LMFDB - Embedded newform 7600.2.a.ch.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 15x^2 - 14*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3800)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.26143$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.26143 q^{3} -4.07225 q^{7} +7.63693 q^{9} +O(q^{10})$	$q-3.26143 q^{3} -4.07225 q^{7} +7.63693 q^{9} +0.786366 q^{11} +1.07974 q^{13} -1.90793 q^{17} +1.00000 q^{19} +13.2813 q^{21} +1.41383 q^{23} -15.1230 q^{27} -7.26439 q^{29} -2.22003 q^{31} -2.56468 q^{33} +9.14283 q^{37} -3.52151 q^{39} -6.11703 q^{41} -8.40634 q^{43} -3.56468 q^{47} +9.58319 q^{49} +6.22257 q^{51} -8.57472 q^{53} -3.26143 q^{57} +13.4043 q^{59} +12.7768 q^{61} -31.0994 q^{63} +5.10008 q^{67} -4.61112 q^{69} -1.65535 q^{71} +10.3302 q^{73} -3.20228 q^{77} +16.2256 q^{79} +26.4119 q^{81} -9.35104 q^{83} +23.6923 q^{87} +3.10419 q^{89} -4.39698 q^{91} +7.24046 q^{93} +4.55874 q^{97} +6.00542 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} - 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} - 19 q^{71} + 3 q^{73} - 36 q^{77} + 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} - 42 q^{91} + 39 q^{93} - 11 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 - 3 * q^11 - q^13 - 14 * q^17 + 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 - 5 * q^31 + 2 * q^33 - 8 * q^37 - 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 - 33 * q^51 + 13 * q^53 - 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 - 19 * q^71 + 3 * q^73 - 36 * q^77 + 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 - 42 * q^91 + 39 * q^93 - 11 * q^97 + 6 * 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
$3$	−3.26143	−1.88299	−0.941494	−	0.337031i	$-0.890577\pi$
$3$	−3.26143	−1.88299	−0.941494	+	0.337031i	$0.890577\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.07225	−1.53916	−0.769582	−	0.638548i	$-0.779536\pi$
$7$	−4.07225	−1.53916	−0.769582	+	0.638548i	$0.779536\pi$
$8$	0	0
$9$	7.63693	2.54564
$10$	0	0
$11$	0.786366	0.237098	0.118549	−	0.992948i	$-0.462176\pi$
$11$	0.786366	0.237098	0.118549	+	0.992948i	$0.462176\pi$
$12$	0	0
$13$	1.07974	0.299467	0.149733	−	0.988726i	$-0.452158\pi$
$13$	1.07974	0.299467	0.149733	+	0.988726i	$0.452158\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.90793	−0.462741	−0.231370	−	0.972866i	$-0.574321\pi$
$17$	−1.90793	−0.462741	−0.231370	+	0.972866i	$0.574321\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	13.2813	2.89823
$22$	0	0
$23$	1.41383	0.294805	0.147402	−	0.989077i	$-0.452909\pi$
$23$	1.41383	0.294805	0.147402	+	0.989077i	$0.452909\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−15.1230	−2.91042
$28$	0	0
$29$	−7.26439	−1.34896	−0.674482	−	0.738291i	$-0.735633\pi$
$29$	−7.26439	−1.34896	−0.674482	+	0.738291i	$0.735633\pi$
$30$	0	0
$31$	−2.22003	−0.398728	−0.199364	−	0.979925i	$-0.563888\pi$
$31$	−2.22003	−0.398728	−0.199364	+	0.979925i	$0.563888\pi$
$32$	0	0
$33$	−2.56468	−0.446453
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.14283	1.50307	0.751536	−	0.659692i	$-0.229313\pi$
$37$	9.14283	1.50307	0.751536	+	0.659692i	$0.229313\pi$
$38$	0	0
$39$	−3.52151	−0.563893
$40$	0	0
$41$	−6.11703	−0.955319	−0.477660	−	0.878545i	$-0.658515\pi$
$41$	−6.11703	−0.955319	−0.477660	+	0.878545i	$0.658515\pi$
$42$	0	0
$43$	−8.40634	−1.28195	−0.640977	−	0.767560i	$-0.721471\pi$
$43$	−8.40634	−1.28195	−0.640977	+	0.767560i	$0.721471\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.56468	−0.519962	−0.259981	−	0.965614i	$-0.583716\pi$
$47$	−3.56468	−0.519962	−0.259981	+	0.965614i	$0.583716\pi$
$48$	0	0
$49$	9.58319	1.36903
$50$	0	0
$51$	6.22257	0.871335
$52$	0	0
$53$	−8.57472	−1.17783	−0.588914	−	0.808195i	$-0.700445\pi$
$53$	−8.57472	−1.17783	−0.588914	+	0.808195i	$0.700445\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.26143	−0.431987
$58$	0	0
$59$	13.4043	1.74509	0.872543	−	0.488537i	$-0.162469\pi$
$59$	13.4043	1.74509	0.872543	+	0.488537i	$0.162469\pi$
$60$	0	0
$61$	12.7768	1.63590	0.817950	−	0.575289i	$-0.195110\pi$
$61$	12.7768	1.63590	0.817950	+	0.575289i	$0.195110\pi$
$62$	0	0
$63$	−31.0994	−3.91816
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.10008	0.623073	0.311537	−	0.950234i	$-0.399156\pi$
$67$	5.10008	0.623073	0.311537	+	0.950234i	$0.399156\pi$
$68$	0	0
$69$	−4.61112	−0.555114
$70$	0	0
$71$	−1.65535	−0.196454	−0.0982268	−	0.995164i	$-0.531317\pi$
$71$	−1.65535	−0.196454	−0.0982268	+	0.995164i	$0.531317\pi$
$72$	0	0
$73$	10.3302	1.20906	0.604532	−	0.796581i	$-0.293360\pi$
$73$	10.3302	1.20906	0.604532	+	0.796581i	$0.293360\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.20228	−0.364933
$78$	0	0
$79$	16.2256	1.82553	0.912764	−	0.408488i	$-0.133944\pi$
$79$	16.2256	1.82553	0.912764	+	0.408488i	$0.133944\pi$
$80$	0	0
$81$	26.4119	2.93465
$82$	0	0
$83$	−9.35104	−1.02641	−0.513205	−	0.858266i	$-0.671542\pi$
$83$	−9.35104	−1.02641	−0.513205	+	0.858266i	$0.671542\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	23.6923	2.54008
$88$	0	0
$89$	3.10419	0.329044	0.164522	−	0.986373i	$-0.447392\pi$
$89$	3.10419	0.329044	0.164522	+	0.986373i	$0.447392\pi$
$90$	0	0
$91$	−4.39698	−0.460929
$92$	0	0
$93$	7.24046	0.750801
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.55874	0.462870	0.231435	−	0.972850i	$-0.425658\pi$
$97$	4.55874	0.462870	0.231435	+	0.972850i	$0.425658\pi$
$98$	0	0
$99$	6.00542	0.603567
$100$	0	0
$101$	0.743401	0.0739712	0.0369856	−	0.999316i	$-0.488224\pi$
$101$	0.743401	0.0739712	0.0369856	+	0.999316i	$0.488224\pi$
$102$	0	0
$103$	10.3710	1.02188	0.510941	−	0.859616i	$-0.329297\pi$
$103$	10.3710	1.02188	0.510941	+	0.859616i	$0.329297\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.1400	1.65698	0.828491	−	0.560002i	$-0.189200\pi$
$107$	17.1400	1.65698	0.828491	+	0.560002i	$0.189200\pi$
$108$	0	0
$109$	−17.1878	−1.64629	−0.823144	−	0.567832i	$-0.807782\pi$
$109$	−17.1878	−1.64629	−0.823144	+	0.567832i	$0.807782\pi$
$110$	0	0
$111$	−29.8187	−2.83027
$112$	0	0
$113$	−1.05696	−0.0994300	−0.0497150	−	0.998763i	$-0.515831\pi$
$113$	−1.05696	−0.0994300	−0.0497150	+	0.998763i	$0.515831\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.24592	0.762336
$118$	0	0
$119$	7.76956	0.712234
$120$	0	0
$121$	−10.3816	−0.943784
$122$	0	0
$123$	19.9503	1.79885
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.22444	0.463594	0.231797	−	0.972764i	$-0.425539\pi$
$127$	5.22444	0.463594	0.231797	+	0.972764i	$0.425539\pi$
$128$	0	0
$129$	27.4167	2.41390
$130$	0	0
$131$	−3.00267	−0.262344	−0.131172	−	0.991360i	$-0.541874\pi$
$131$	−3.00267	−0.262344	−0.131172	+	0.991360i	$0.541874\pi$
$132$	0	0
$133$	−4.07225	−0.353109
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−23.0922	−1.97290	−0.986448	−	0.164072i	$-0.947537\pi$
$137$	−23.0922	−1.97290	−0.986448	+	0.164072i	$0.947537\pi$
$138$	0	0
$139$	11.9630	1.01469	0.507343	−	0.861744i	$-0.330628\pi$
$139$	11.9630	1.01469	0.507343	+	0.861744i	$0.330628\pi$
$140$	0	0
$141$	11.6259	0.979082
$142$	0	0
$143$	0.849074	0.0710031
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−31.2549	−2.57786
$148$	0	0
$149$	23.5411	1.92856	0.964282	−	0.264879i	$-0.0853320\pi$
$149$	23.5411	1.92856	0.964282	+	0.264879i	$0.0853320\pi$
$150$	0	0
$151$	−10.6136	−0.863721	−0.431860	−	0.901940i	$-0.642143\pi$
$151$	−10.6136	−0.863721	−0.431860	+	0.901940i	$0.642143\pi$
$152$	0	0
$153$	−14.5707	−1.17797
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.9997	1.11730	0.558648	−	0.829405i	$-0.311320\pi$
$157$	13.9997	1.11730	0.558648	+	0.829405i	$0.311320\pi$
$158$	0	0
$159$	27.9659	2.21784
$160$	0	0
$161$	−5.75748	−0.453753
$162$	0	0
$163$	10.6662	0.835442	0.417721	−	0.908575i	$-0.362829\pi$
$163$	10.6662	0.835442	0.417721	+	0.908575i	$0.362829\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.97856	0.307870	0.153935	−	0.988081i	$-0.450805\pi$
$167$	3.97856	0.307870	0.153935	+	0.988081i	$0.450805\pi$
$168$	0	0
$169$	−11.8342	−0.910320
$170$	0	0
$171$	7.63693	0.584010
$172$	0	0
$173$	22.1310	1.68259	0.841295	−	0.540577i	$-0.181794\pi$
$173$	22.1310	1.68259	0.841295	+	0.540577i	$0.181794\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−43.7171	−3.28598
$178$	0	0
$179$	−4.80086	−0.358833	−0.179417	−	0.983773i	$-0.557421\pi$
$179$	−4.80086	−0.358833	−0.179417	+	0.983773i	$0.557421\pi$
$180$	0	0
$181$	−16.4333	−1.22148	−0.610740	−	0.791831i	$-0.709128\pi$
$181$	−16.4333	−1.22148	−0.610740	+	0.791831i	$0.709128\pi$
$182$	0	0
$183$	−41.6706	−3.08038
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.50033	−0.109715
$188$	0	0
$189$	61.5846	4.47962
$190$	0	0
$191$	−5.45151	−0.394457	−0.197229	−	0.980358i	$-0.563194\pi$
$191$	−5.45151	−0.394457	−0.197229	+	0.980358i	$0.563194\pi$
$192$	0	0
$193$	−16.3968	−1.18026	−0.590132	−	0.807306i	$-0.700924\pi$
$193$	−16.3968	−1.18026	−0.590132	+	0.807306i	$0.700924\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.2546	1.37183	0.685916	−	0.727681i	$-0.259402\pi$
$197$	19.2546	1.37183	0.685916	+	0.727681i	$0.259402\pi$
$198$	0	0
$199$	−7.95572	−0.563966	−0.281983	−	0.959419i	$-0.590992\pi$
$199$	−7.95572	−0.563966	−0.281983	+	0.959419i	$0.590992\pi$
$200$	0	0
$201$	−16.6335	−1.17324
$202$	0	0
$203$	29.5824	2.07628
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10.7973	0.750468
$208$	0	0
$209$	0.786366	0.0543941
$210$	0	0
$211$	−18.2270	−1.25480	−0.627400	−	0.778697i	$-0.715881\pi$
$211$	−18.2270	−1.25480	−0.627400	+	0.778697i	$0.715881\pi$
$212$	0	0
$213$	5.39880	0.369920
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.04049	0.613709
$218$	0	0
$219$	−33.6914	−2.27665
$220$	0	0
$221$	−2.06007	−0.138576
$222$	0	0
$223$	0.430994	0.0288615	0.0144307	−	0.999896i	$-0.495406\pi$
$223$	0.430994	0.0288615	0.0144307	+	0.999896i	$0.495406\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.63334	0.506643	0.253321	−	0.967382i	$-0.418477\pi$
$227$	7.63334	0.506643	0.253321	+	0.967382i	$0.418477\pi$
$228$	0	0
$229$	20.3914	1.34750	0.673750	−	0.738959i	$-0.264682\pi$
$229$	20.3914	1.34750	0.673750	+	0.738959i	$0.264682\pi$
$230$	0	0
$231$	10.4440	0.687165
$232$	0	0
$233$	−20.6308	−1.35157	−0.675785	−	0.737099i	$-0.736195\pi$
$233$	−20.6308	−1.35157	−0.675785	+	0.737099i	$0.736195\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−52.9188	−3.43744
$238$	0	0
$239$	4.61247	0.298356	0.149178	−	0.988810i	$-0.452337\pi$
$239$	4.61247	0.298356	0.149178	+	0.988810i	$0.452337\pi$
$240$	0	0
$241$	3.92363	0.252744	0.126372	−	0.991983i	$-0.459667\pi$
$241$	3.92363	0.252744	0.126372	+	0.991983i	$0.459667\pi$
$242$	0	0
$243$	−40.7714	−2.61549
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.07974	0.0687024
$248$	0	0
$249$	30.4978	1.93272
$250$	0	0
$251$	−23.9491	−1.51165	−0.755827	−	0.654771i	$-0.772765\pi$
$251$	−23.9491	−1.51165	−0.755827	+	0.654771i	$0.772765\pi$
$252$	0	0
$253$	1.11179	0.0698978
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.05331	0.564730	0.282365	−	0.959307i	$-0.408881\pi$
$257$	9.05331	0.564730	0.282365	+	0.959307i	$0.408881\pi$
$258$	0	0
$259$	−37.2319	−2.31348
$260$	0	0
$261$	−55.4776	−3.43398
$262$	0	0
$263$	−15.4483	−0.952583	−0.476291	−	0.879287i	$-0.658019\pi$
$263$	−15.4483	−0.952583	−0.476291	+	0.879287i	$0.658019\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−10.1241	−0.619586
$268$	0	0
$269$	−4.44718	−0.271150	−0.135575	−	0.990767i	$-0.543288\pi$
$269$	−4.44718	−0.271150	−0.135575	+	0.990767i	$0.543288\pi$
$270$	0	0
$271$	−23.8511	−1.44885	−0.724425	−	0.689354i	$-0.757895\pi$
$271$	−23.8511	−1.44885	−0.724425	+	0.689354i	$0.757895\pi$
$272$	0	0
$273$	14.3404	0.867923
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.86751	0.412629	0.206314	−	0.978486i	$-0.433853\pi$
$277$	6.86751	0.412629	0.206314	+	0.978486i	$0.433853\pi$
$278$	0	0
$279$	−16.9542	−1.01502
$280$	0	0
$281$	26.7419	1.59529	0.797645	−	0.603128i	$-0.206079\pi$
$281$	26.7419	1.59529	0.797645	+	0.603128i	$0.206079\pi$
$282$	0	0
$283$	17.4698	1.03847	0.519235	−	0.854632i	$-0.326217\pi$
$283$	17.4698	1.03847	0.519235	+	0.854632i	$0.326217\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.9101	1.47039
$288$	0	0
$289$	−13.3598	−0.785871
$290$	0	0
$291$	−14.8680	−0.871579
$292$	0	0
$293$	−14.8968	−0.870282	−0.435141	−	0.900362i	$-0.643302\pi$
$293$	−14.8968	−0.870282	−0.435141	+	0.900362i	$0.643302\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−11.8922	−0.690057
$298$	0	0
$299$	1.52658	0.0882843
$300$	0	0
$301$	34.2327	1.97314
$302$	0	0
$303$	−2.42455	−0.139287
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−11.2895	−0.644326	−0.322163	−	0.946684i	$-0.604410\pi$
$307$	−11.2895	−0.644326	−0.322163	+	0.946684i	$0.604410\pi$
$308$	0	0
$309$	−33.8242	−1.92419
$310$	0	0
$311$	5.65634	0.320741	0.160371	−	0.987057i	$-0.448731\pi$
$311$	5.65634	0.320741	0.160371	+	0.987057i	$0.448731\pi$
$312$	0	0
$313$	−29.9564	−1.69324	−0.846618	−	0.532201i	$-0.821365\pi$
$313$	−29.9564	−1.69324	−0.846618	+	0.532201i	$0.821365\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.7992	0.662706	0.331353	−	0.943507i	$-0.392495\pi$
$317$	11.7992	0.662706	0.331353	+	0.943507i	$0.392495\pi$
$318$	0	0
$319$	−5.71247	−0.319837
$320$	0	0
$321$	−55.9008	−3.12008
$322$	0	0
$323$	−1.90793	−0.106160
$324$	0	0
$325$	0	0
$326$	0	0
$327$	56.0567	3.09994
$328$	0	0
$329$	14.5163	0.800307
$330$	0	0
$331$	21.7197	1.19382	0.596912	−	0.802307i	$-0.296394\pi$
$331$	21.7197	1.19382	0.596912	+	0.802307i	$0.296394\pi$
$332$	0	0
$333$	69.8231	3.82628
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−32.7221	−1.78249	−0.891245	−	0.453523i	$-0.850167\pi$
$337$	−32.7221	−1.78249	−0.891245	+	0.453523i	$0.850167\pi$
$338$	0	0
$339$	3.44718	0.187225
$340$	0	0
$341$	−1.74575	−0.0945378
$342$	0	0
$343$	−10.5194	−0.567994
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.5890	1.53474	0.767369	−	0.641206i	$-0.221565\pi$
$347$	28.5890	1.53474	0.767369	+	0.641206i	$0.221565\pi$
$348$	0	0
$349$	22.4911	1.20392	0.601961	−	0.798525i	$-0.294386\pi$
$349$	22.4911	1.20392	0.601961	+	0.798525i	$0.294386\pi$
$350$	0	0
$351$	−16.3290	−0.871576
$352$	0	0
$353$	−21.2381	−1.13039	−0.565196	−	0.824957i	$-0.691199\pi$
$353$	−21.2381	−1.13039	−0.565196	+	0.824957i	$0.691199\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−25.3399	−1.34113
$358$	0	0
$359$	5.30705	0.280095	0.140048	−	0.990145i	$-0.455274\pi$
$359$	5.30705	0.280095	0.140048	+	0.990145i	$0.455274\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	33.8590	1.77713
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.12291	−0.0586156	−0.0293078	−	0.999570i	$-0.509330\pi$
$367$	−1.12291	−0.0586156	−0.0293078	+	0.999570i	$0.509330\pi$
$368$	0	0
$369$	−46.7153	−2.43190
$370$	0	0
$371$	34.9184	1.81287
$372$	0	0
$373$	3.99673	0.206943	0.103471	−	0.994632i	$-0.467005\pi$
$373$	3.99673	0.206943	0.103471	+	0.994632i	$0.467005\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.84368	−0.403970
$378$	0	0
$379$	−3.36846	−0.173026	−0.0865132	−	0.996251i	$-0.527572\pi$
$379$	−3.36846	−0.173026	−0.0865132	+	0.996251i	$0.527572\pi$
$380$	0	0
$381$	−17.0392	−0.872942
$382$	0	0
$383$	−17.8476	−0.911969	−0.455985	−	0.889988i	$-0.650713\pi$
$383$	−17.8476	−0.911969	−0.455985	+	0.889988i	$0.650713\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−64.1986	−3.26340
$388$	0	0
$389$	−19.1690	−0.971907	−0.485953	−	0.873985i	$-0.661528\pi$
$389$	−19.1690	−0.971907	−0.485953	+	0.873985i	$0.661528\pi$
$390$	0	0
$391$	−2.69750	−0.136418
$392$	0	0
$393$	9.79298	0.493991
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.04085	0.453747	0.226874	−	0.973924i	$-0.427150\pi$
$397$	9.04085	0.453747	0.226874	+	0.973924i	$0.427150\pi$
$398$	0	0
$399$	13.2813	0.664899
$400$	0	0
$401$	−15.0634	−0.752230	−0.376115	−	0.926573i	$-0.622740\pi$
$401$	−15.0634	−0.752230	−0.376115	+	0.926573i	$0.622740\pi$
$402$	0	0
$403$	−2.39706	−0.119406
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.18961	0.356376
$408$	0	0
$409$	−7.22091	−0.357051	−0.178525	−	0.983935i	$-0.557133\pi$
$409$	−7.22091	−0.357051	−0.178525	+	0.983935i	$0.557133\pi$
$410$	0	0
$411$	75.3135	3.71494
$412$	0	0
$413$	−54.5855	−2.68597
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−39.0164	−1.91064
$418$	0	0
$419$	21.0968	1.03065	0.515323	−	0.856996i	$-0.327672\pi$
$419$	21.0968	1.03065	0.515323	+	0.856996i	$0.327672\pi$
$420$	0	0
$421$	5.50321	0.268210	0.134105	−	0.990967i	$-0.457184\pi$
$421$	5.50321	0.268210	0.134105	+	0.990967i	$0.457184\pi$
$422$	0	0
$423$	−27.2232	−1.32364
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−52.0303	−2.51792
$428$	0	0
$429$	−2.76920	−0.133698
$430$	0	0
$431$	10.9748	0.528640	0.264320	−	0.964435i	$-0.414853\pi$
$431$	10.9748	0.528640	0.264320	+	0.964435i	$0.414853\pi$
$432$	0	0
$433$	14.2313	0.683915	0.341957	−	0.939715i	$-0.388910\pi$
$433$	14.2313	0.683915	0.341957	+	0.939715i	$0.388910\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.41383	0.0676329
$438$	0	0
$439$	−12.0446	−0.574859	−0.287429	−	0.957802i	$-0.592801\pi$
$439$	−12.0446	−0.574859	−0.287429	+	0.957802i	$0.592801\pi$
$440$	0	0
$441$	73.1861	3.48505
$442$	0	0
$443$	−11.1718	−0.530789	−0.265394	−	0.964140i	$-0.585502\pi$
$443$	−11.1718	−0.530789	−0.265394	+	0.964140i	$0.585502\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−76.7777	−3.63146
$448$	0	0
$449$	−9.07650	−0.428347	−0.214173	−	0.976796i	$-0.568706\pi$
$449$	−9.07650	−0.428347	−0.214173	+	0.976796i	$0.568706\pi$
$450$	0	0
$451$	−4.81023	−0.226505
$452$	0	0
$453$	34.6154	1.62638
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.6795	−0.780235	−0.390118	−	0.920765i	$-0.627566\pi$
$457$	−16.6795	−0.780235	−0.390118	+	0.920765i	$0.627566\pi$
$458$	0	0
$459$	28.8536	1.34677
$460$	0	0
$461$	−31.2657	−1.45619	−0.728095	−	0.685476i	$-0.759594\pi$
$461$	−31.2657	−1.45619	−0.728095	+	0.685476i	$0.759594\pi$
$462$	0	0
$463$	10.1802	0.473116	0.236558	−	0.971617i	$-0.423981\pi$
$463$	10.1802	0.473116	0.236558	+	0.971617i	$0.423981\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.0731	−1.53044	−0.765220	−	0.643769i	$-0.777370\pi$
$467$	−33.0731	−1.53044	−0.765220	+	0.643769i	$0.777370\pi$
$468$	0	0
$469$	−20.7688	−0.959012
$470$	0	0
$471$	−45.6590	−2.10385
$472$	0	0
$473$	−6.61046	−0.303949
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−65.4845	−2.99833
$478$	0	0
$479$	−7.80955	−0.356827	−0.178414	−	0.983956i	$-0.557097\pi$
$479$	−7.80955	−0.356827	−0.178414	+	0.983956i	$0.557097\pi$
$480$	0	0
$481$	9.87191	0.450120
$482$	0	0
$483$	18.7776	0.854412
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.37584	0.198288	0.0991442	−	0.995073i	$-0.468389\pi$
$487$	4.37584	0.198288	0.0991442	+	0.995073i	$0.468389\pi$
$488$	0	0
$489$	−34.7871	−1.57313
$490$	0	0
$491$	−42.0821	−1.89914	−0.949568	−	0.313560i	$-0.898478\pi$
$491$	−42.0821	−1.89914	−0.949568	+	0.313560i	$0.898478\pi$
$492$	0	0
$493$	13.8599	0.624220
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.74098	0.302374
$498$	0	0
$499$	8.59546	0.384786	0.192393	−	0.981318i	$-0.438375\pi$
$499$	8.59546	0.384786	0.192393	+	0.981318i	$0.438375\pi$
$500$	0	0
$501$	−12.9758	−0.579716
$502$	0	0
$503$	−6.30722	−0.281225	−0.140612	−	0.990065i	$-0.544907\pi$
$503$	−6.30722	−0.281225	−0.140612	+	0.990065i	$0.544907\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	38.5963	1.71412
$508$	0	0
$509$	−31.0352	−1.37561	−0.687805	−	0.725896i	$-0.741426\pi$
$509$	−31.0352	−1.37561	−0.687805	+	0.725896i	$0.741426\pi$
$510$	0	0
$511$	−42.0673	−1.86095
$512$	0	0
$513$	−15.1230	−0.667697
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.80314	−0.123282
$518$	0	0
$519$	−72.1787	−3.16830
$520$	0	0
$521$	32.8924	1.44104	0.720522	−	0.693432i	$-0.243902\pi$
$521$	32.8924	1.44104	0.720522	+	0.693432i	$0.243902\pi$
$522$	0	0
$523$	37.6201	1.64501	0.822506	−	0.568757i	$-0.192575\pi$
$523$	37.6201	1.64501	0.822506	+	0.568757i	$0.192575\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.23565	0.184508
$528$	0	0
$529$	−21.0011	−0.913090
$530$	0	0
$531$	102.367	4.44236
$532$	0	0
$533$	−6.60482	−0.286087
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.6577	0.675679
$538$	0	0
$539$	7.53590	0.324594
$540$	0	0
$541$	−33.6736	−1.44774	−0.723871	−	0.689936i	$-0.757639\pi$
$541$	−33.6736	−1.44774	−0.723871	+	0.689936i	$0.757639\pi$
$542$	0	0
$543$	53.5962	2.30003
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−14.0295	−0.599859	−0.299930	−	0.953961i	$-0.596963\pi$
$547$	−14.0295	−0.599859	−0.299930	+	0.953961i	$0.596963\pi$
$548$	0	0
$549$	97.5754	4.16442
$550$	0	0
$551$	−7.26439	−0.309474
$552$	0	0
$553$	−66.0748	−2.80979
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.5434	−0.700968	−0.350484	−	0.936569i	$-0.613983\pi$
$557$	−16.5434	−0.700968	−0.350484	+	0.936569i	$0.613983\pi$
$558$	0	0
$559$	−9.07669	−0.383903
$560$	0	0
$561$	4.89322	0.206592
$562$	0	0
$563$	−36.3727	−1.53293	−0.766463	−	0.642288i	$-0.777985\pi$
$563$	−36.3727	−1.53293	−0.766463	+	0.642288i	$0.777985\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−107.556	−4.51691
$568$	0	0
$569$	21.8270	0.915035	0.457517	−	0.889201i	$-0.348739\pi$
$569$	21.8270	0.915035	0.457517	+	0.889201i	$0.348739\pi$
$570$	0	0
$571$	−41.2781	−1.72743	−0.863717	−	0.503977i	$-0.831870\pi$
$571$	−41.2781	−1.72743	−0.863717	+	0.503977i	$0.831870\pi$
$572$	0	0
$573$	17.7797	0.742758
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.6240	0.983479	0.491739	−	0.870742i	$-0.336361\pi$
$577$	23.6240	0.983479	0.491739	+	0.870742i	$0.336361\pi$
$578$	0	0
$579$	53.4769	2.22242
$580$	0	0
$581$	38.0798	1.57981
$582$	0	0
$583$	−6.74287	−0.279261
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.48730	−0.102662	−0.0513309	−	0.998682i	$-0.516346\pi$
$587$	−2.48730	−0.102662	−0.0513309	+	0.998682i	$0.516346\pi$
$588$	0	0
$589$	−2.22003	−0.0914746
$590$	0	0
$591$	−62.7975	−2.58314
$592$	0	0
$593$	−11.6543	−0.478587	−0.239293	−	0.970947i	$-0.576916\pi$
$593$	−11.6543	−0.478587	−0.239293	+	0.970947i	$0.576916\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.9470	1.06194
$598$	0	0
$599$	0.149185	0.00609555	0.00304777	−	0.999995i	$-0.499030\pi$
$599$	0.149185	0.00609555	0.00304777	+	0.999995i	$0.499030\pi$
$600$	0	0
$601$	29.2954	1.19498	0.597492	−	0.801875i	$-0.296164\pi$
$601$	29.2954	1.19498	0.597492	+	0.801875i	$0.296164\pi$
$602$	0	0
$603$	38.9489	1.58612
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.55466	0.103691	0.0518453	−	0.998655i	$-0.483490\pi$
$607$	2.55466	0.103691	0.0518453	+	0.998655i	$0.483490\pi$
$608$	0	0
$609$	−96.4809	−3.90960
$610$	0	0
$611$	−3.84894	−0.155711
$612$	0	0
$613$	−42.6703	−1.72344	−0.861718	−	0.507387i	$-0.830611\pi$
$613$	−42.6703	−1.72344	−0.861718	+	0.507387i	$0.830611\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.8058	−0.797353	−0.398676	−	0.917092i	$-0.630530\pi$
$617$	−19.8058	−0.797353	−0.398676	+	0.917092i	$0.630530\pi$
$618$	0	0
$619$	23.6180	0.949287	0.474643	−	0.880178i	$-0.342577\pi$
$619$	23.6180	0.949287	0.474643	+	0.880178i	$0.342577\pi$
$620$	0	0
$621$	−21.3814	−0.858007
$622$	0	0
$623$	−12.6410	−0.506453
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.56468	−0.102423
$628$	0	0
$629$	−17.4439	−0.695533
$630$	0	0
$631$	−33.3007	−1.32568	−0.662841	−	0.748760i	$-0.730649\pi$
$631$	−33.3007	−1.32568	−0.662841	+	0.748760i	$0.730649\pi$
$632$	0	0
$633$	59.4462	2.36277
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.3474	0.409979
$638$	0	0
$639$	−12.6418	−0.500100
$640$	0	0
$641$	−37.5897	−1.48470	−0.742352	−	0.670010i	$-0.766290\pi$
$641$	−37.5897	−1.48470	−0.742352	+	0.670010i	$0.766290\pi$
$642$	0	0
$643$	2.61556	0.103148	0.0515739	−	0.998669i	$-0.483576\pi$
$643$	2.61556	0.103148	0.0515739	+	0.998669i	$0.483576\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.3725	−0.407784	−0.203892	−	0.978993i	$-0.565359\pi$
$647$	−10.3725	−0.407784	−0.203892	+	0.978993i	$0.565359\pi$
$648$	0	0
$649$	10.5407	0.413757
$650$	0	0
$651$	−29.4849	−1.15561
$652$	0	0
$653$	0.949620	0.0371615	0.0185808	−	0.999827i	$-0.494085\pi$
$653$	0.949620	0.0371615	0.0185808	+	0.999827i	$0.494085\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	78.8913	3.07784
$658$	0	0
$659$	−32.1009	−1.25047	−0.625237	−	0.780435i	$-0.714998\pi$
$659$	−32.1009	−1.25047	−0.625237	+	0.780435i	$0.714998\pi$
$660$	0	0
$661$	−32.8796	−1.27887	−0.639434	−	0.768846i	$-0.720831\pi$
$661$	−32.8796	−1.27887	−0.639434	+	0.768846i	$0.720831\pi$
$662$	0	0
$663$	6.71879	0.260936
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.2707	−0.397681
$668$	0	0
$669$	−1.40566	−0.0543458
$670$	0	0
$671$	10.0472	0.387869
$672$	0	0
$673$	25.9466	1.00017	0.500084	−	0.865977i	$-0.333302\pi$
$673$	25.9466	1.00017	0.500084	+	0.865977i	$0.333302\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.0807	1.80946	0.904729	−	0.425987i	$-0.140073\pi$
$677$	47.0807	1.80946	0.904729	+	0.425987i	$0.140073\pi$
$678$	0	0
$679$	−18.5643	−0.712433
$680$	0	0
$681$	−24.8956	−0.954002
$682$	0	0
$683$	−29.9429	−1.14573	−0.572867	−	0.819648i	$-0.694169\pi$
$683$	−29.9429	−1.14573	−0.572867	+	0.819648i	$0.694169\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−66.5050	−2.53733
$688$	0	0
$689$	−9.25850	−0.352721
$690$	0	0
$691$	30.4205	1.15725	0.578625	−	0.815594i	$-0.303589\pi$
$691$	30.4205	1.15725	0.578625	+	0.815594i	$0.303589\pi$
$692$	0	0
$693$	−24.4556	−0.928990
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.6709	0.442065
$698$	0	0
$699$	67.2859	2.54499
$700$	0	0
$701$	25.0769	0.947141	0.473570	−	0.880756i	$-0.342965\pi$
$701$	25.0769	0.947141	0.473570	+	0.880756i	$0.342965\pi$
$702$	0	0
$703$	9.14283	0.344828
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.02731	−0.113854
$708$	0	0
$709$	13.5057	0.507219	0.253610	−	0.967307i	$-0.418382\pi$
$709$	13.5057	0.507219	0.253610	+	0.967307i	$0.418382\pi$
$710$	0	0
$711$	123.914	4.64714
$712$	0	0
$713$	−3.13875	−0.117547
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.0433	−0.561801
$718$	0	0
$719$	2.27969	0.0850179	0.0425090	−	0.999096i	$-0.486465\pi$
$719$	2.27969	0.0850179	0.0425090	+	0.999096i	$0.486465\pi$
$720$	0	0
$721$	−42.2331	−1.57284
$722$	0	0
$723$	−12.7967	−0.475913
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−0.925937	−0.0343411	−0.0171705	−	0.999853i	$-0.505466\pi$
$727$	−0.925937	−0.0343411	−0.0171705	+	0.999853i	$0.505466\pi$
$728$	0	0
$729$	53.7374	1.99028
$730$	0	0
$731$	16.0387	0.593212
$732$	0	0
$733$	20.9455	0.773639	0.386820	−	0.922155i	$-0.373574\pi$
$733$	20.9455	0.773639	0.386820	+	0.922155i	$0.373574\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.01053	0.147730
$738$	0	0
$739$	−36.9385	−1.35880	−0.679402	−	0.733766i	$-0.737761\pi$
$739$	−36.9385	−1.35880	−0.679402	+	0.733766i	$0.737761\pi$
$740$	0	0
$741$	−3.52151	−0.129366
$742$	0	0
$743$	48.3612	1.77420	0.887100	−	0.461577i	$-0.152716\pi$
$743$	48.3612	1.77420	0.887100	+	0.461577i	$0.152716\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−71.4132	−2.61287
$748$	0	0
$749$	−69.7981	−2.55037
$750$	0	0
$751$	19.9871	0.729340	0.364670	−	0.931137i	$-0.381182\pi$
$751$	19.9871	0.729340	0.364670	+	0.931137i	$0.381182\pi$
$752$	0	0
$753$	78.1084	2.84643
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−36.3938	−1.32276	−0.661378	−	0.750053i	$-0.730028\pi$
$757$	−36.3938	−1.32276	−0.661378	+	0.750053i	$0.730028\pi$
$758$	0	0
$759$	−3.62603	−0.131617
$760$	0	0
$761$	−24.2163	−0.877839	−0.438920	−	0.898526i	$-0.644639\pi$
$761$	−24.2163	−0.877839	−0.438920	+	0.898526i	$0.644639\pi$
$762$	0	0
$763$	69.9928	2.53391
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.4732	0.522596
$768$	0	0
$769$	30.7013	1.10712	0.553559	−	0.832810i	$-0.313269\pi$
$769$	30.7013	1.10712	0.553559	+	0.832810i	$0.313269\pi$
$770$	0	0
$771$	−29.5267	−1.06338
$772$	0	0
$773$	21.9986	0.791234	0.395617	−	0.918416i	$-0.370531\pi$
$773$	21.9986	0.791234	0.395617	+	0.918416i	$0.370531\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	121.429	4.35625
$778$	0	0
$779$	−6.11703	−0.219165
$780$	0	0
$781$	−1.30171	−0.0465788
$782$	0	0
$783$	109.859	3.92606
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.9830	0.747962	0.373981	−	0.927436i	$-0.377993\pi$
$787$	20.9830	0.747962	0.373981	+	0.927436i	$0.377993\pi$
$788$	0	0
$789$	50.3835	1.79370
$790$	0	0
$791$	4.30418	0.153039
$792$	0	0
$793$	13.7957	0.489898
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.1150	−1.03131	−0.515653	−	0.856798i	$-0.672451\pi$
$797$	−29.1150	−1.03131	−0.515653	+	0.856798i	$0.672451\pi$
$798$	0	0
$799$	6.80115	0.240607
$800$	0	0
$801$	23.7065	0.837628
$802$	0	0
$803$	8.12336	0.286667
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.5042	0.510571
$808$	0	0
$809$	−47.5139	−1.67050	−0.835250	−	0.549871i	$-0.814677\pi$
$809$	−47.5139	−1.67050	−0.835250	+	0.549871i	$0.814677\pi$
$810$	0	0
$811$	−11.7957	−0.414204	−0.207102	−	0.978319i	$-0.566403\pi$
$811$	−11.7957	−0.414204	−0.207102	+	0.978319i	$0.566403\pi$
$812$	0	0
$813$	77.7886	2.72817
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.40634	−0.294101
$818$	0	0
$819$	−33.5794	−1.17336
$820$	0	0
$821$	28.0549	0.979123	0.489562	−	0.871969i	$-0.337157\pi$
$821$	28.0549	0.979123	0.489562	+	0.871969i	$0.337157\pi$
$822$	0	0
$823$	−16.7826	−0.585006	−0.292503	−	0.956265i	$-0.594488\pi$
$823$	−16.7826	−0.585006	−0.292503	+	0.956265i	$0.594488\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.1984	−0.424180	−0.212090	−	0.977250i	$-0.568027\pi$
$827$	−12.1984	−0.424180	−0.212090	+	0.977250i	$0.568027\pi$
$828$	0	0
$829$	3.30978	0.114954	0.0574768	−	0.998347i	$-0.481694\pi$
$829$	3.30978	0.114954	0.0574768	+	0.998347i	$0.481694\pi$
$830$	0	0
$831$	−22.3979	−0.776975
$832$	0	0
$833$	−18.2840	−0.633505
$834$	0	0
$835$	0	0
$836$	0	0
$837$	33.5735	1.16047
$838$	0	0
$839$	27.9925	0.966408	0.483204	−	0.875508i	$-0.339473\pi$
$839$	27.9925	0.966408	0.483204	+	0.875508i	$0.339473\pi$
$840$	0	0
$841$	23.7714	0.819704
$842$	0	0
$843$	−87.2169	−3.00391
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.2766	1.45264
$848$	0	0
$849$	−56.9764	−1.95543
$850$	0	0
$851$	12.9265	0.443113
$852$	0	0
$853$	−11.5191	−0.394408	−0.197204	−	0.980362i	$-0.563186\pi$
$853$	−11.5191	−0.394408	−0.197204	+	0.980362i	$0.563186\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.40616	0.184671	0.0923355	−	0.995728i	$-0.470567\pi$
$857$	5.40616	0.184671	0.0923355	+	0.995728i	$0.470567\pi$
$858$	0	0
$859$	17.4306	0.594725	0.297363	−	0.954765i	$-0.403893\pi$
$859$	17.4306	0.594725	0.297363	+	0.954765i	$0.403893\pi$
$860$	0	0
$861$	−81.2424	−2.76873
$862$	0	0
$863$	8.83048	0.300593	0.150297	−	0.988641i	$-0.451977\pi$
$863$	8.83048	0.300593	0.150297	+	0.988641i	$0.451977\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	43.5721	1.47979
$868$	0	0
$869$	12.7593	0.432829
$870$	0	0
$871$	5.50677	0.186590
$872$	0	0
$873$	34.8148	1.17830
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.840573	−0.0283841	−0.0141921	−	0.999899i	$-0.504518\pi$
$877$	−0.840573	−0.0283841	−0.0141921	+	0.999899i	$0.504518\pi$
$878$	0	0
$879$	48.5850	1.63873
$880$	0	0
$881$	6.78562	0.228613	0.114307	−	0.993446i	$-0.463535\pi$
$881$	6.78562	0.228613	0.114307	+	0.993446i	$0.463535\pi$
$882$	0	0
$883$	10.0799	0.339217	0.169608	−	0.985512i	$-0.445750\pi$
$883$	10.0799	0.339217	0.169608	+	0.985512i	$0.445750\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.41650	0.316175	0.158088	−	0.987425i	$-0.449467\pi$
$887$	9.41650	0.316175	0.158088	+	0.987425i	$0.449467\pi$
$888$	0	0
$889$	−21.2752	−0.713548
$890$	0	0
$891$	20.7694	0.695801
$892$	0	0
$893$	−3.56468	−0.119287
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.97883	−0.166238
$898$	0	0
$899$	16.1271	0.537870
$900$	0	0
$901$	16.3600	0.545029
$902$	0	0
$903$	−111.647	−3.71540
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.2639	0.772466	0.386233	−	0.922401i	$-0.373776\pi$
$907$	23.2639	0.772466	0.386233	+	0.922401i	$0.373776\pi$
$908$	0	0
$909$	5.67730	0.188304
$910$	0	0
$911$	−15.4610	−0.512245	−0.256123	−	0.966644i	$-0.582445\pi$
$911$	−15.4610	−0.512245	−0.256123	+	0.966644i	$0.582445\pi$
$912$	0	0
$913$	−7.35335	−0.243360
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.2276	0.403791
$918$	0	0
$919$	−20.1775	−0.665594	−0.332797	−	0.942998i	$-0.607992\pi$
$919$	−20.1775	−0.665594	−0.332797	+	0.942998i	$0.607992\pi$
$920$	0	0
$921$	36.8199	1.21326
$922$	0	0
$923$	−1.78735	−0.0588314
$924$	0	0
$925$	0	0
$926$	0	0
$927$	79.2023	2.60134
$928$	0	0
$929$	11.3148	0.371227	0.185613	−	0.982623i	$-0.440573\pi$
$929$	11.3148	0.371227	0.185613	+	0.982623i	$0.440573\pi$
$930$	0	0
$931$	9.58319	0.314076
$932$	0	0
$933$	−18.4477	−0.603952
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.8943	−0.551911	−0.275956	−	0.961170i	$-0.588994\pi$
$937$	−16.8943	−0.551911	−0.275956	+	0.961170i	$0.588994\pi$
$938$	0	0
$939$	97.7007	3.18834
$940$	0	0
$941$	−15.0272	−0.489871	−0.244936	−	0.969539i	$-0.578767\pi$
$941$	−15.0272	−0.489871	−0.244936	+	0.969539i	$0.578767\pi$
$942$	0	0
$943$	−8.64847	−0.281633
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.208895	0.00678818	0.00339409	−	0.999994i	$-0.498920\pi$
$947$	0.208895	0.00678818	0.00339409	+	0.999994i	$0.498920\pi$
$948$	0	0
$949$	11.1540	0.362075
$950$	0	0
$951$	−38.4821	−1.24787
$952$	0	0
$953$	18.0875	0.585910	0.292955	−	0.956126i	$-0.405361\pi$
$953$	18.0875	0.585910	0.292955	+	0.956126i	$0.405361\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	18.6308	0.602249
$958$	0	0
$959$	94.0370	3.03661
$960$	0	0
$961$	−26.0715	−0.841016
$962$	0	0
$963$	130.897	4.21808
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.5540	1.30413	0.652064	−	0.758164i	$-0.273903\pi$
$967$	40.5540	1.30413	0.652064	+	0.758164i	$0.273903\pi$
$968$	0	0
$969$	6.22257	0.199898
$970$	0	0
$971$	51.7597	1.66105	0.830523	−	0.556984i	$-0.188041\pi$
$971$	51.7597	1.66105	0.830523	+	0.556984i	$0.188041\pi$
$972$	0	0
$973$	−48.7161	−1.56177
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.6104	−0.755364	−0.377682	−	0.925935i	$-0.623279\pi$
$977$	−23.6104	−0.755364	−0.377682	+	0.925935i	$0.623279\pi$
$978$	0	0
$979$	2.44103	0.0780158
$980$	0	0
$981$	−131.262	−4.19086
$982$	0	0
$983$	−34.0532	−1.08613	−0.543065	−	0.839691i	$-0.682736\pi$
$983$	−34.0532	−1.08613	−0.543065	+	0.839691i	$0.682736\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−47.3437	−1.50697
$988$	0	0
$989$	−11.8852	−0.377926
$990$	0	0
$991$	−42.7936	−1.35938	−0.679692	−	0.733497i	$-0.737887\pi$
$991$	−42.7936	−1.35938	−0.679692	+	0.733497i	$0.737887\pi$
$992$	0	0
$993$	−70.8373	−2.24796
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−30.1937	−0.956243	−0.478122	−	0.878294i	$-0.658682\pi$
$997$	−30.1937	−0.956243	−0.478122	+	0.878294i	$0.658682\pi$
$998$	0	0
$999$	−138.267	−4.37458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ch.1.1		6
4.3	odd	2		3800.2.a.bc.1.6	yes	6
5.4	even	2		7600.2.a.cl.1.6		6
20.3	even	4		3800.2.d.q.3649.12		12
20.7	even	4		3800.2.d.q.3649.1		12
20.19	odd	2		3800.2.a.ba.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.1	✓	6	20.19	odd	2
3800.2.a.bc.1.6	yes	6	4.3	odd	2
3800.2.d.q.3649.1		12	20.7	even	4
3800.2.d.q.3649.12		12	20.3	even	4
7600.2.a.ch.1.1		6	1.1	even	1	trivial
7600.2.a.cl.1.6		6	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-3.26143 q^{3} -4.07225 q^{7} +7.63693 q^{9} +O(q^{10})\)	\(q-3.26143 q^{3} -4.07225 q^{7} +7.63693 q^{9} +0.786366 q^{11} +1.07974 q^{13} -1.90793 q^{17} +1.00000 q^{19} +13.2813 q^{21} +1.41383 q^{23} -15.1230 q^{27} -7.26439 q^{29} -2.22003 q^{31} -2.56468 q^{33} +9.14283 q^{37} -3.52151 q^{39} -6.11703 q^{41} -8.40634 q^{43} -3.56468 q^{47} +9.58319 q^{49} +6.22257 q^{51} -8.57472 q^{53} -3.26143 q^{57} +13.4043 q^{59} +12.7768 q^{61} -31.0994 q^{63} +5.10008 q^{67} -4.61112 q^{69} -1.65535 q^{71} +10.3302 q^{73} -3.20228 q^{77} +16.2256 q^{79} +26.4119 q^{81} -9.35104 q^{83} +23.6923 q^{87} +3.10419 q^{89} -4.39698 q^{91} +7.24046 q^{93} +4.55874 q^{97} +6.00542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} - 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} - 19 q^{71} + 3 q^{73} - 36 q^{77} + 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} - 42 q^{91} + 39 q^{93} - 11 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 - 3 * q^11 - q^13 - 14 * q^17 + 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 - 5 * q^31 + 2 * q^33 - 8 * q^37 - 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 - 33 * q^51 + 13 * q^53 - 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 - 19 * q^71 + 3 * q^73 - 36 * q^77 + 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 - 42 * q^91 + 39 * q^93 - 11 * q^97 + 6 * q^99

Introduction

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