$0.758867\pi$
$942$	0	0
$943$	−3.88592	−0.126543
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.3493	−0.921229	−0.460615	−	0.887600i	$-0.652371\pi$
$947$	−28.3493	−0.921229	−0.460615	+	0.887600i	$0.652371\pi$
$948$	0	0
$949$	3.32962	0.108084
$950$	0	0
$951$	−23.3296	−0.756515
$952$	0	0
$953$	49.7056	1.61012	0.805062	−	0.593191i	$-0.202132\pi$
$953$	49.7056	1.61012	0.805062	+	0.593191i	$0.202132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−25.8879	−0.836837
$958$	0	0
$959$	−3.78899	−0.122353
$960$	0	0
$961$	4.70979	0.151929
$962$	0	0
$963$	−4.87026	−0.156942
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.21959	0.167851	0.0839253	−	0.996472i	$-0.473254\pi$
$967$	5.21959	0.167851	0.0839253	+	0.996472i	$0.473254\pi$
$968$	0	0
$969$	−4.24835	−0.136477
$970$	0	0
$971$	48.1978	1.54674	0.773371	−	0.633954i	$-0.218569\pi$
$971$	48.1978	1.54674	0.773371	+	0.633954i	$0.218569\pi$
$972$	0	0
$973$	13.9606	0.447556
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.9187	0.797221	0.398610	−	0.917120i	$-0.369493\pi$
$977$	24.9187	0.797221	0.398610	+	0.917120i	$0.369493\pi$
$978$	0	0
$979$	−14.4482	−0.461767
$980$	0	0
$981$	1.32962	0.0424516
$982$	0	0
$983$	37.5537	1.19778	0.598889	−	0.800832i	$-0.295609\pi$
$983$	37.5537	1.19778	0.598889	+	0.800832i	$0.295609\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30.6638	0.976039
$988$	0	0
$989$	−9.62644	−0.306103
$990$	0	0
$991$	−47.8354	−1.51954	−0.759770	−	0.650192i	$-0.774689\pi$
$991$	−47.8354	−1.51954	−0.759770	+	0.650192i	$0.774689\pi$
$992$	0	0
$993$	7.32962	0.232599
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−40.0509	−1.26843	−0.634213	−	0.773159i	$-0.718676\pi$
$997$	−40.0509	−1.26843	−0.634213	+	0.773159i	$0.718676\pi$
$998$	0	0
$999$	−4.45936	−0.141088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bg.1.1	✓	3
5.4	even	2		7800.2.a.br.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bg.1.1	✓	3	1.1	even	1	trivial
7800.2.a.br.1.3	yes	3	5.4	even	2

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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.35386 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.35386 q^{7} +1.00000 q^{9} -3.81322 q^{11} +1.00000 q^{13} +1.00000 q^{17} +4.24835 q^{19} +3.35386 q^{21} +6.24835 q^{23} -1.00000 q^{27} -6.78899 q^{29} -5.97577 q^{31} +3.81322 q^{33} +4.45936 q^{37} -1.00000 q^{39} -0.621911 q^{41} -1.54064 q^{43} +9.14284 q^{47} +4.24835 q^{49} -1.00000 q^{51} +7.08580 q^{53} -4.24835 q^{57} +0.272582 q^{59} +4.78899 q^{61} -3.35386 q^{63} -1.81322 q^{67} -6.24835 q^{69} -9.16707 q^{71} +3.32962 q^{73} +12.7890 q^{77} -5.54064 q^{79} +1.00000 q^{81} +6.27258 q^{83} +6.78899 q^{87} +3.78899 q^{89} -3.35386 q^{91} +5.97577 q^{93} +14.7077 q^{97} -3.81322 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} - 6 q^{19} + q^{21} - 3 q^{27} - q^{29} - 7 q^{31} + 3 q^{33} + 14 q^{37} - 3 q^{39} - 4 q^{43} - q^{47} - 6 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{57} - 7 q^{59} - 5 q^{61} - q^{63} + 3 q^{67} - 10 q^{71} - 10 q^{73} + 19 q^{77} - 16 q^{79} + 3 q^{81} + 11 q^{83} + q^{87} - 8 q^{89} - q^{91} + 7 q^{93} + 26 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^17 - 6 * q^19 + q^21 - 3 * q^27 - q^29 - 7 * q^31 + 3 * q^33 + 14 * q^37 - 3 * q^39 - 4 * q^43 - q^47 - 6 * q^49 - 3 * q^51 + 5 * q^53 + 6 * q^57 - 7 * q^59 - 5 * q^61 - q^63 + 3 * q^67 - 10 * q^71 - 10 * q^73 + 19 * q^77 - 16 * q^79 + 3 * q^81 + 11 * q^83 + q^87 - 8 * q^89 - q^91 + 7 * q^93 + 26 * q^97 - 3 * q^99

Introduction

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