LMFDB - Embedded newform 4.18.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,18,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	4.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:	$7.32888349378$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{9361}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2340 $ x^2 - x - 2340
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$48.8761$ of defining polynomial
Character	$\chi$	$=$	4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21516.4 q^{3} +970774. q^{5} +1.64068e6 q^{7} +3.33817e8 q^{9} +O(q^{10})$	$q-21516.4 q^{3} +970774. q^{5} +1.64068e6 q^{7} +3.33817e8 q^{9} -7.82166e7 q^{11} +2.36340e9 q^{13} -2.08876e10 q^{15} -1.04976e10 q^{17} +1.03725e11 q^{19} -3.53015e10 q^{21} +3.30946e11 q^{23} +1.79462e11 q^{25} -4.40391e12 q^{27} +3.69924e12 q^{29} +2.53459e12 q^{31} +1.68294e12 q^{33} +1.59273e12 q^{35} +1.96434e13 q^{37} -5.08519e13 q^{39} -3.32602e13 q^{41} -8.49511e13 q^{43} +3.24061e14 q^{45} +2.18640e12 q^{47} -2.29939e14 q^{49} +2.25872e14 q^{51} -1.81963e13 q^{53} -7.59307e13 q^{55} -2.23179e15 q^{57} +1.24801e15 q^{59} +2.09807e15 q^{61} +5.47686e14 q^{63} +2.29433e15 q^{65} -2.53819e15 q^{67} -7.12077e15 q^{69} +8.39308e15 q^{71} -6.50233e15 q^{73} -3.86138e15 q^{75} -1.28328e14 q^{77} -1.41887e16 q^{79} +5.16473e16 q^{81} +1.76295e16 q^{83} -1.01908e16 q^{85} -7.95945e16 q^{87} +2.17799e16 q^{89} +3.87758e15 q^{91} -5.45352e16 q^{93} +1.00694e17 q^{95} -8.31143e16 q^{97} -2.61100e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5880 q^{3} + 604044 q^{5} + 25350160 q^{7} + 449174682 q^{9}+O(q^{10}) $ 2 * q - 5880 * q^3 + 604044 * q^5 + 25350160 * q^7 + 449174682 * q^9	$ 2 q - 5880 q^{3} + 604044 q^{5} + 25350160 q^{7} + 449174682 q^{9} + 1259648280 q^{11} - 1320052580 q^{13} - 26621930448 q^{15} - 27498226140 q^{17} + 101133633832 q^{19} + 335430207552 q^{21} + 134767491120 q^{23} - 448986850114 q^{25} - 4619416117680 q^{27} + 2337155582652 q^{29} + 278836113472 q^{31} + 22602380058720 q^{33} - 7102242382752 q^{35} + 20929802888140 q^{37} - 108447997173648 q^{39} + 4166592315732 q^{41} - 111143148534440 q^{43} + 281755357404444 q^{45} + 196772651157600 q^{47} + 99570326741874 q^{49} - 39956666918256 q^{51} - 487965122736660 q^{53} - 566565363246960 q^{55} - 22\!\cdots\!40 q^{57}+ \cdots + 12\!\cdots\!80 q^{99}+O(q^{100}) $ 2 * q - 5880 * q^3 + 604044 * q^5 + 25350160 * q^7 + 449174682 * q^9 + 1259648280 * q^11 - 1320052580 * q^13 - 26621930448 * q^15 - 27498226140 * q^17 + 101133633832 * q^19 + 335430207552 * q^21 + 134767491120 * q^23 - 448986850114 * q^25 - 4619416117680 * q^27 + 2337155582652 * q^29 + 278836113472 * q^31 + 22602380058720 * q^33 - 7102242382752 * q^35 + 20929802888140 * q^37 - 108447997173648 * q^39 + 4166592315732 * q^41 - 111143148534440 * q^43 + 281755357404444 * q^45 + 196772651157600 * q^47 + 99570326741874 * q^49 - 39956666918256 * q^51 - 487965122736660 * q^53 - 566565363246960 * q^55 - 2272313631999840 * q^57 + 2835904197813624 * q^59 - 67544034994436 * q^61 + 3282762121966800 * q^63 + 3645157343001768 * q^65 - 995171321546360 * q^67 - 10188302100105024 * q^69 + 3882245493215376 * q^71 - 12746425881769580 * q^73 - 13688080703624712 * q^75 + 31591755846002880 * q^77 - 14984271534065504 * q^79 + 33380230287529458 * q^81 + 43899417809893800 * q^83 - 3956216991540264 * q^85 - 100892756486768400 * q^87 + 51909007958846388 * q^89 - 83455169400462112 * q^91 - 89807116632334080 * q^93 + 101643889304423664 * q^95 - 49281007789848380 * q^97 + 128223271309133880 * 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
$3$	−21516.4	−1.89339	−0.946694	−	0.322134i	$-0.895600\pi$
$3$	−21516.4	−1.89339	−0.946694	+	0.322134i	$0.895600\pi$
$4$	0	0
$5$	970774.	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$5$	970774.	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$6$	0	0
$7$	1.64068e6	0.107570	0.0537849	−	0.998553i	$-0.482871\pi$
$7$	1.64068e6	0.107570	0.0537849	+	0.998553i	$0.482871\pi$
$8$	0	0
$9$	3.33817e8	2.58492
$10$	0	0
$11$	−7.82166e7	−0.110017	−0.0550087	−	0.998486i	$-0.517519\pi$
$11$	−7.82166e7	−0.110017	−0.0550087	+	0.998486i	$0.517519\pi$
$12$	0	0
$13$	2.36340e9	0.803561	0.401780	−	0.915736i	$-0.368392\pi$
$13$	2.36340e9	0.803561	0.401780	+	0.915736i	$0.368392\pi$
$14$	0	0
$15$	−2.08876e10	−2.10432
$16$	0	0
$17$	−1.04976e10	−0.364986	−0.182493	−	0.983207i	$-0.558417\pi$
$17$	−1.04976e10	−0.364986	−0.182493	+	0.983207i	$0.558417\pi$
$18$	0	0
$19$	1.03725e11	1.40113	0.700564	−	0.713589i	$-0.252932\pi$
$19$	1.03725e11	1.40113	0.700564	+	0.713589i	$0.252932\pi$
$20$	0	0
$21$	−3.53015e10	−0.203671
$22$	0	0
$23$	3.30946e11	0.881191	0.440596	−	0.897706i	$-0.354767\pi$
$23$	3.30946e11	0.881191	0.440596	+	0.897706i	$0.354767\pi$
$24$	0	0
$25$	1.79462e11	0.235224
$26$	0	0
$27$	−4.40391e12	−3.00087
$28$	0	0
$29$	3.69924e12	1.37319	0.686594	−	0.727041i	$-0.259105\pi$
$29$	3.69924e12	1.37319	0.686594	+	0.727041i	$0.259105\pi$
$30$	0	0
$31$	2.53459e12	0.533743	0.266872	−	0.963732i	$-0.414010\pi$
$31$	2.53459e12	0.533743	0.266872	+	0.963732i	$0.414010\pi$
$32$	0	0
$33$	1.68294e12	0.208306
$34$	0	0
$35$	1.59273e12	0.119554
$36$	0	0
$37$	1.96434e13	0.919393	0.459697	−	0.888076i	$-0.347958\pi$
$37$	1.96434e13	0.919393	0.459697	+	0.888076i	$0.347958\pi$
$38$	0	0
$39$	−5.08519e13	−1.52145
$40$	0	0
$41$	−3.32602e13	−0.650522	−0.325261	−	0.945624i	$-0.605452\pi$
$41$	−3.32602e13	−0.650522	−0.325261	+	0.945624i	$0.605452\pi$
$42$	0	0
$43$	−8.49511e13	−1.10838	−0.554188	−	0.832392i	$-0.686971\pi$
$43$	−8.49511e13	−1.10838	−0.554188	+	0.832392i	$0.686971\pi$
$44$	0	0
$45$	3.24061e14	2.87290
$46$	0	0
$47$	2.18640e12	0.0133936	0.00669681	−	0.999978i	$-0.497868\pi$
$47$	2.18640e12	0.0133936	0.00669681	+	0.999978i	$0.497868\pi$
$48$	0	0
$49$	−2.29939e14	−0.988429
$50$	0	0
$51$	2.25872e14	0.691060
$52$	0	0
$53$	−1.81963e13	−0.0401456	−0.0200728	−	0.999799i	$-0.506390\pi$
$53$	−1.81963e13	−0.0401456	−0.0200728	+	0.999799i	$0.506390\pi$
$54$	0	0
$55$	−7.59307e13	−0.122274
$56$	0	0
$57$	−2.23179e15	−2.65288
$58$	0	0
$59$	1.24801e15	1.10656	0.553282	−	0.832994i	$-0.313375\pi$
$59$	1.24801e15	1.10656	0.553282	+	0.832994i	$0.313375\pi$
$60$	0	0
$61$	2.09807e15	1.40126	0.700628	−	0.713527i	$-0.252903\pi$
$61$	2.09807e15	1.40126	0.700628	+	0.713527i	$0.252903\pi$
$62$	0	0
$63$	5.47686e14	0.278059
$64$	0	0
$65$	2.29433e15	0.893083
$66$	0	0
$67$	−2.53819e15	−0.763639	−0.381820	−	0.924237i	$-0.624702\pi$
$67$	−2.53819e15	−0.763639	−0.381820	+	0.924237i	$0.624702\pi$
$68$	0	0
$69$	−7.12077e15	−1.66844
$70$	0	0
$71$	8.39308e15	1.54250	0.771250	−	0.636532i	$-0.219632\pi$
$71$	8.39308e15	1.54250	0.771250	+	0.636532i	$0.219632\pi$
$72$	0	0
$73$	−6.50233e15	−0.943680	−0.471840	−	0.881684i	$-0.656410\pi$
$73$	−6.50233e15	−0.943680	−0.471840	+	0.881684i	$0.656410\pi$
$74$	0	0
$75$	−3.86138e15	−0.445371
$76$	0	0
$77$	−1.28328e14	−0.0118345
$78$	0	0
$79$	−1.41887e16	−1.05223	−0.526116	−	0.850413i	$-0.676352\pi$
$79$	−1.41887e16	−1.05223	−0.526116	+	0.850413i	$0.676352\pi$
$80$	0	0
$81$	5.16473e16	3.09688
$82$	0	0
$83$	1.76295e16	0.859165	0.429583	−	0.903028i	$-0.358661\pi$
$83$	1.76295e16	0.859165	0.429583	+	0.903028i	$0.358661\pi$
$84$	0	0
$85$	−1.01908e16	−0.405648
$86$	0	0
$87$	−7.95945e16	−2.59998
$88$	0	0
$89$	2.17799e16	0.586464	0.293232	−	0.956041i	$-0.405269\pi$
$89$	2.17799e16	0.586464	0.293232	+	0.956041i	$0.405269\pi$
$90$	0	0
$91$	3.87758e15	0.0864388
$92$	0	0
$93$	−5.45352e16	−1.01058
$94$	0	0
$95$	1.00694e17	1.55722
$96$	0	0
$97$	−8.31143e16	−1.07675	−0.538377	−	0.842704i	$-0.680962\pi$
$97$	−8.31143e16	−1.07675	−0.538377	+	0.842704i	$0.680962\pi$
$98$	0	0
$99$	−2.61100e16	−0.284386
$100$	0	0
$101$	1.52559e17	1.40186	0.700931	−	0.713229i	$-0.252768\pi$
$101$	1.52559e17	1.40186	0.700931	+	0.713229i	$0.252768\pi$
$102$	0	0
$103$	1.07476e17	0.835977	0.417989	−	0.908452i	$-0.362735\pi$
$103$	1.07476e17	0.835977	0.417989	+	0.908452i	$0.362735\pi$
$104$	0	0
$105$	−3.42698e16	−0.226362
$106$	0	0
$107$	1.26948e17	0.714274	0.357137	−	0.934052i	$-0.383753\pi$
$107$	1.26948e17	0.714274	0.357137	+	0.934052i	$0.383753\pi$
$108$	0	0
$109$	−1.60767e17	−0.772807	−0.386404	−	0.922330i	$-0.626283\pi$
$109$	−1.60767e17	−0.772807	−0.386404	+	0.922330i	$0.626283\pi$
$110$	0	0
$111$	−4.22655e17	−1.74077
$112$	0	0
$113$	−4.76977e17	−1.68784	−0.843919	−	0.536471i	$-0.819757\pi$
$113$	−4.76977e17	−1.68784	−0.843919	+	0.536471i	$0.819757\pi$
$114$	0	0
$115$	3.21273e17	0.979362
$116$	0	0
$117$	7.88943e17	2.07714
$118$	0	0
$119$	−1.72233e16	−0.0392614
$120$	0	0
$121$	−4.99329e17	−0.987896
$122$	0	0
$123$	7.15641e17	1.23169
$124$	0	0
$125$	−5.66425e17	−0.849977
$126$	0	0
$127$	8.84957e16	0.116035	0.0580177	−	0.998316i	$-0.481522\pi$
$127$	8.84957e16	0.116035	0.0580177	+	0.998316i	$0.481522\pi$
$128$	0	0
$129$	1.82784e18	2.09859
$130$	0	0
$131$	−9.61186e17	−0.968281	−0.484141	−	0.874990i	$-0.660868\pi$
$131$	−9.61186e17	−0.968281	−0.484141	+	0.874990i	$0.660868\pi$
$132$	0	0
$133$	1.70179e17	0.150719
$134$	0	0
$135$	−4.27520e18	−3.33518
$136$	0	0
$137$	1.92685e17	0.132655	0.0663273	−	0.997798i	$-0.478872\pi$
$137$	1.92685e17	0.132655	0.0663273	+	0.997798i	$0.478872\pi$
$138$	0	0
$139$	1.69703e18	1.03292	0.516458	−	0.856313i	$-0.327250\pi$
$139$	1.69703e18	1.03292	0.516458	+	0.856313i	$0.327250\pi$
$140$	0	0
$141$	−4.70436e16	−0.0253593
$142$	0	0
$143$	−1.84857e17	−0.0884057
$144$	0	0
$145$	3.59113e18	1.52617
$146$	0	0
$147$	4.94746e18	1.87148
$148$	0	0
$149$	−5.35576e18	−1.80608	−0.903042	−	0.429551i	$-0.858672\pi$
$149$	−5.35576e18	−1.80608	−0.903042	+	0.429551i	$0.858672\pi$
$150$	0	0
$151$	4.78142e17	0.143964	0.0719819	−	0.997406i	$-0.477068\pi$
$151$	4.78142e17	0.143964	0.0719819	+	0.997406i	$0.477068\pi$
$152$	0	0
$153$	−3.50429e18	−0.943458
$154$	0	0
$155$	2.46051e18	0.593206
$156$	0	0
$157$	−8.86680e17	−0.191699	−0.0958495	−	0.995396i	$-0.530557\pi$
$157$	−8.86680e17	−0.191699	−0.0958495	+	0.995396i	$0.530557\pi$
$158$	0	0
$159$	3.91519e17	0.0760112
$160$	0	0
$161$	5.42976e17	0.0947895
$162$	0	0
$163$	−9.72864e18	−1.52918	−0.764588	−	0.644519i	$-0.777058\pi$
$163$	−9.72864e18	−1.52918	−0.764588	+	0.644519i	$0.777058\pi$
$164$	0	0
$165$	1.63376e18	0.231512
$166$	0	0
$167$	−1.04608e19	−1.33805	−0.669027	−	0.743238i	$-0.733289\pi$
$167$	−1.04608e19	−1.33805	−0.669027	+	0.743238i	$0.733289\pi$
$168$	0	0
$169$	−3.06476e18	−0.354290
$170$	0	0
$171$	3.46252e19	3.62180
$172$	0	0
$173$	1.50783e19	1.42876	0.714381	−	0.699756i	$-0.246708\pi$
$173$	1.50783e19	1.42876	0.714381	+	0.699756i	$0.246708\pi$
$174$	0	0
$175$	2.94439e17	0.0253030
$176$	0	0
$177$	−2.68528e19	−2.09515
$178$	0	0
$179$	4.59833e18	0.326099	0.163049	−	0.986618i	$-0.447867\pi$
$179$	4.59833e18	0.326099	0.163049	+	0.986618i	$0.447867\pi$
$180$	0	0
$181$	8.29716e18	0.535379	0.267690	−	0.963505i	$-0.413740\pi$
$181$	8.29716e18	0.535379	0.267690	+	0.963505i	$0.413740\pi$
$182$	0	0
$183$	−4.51431e19	−2.65312
$184$	0	0
$185$	1.90693e19	1.02182
$186$	0	0
$187$	8.21090e17	0.0401548
$188$	0	0
$189$	−7.22540e18	−0.322802
$190$	0	0
$191$	1.62722e19	0.664756	0.332378	−	0.943146i	$-0.392149\pi$
$191$	1.62722e19	0.664756	0.332378	+	0.943146i	$0.392149\pi$
$192$	0	0
$193$	−7.66981e18	−0.286779	−0.143390	−	0.989666i	$-0.545800\pi$
$193$	−7.66981e18	−0.286779	−0.143390	+	0.989666i	$0.545800\pi$
$194$	0	0
$195$	−4.93657e19	−1.69095
$196$	0	0
$197$	4.94003e19	1.55155	0.775776	−	0.631008i	$-0.217359\pi$
$197$	4.94003e19	1.55155	0.775776	+	0.631008i	$0.217359\pi$
$198$	0	0
$199$	−5.51277e18	−0.158898	−0.0794491	−	0.996839i	$-0.525316\pi$
$199$	−5.51277e18	−0.158898	−0.0794491	+	0.996839i	$0.525316\pi$
$200$	0	0
$201$	5.46128e19	1.44587
$202$	0	0
$203$	6.06927e18	0.147713
$204$	0	0
$205$	−3.22881e19	−0.722994
$206$	0	0
$207$	1.10475e20	2.27781
$208$	0	0
$209$	−8.11303e18	−0.154149
$210$	0	0
$211$	−3.73068e19	−0.653712	−0.326856	−	0.945074i	$-0.605989\pi$
$211$	−3.73068e19	−0.653712	−0.326856	+	0.945074i	$0.605989\pi$
$212$	0	0
$213$	−1.80589e20	−2.92055
$214$	0	0
$215$	−8.24683e19	−1.23186
$216$	0	0
$217$	4.15844e18	0.0574146
$218$	0	0
$219$	1.39907e20	1.78675
$220$	0	0
$221$	−2.48101e19	−0.293288
$222$	0	0
$223$	7.68538e19	0.841538	0.420769	−	0.907168i	$-0.361760\pi$
$223$	7.68538e19	0.841538	0.420769	+	0.907168i	$0.361760\pi$
$224$	0	0
$225$	5.99074e19	0.608036
$226$	0	0
$227$	−7.77271e19	−0.731733	−0.365867	−	0.930667i	$-0.619227\pi$
$227$	−7.77271e19	−0.731733	−0.365867	+	0.930667i	$0.619227\pi$
$228$	0	0
$229$	3.74051e19	0.326835	0.163418	−	0.986557i	$-0.447748\pi$
$229$	3.74051e19	0.326835	0.163418	+	0.986557i	$0.447748\pi$
$230$	0	0
$231$	2.76117e18	0.0224074
$232$	0	0
$233$	−6.95476e19	−0.524514	−0.262257	−	0.964998i	$-0.584467\pi$
$233$	−6.95476e19	−0.524514	−0.262257	+	0.964998i	$0.584467\pi$
$234$	0	0
$235$	2.12250e18	0.0148858
$236$	0	0
$237$	3.05290e20	1.99228
$238$	0	0
$239$	1.27125e20	0.772410	0.386205	−	0.922413i	$-0.373786\pi$
$239$	1.27125e20	0.772410	0.386205	+	0.922413i	$0.373786\pi$
$240$	0	0
$241$	4.16245e19	0.235616	0.117808	−	0.993036i	$-0.462413\pi$
$241$	4.16245e19	0.235616	0.117808	+	0.993036i	$0.462413\pi$
$242$	0	0
$243$	−5.42544e20	−2.86274
$244$	0	0
$245$	−2.23218e20	−1.09855
$246$	0	0
$247$	2.45144e20	1.12589
$248$	0	0
$249$	−3.79324e20	−1.62673
$250$	0	0
$251$	−6.23721e18	−0.0249899	−0.0124949	−	0.999922i	$-0.503977\pi$
$251$	−6.23721e18	−0.0249899	−0.0124949	+	0.999922i	$0.503977\pi$
$252$	0	0
$253$	−2.58855e19	−0.0969464
$254$	0	0
$255$	2.19270e20	0.768048
$256$	0	0
$257$	1.25865e20	0.412547	0.206274	−	0.978494i	$-0.433866\pi$
$257$	1.25865e20	0.412547	0.206274	+	0.978494i	$0.433866\pi$
$258$	0	0
$259$	3.22284e19	0.0988989
$260$	0	0
$261$	1.23487e21	3.54958
$262$	0	0
$263$	−5.68084e20	−1.53034	−0.765171	−	0.643827i	$-0.777345\pi$
$263$	−5.68084e20	−1.53034	−0.765171	+	0.643827i	$0.777345\pi$
$264$	0	0
$265$	−1.76645e19	−0.0446181
$266$	0	0
$267$	−4.68626e20	−1.11040
$268$	0	0
$269$	−1.43474e20	−0.319066	−0.159533	−	0.987193i	$-0.550999\pi$
$269$	−1.43474e20	−0.319066	−0.159533	+	0.987193i	$0.550999\pi$
$270$	0	0
$271$	3.74752e20	0.782537	0.391269	−	0.920277i	$-0.372036\pi$
$271$	3.74752e20	0.782537	0.391269	+	0.920277i	$0.372036\pi$
$272$	0	0
$273$	−8.34317e19	−0.163662
$274$	0	0
$275$	−1.40369e19	−0.0258788
$276$	0	0
$277$	−8.63656e20	−1.49714	−0.748571	−	0.663055i	$-0.769259\pi$
$277$	−8.63656e20	−1.49714	−0.748571	+	0.663055i	$0.769259\pi$
$278$	0	0
$279$	8.46087e20	1.37968
$280$	0	0
$281$	6.30753e20	0.967956	0.483978	−	0.875080i	$-0.339191\pi$
$281$	6.30753e20	0.967956	0.483978	+	0.875080i	$0.339191\pi$
$282$	0	0
$283$	−6.12217e20	−0.884546	−0.442273	−	0.896880i	$-0.645828\pi$
$283$	−6.12217e20	−0.884546	−0.442273	+	0.896880i	$0.645828\pi$
$284$	0	0
$285$	−2.16657e21	−2.94843
$286$	0	0
$287$	−5.45693e19	−0.0699765
$288$	0	0
$289$	−7.17040e20	−0.866785
$290$	0	0
$291$	1.78832e21	2.03871
$292$	0	0
$293$	1.44992e21	1.55945	0.779723	−	0.626125i	$-0.215360\pi$
$293$	1.44992e21	1.55945	0.779723	+	0.626125i	$0.215360\pi$
$294$	0	0
$295$	1.21154e21	1.22984
$296$	0	0
$297$	3.44459e20	0.330147
$298$	0	0
$299$	7.82157e20	0.708091
$300$	0	0
$301$	−1.39377e20	−0.119228
$302$	0	0
$303$	−3.28252e21	−2.65427
$304$	0	0
$305$	2.03675e21	1.55736
$306$	0	0
$307$	−2.06695e19	−0.0149505	−0.00747523	−	0.999972i	$-0.502379\pi$
$307$	−2.06695e19	−0.0149505	−0.00747523	+	0.999972i	$0.502379\pi$
$308$	0	0
$309$	−2.31250e21	−1.58283
$310$	0	0
$311$	−1.75766e21	−1.13886	−0.569431	−	0.822039i	$-0.692837\pi$
$311$	−1.75766e21	−1.13886	−0.569431	+	0.822039i	$0.692837\pi$
$312$	0	0
$313$	7.56307e20	0.464057	0.232028	−	0.972709i	$-0.425464\pi$
$313$	7.56307e20	0.464057	0.232028	+	0.972709i	$0.425464\pi$
$314$	0	0
$315$	5.31679e20	0.309037
$316$	0	0
$317$	1.98212e21	1.09176	0.545880	−	0.837864i	$-0.316196\pi$
$317$	1.98212e21	1.09176	0.545880	+	0.837864i	$0.316196\pi$
$318$	0	0
$319$	−2.89342e20	−0.151075
$320$	0	0
$321$	−2.73147e21	−1.35240
$322$	0	0
$323$	−1.08887e21	−0.511392
$324$	0	0
$325$	4.24140e20	0.189017
$326$	0	0
$327$	3.45913e21	1.46322
$328$	0	0
$329$	3.58718e18	0.00144075
$330$	0	0
$331$	−3.67582e21	−1.40222	−0.701109	−	0.713054i	$-0.747312\pi$
$331$	−3.67582e21	−1.40222	−0.701109	+	0.713054i	$0.747312\pi$
$332$	0	0
$333$	6.55729e21	2.37656
$334$	0	0
$335$	−2.46401e21	−0.848714
$336$	0	0
$337$	−5.16622e20	−0.169168	−0.0845842	−	0.996416i	$-0.526956\pi$
$337$	−5.16622e20	−0.169168	−0.0845842	+	0.996416i	$0.526956\pi$
$338$	0	0
$339$	1.02628e22	3.19573
$340$	0	0
$341$	−1.98247e20	−0.0587211
$342$	0	0
$343$	−7.58927e20	−0.213895
$344$	0	0
$345$	−6.91266e21	−1.85431
$346$	0	0
$347$	−6.31869e21	−1.61371	−0.806857	−	0.590747i	$-0.798833\pi$
$347$	−6.31869e21	−1.61371	−0.806857	+	0.590747i	$0.798833\pi$
$348$	0	0
$349$	−9.48315e20	−0.230641	−0.115320	−	0.993328i	$-0.536789\pi$
$349$	−9.48315e20	−0.230641	−0.115320	+	0.993328i	$0.536789\pi$
$350$	0	0
$351$	−1.04082e22	−2.41138
$352$	0	0
$353$	3.35256e21	0.740101	0.370051	−	0.929012i	$-0.379340\pi$
$353$	3.35256e21	0.740101	0.370051	+	0.929012i	$0.379340\pi$
$354$	0	0
$355$	8.14778e21	1.71434
$356$	0	0
$357$	3.70583e20	0.0743371
$358$	0	0
$359$	−4.50173e21	−0.861146	−0.430573	−	0.902556i	$-0.641689\pi$
$359$	−4.50173e21	−0.861146	−0.430573	+	0.902556i	$0.641689\pi$
$360$	0	0
$361$	5.27850e21	0.963162
$362$	0	0
$363$	1.07438e22	1.87047
$364$	0	0
$365$	−6.31229e21	−1.04881
$366$	0	0
$367$	6.01056e21	0.953353	0.476676	−	0.879079i	$-0.341841\pi$
$367$	6.01056e21	0.953353	0.476676	+	0.879079i	$0.341841\pi$
$368$	0	0
$369$	−1.11028e22	−1.68155
$370$	0	0
$371$	−2.98542e19	−0.00431845
$372$	0	0
$373$	−4.77088e21	−0.659285	−0.329643	−	0.944106i	$-0.606928\pi$
$373$	−4.77088e21	−0.659285	−0.329643	+	0.944106i	$0.606928\pi$
$374$	0	0
$375$	1.21874e22	1.60934
$376$	0	0
$377$	8.74279e21	1.10344
$378$	0	0
$379$	4.74004e21	0.571938	0.285969	−	0.958239i	$-0.407685\pi$
$379$	4.74004e21	0.571938	0.285969	+	0.958239i	$0.407685\pi$
$380$	0	0
$381$	−1.90411e21	−0.219700
$382$	0	0
$383$	1.20992e21	0.133526	0.0667631	−	0.997769i	$-0.478733\pi$
$383$	1.20992e21	0.133526	0.0667631	+	0.997769i	$0.478733\pi$
$384$	0	0
$385$	−1.24578e20	−0.0131530
$386$	0	0
$387$	−2.83581e22	−2.86506
$388$	0	0
$389$	−4.86368e21	−0.470320	−0.235160	−	0.971957i	$-0.575561\pi$
$389$	−4.86368e21	−0.470320	−0.235160	+	0.971957i	$0.575561\pi$
$390$	0	0
$391$	−3.47415e21	−0.321622
$392$	0	0
$393$	2.06813e22	1.83333
$394$	0	0
$395$	−1.37740e22	−1.16946
$396$	0	0
$397$	−2.33860e22	−1.90212	−0.951058	−	0.309013i	$-0.900001\pi$
$397$	−2.33860e22	−1.90212	−0.951058	+	0.309013i	$0.900001\pi$
$398$	0	0
$399$	−3.66165e21	−0.285370
$400$	0	0
$401$	1.49882e22	1.11949	0.559746	−	0.828664i	$-0.310899\pi$
$401$	1.49882e22	1.11949	0.559746	+	0.828664i	$0.310899\pi$
$402$	0	0
$403$	5.99024e21	0.428895
$404$	0	0
$405$	5.01378e22	3.44190
$406$	0	0
$407$	−1.53644e21	−0.101149
$408$	0	0
$409$	−6.71662e21	−0.424134	−0.212067	−	0.977255i	$-0.568019\pi$
$409$	−6.71662e21	−0.424134	−0.212067	+	0.977255i	$0.568019\pi$
$410$	0	0
$411$	−4.14589e21	−0.251167
$412$	0	0
$413$	2.04758e21	0.119033
$414$	0	0
$415$	1.71143e22	0.954882
$416$	0	0
$417$	−3.65141e22	−1.95571
$418$	0	0
$419$	1.38192e22	0.710663	0.355331	−	0.934740i	$-0.384368\pi$
$419$	1.38192e22	0.710663	0.355331	+	0.934740i	$0.384368\pi$
$420$	0	0
$421$	−2.06280e22	−1.01873	−0.509366	−	0.860550i	$-0.670120\pi$
$421$	−2.06280e22	−1.01873	−0.509366	+	0.860550i	$0.670120\pi$
$422$	0	0
$423$	7.29858e20	0.0346214
$424$	0	0
$425$	−1.88393e21	−0.0858536
$426$	0	0
$427$	3.44226e21	0.150733
$428$	0	0
$429$	3.97747e21	0.167386
$430$	0	0
$431$	2.30892e22	0.934011	0.467005	−	0.884254i	$-0.345333\pi$
$431$	2.30892e22	0.934011	0.467005	+	0.884254i	$0.345333\pi$
$432$	0	0
$433$	1.35763e22	0.527999	0.263999	−	0.964523i	$-0.414958\pi$
$433$	1.35763e22	0.527999	0.263999	+	0.964523i	$0.414958\pi$
$434$	0	0
$435$	−7.72683e22	−2.88963
$436$	0	0
$437$	3.43274e22	1.23466
$438$	0	0
$439$	−4.41799e22	−1.52854	−0.764268	−	0.644898i	$-0.776900\pi$
$439$	−4.41799e22	−1.52854	−0.764268	+	0.644898i	$0.776900\pi$
$440$	0	0
$441$	−7.67574e22	−2.55501
$442$	0	0
$443$	1.93958e22	0.621263	0.310632	−	0.950530i	$-0.399459\pi$
$443$	1.93958e22	0.621263	0.310632	+	0.950530i	$0.399459\pi$
$444$	0	0
$445$	2.11434e22	0.651800
$446$	0	0
$447$	1.15237e23	3.41962
$448$	0	0
$449$	3.18755e22	0.910676	0.455338	−	0.890319i	$-0.349518\pi$
$449$	3.18755e22	0.910676	0.455338	+	0.890319i	$0.349518\pi$
$450$	0	0
$451$	2.60150e21	0.0715687
$452$	0	0
$453$	−1.02879e22	−0.272579
$454$	0	0
$455$	3.76425e21	0.0960687
$456$	0	0
$457$	2.48900e22	0.611981	0.305990	−	0.952035i	$-0.401012\pi$
$457$	2.48900e22	0.611981	0.305990	+	0.952035i	$0.401012\pi$
$458$	0	0
$459$	4.62307e22	1.09527
$460$	0	0
$461$	−5.46414e22	−1.24757	−0.623784	−	0.781597i	$-0.714405\pi$
$461$	−5.46414e22	−1.24757	−0.623784	+	0.781597i	$0.714405\pi$
$462$	0	0
$463$	−2.58506e22	−0.568894	−0.284447	−	0.958692i	$-0.591810\pi$
$463$	−2.58506e22	−0.568894	−0.284447	+	0.958692i	$0.591810\pi$
$464$	0	0
$465$	−5.29414e22	−1.12317
$466$	0	0
$467$	−5.73610e22	−1.17334	−0.586669	−	0.809827i	$-0.699561\pi$
$467$	−5.73610e22	−1.17334	−0.586669	+	0.809827i	$0.699561\pi$
$468$	0	0
$469$	−4.16435e21	−0.0821445
$470$	0	0
$471$	1.90782e22	0.362961
$472$	0	0
$473$	6.64459e21	0.121941
$474$	0	0
$475$	1.86147e22	0.329580
$476$	0	0
$477$	−6.07423e21	−0.103773
$478$	0	0
$479$	−4.01692e22	−0.662280	−0.331140	−	0.943582i	$-0.607433\pi$
$479$	−4.01692e22	−0.662280	−0.331140	+	0.943582i	$0.607433\pi$
$480$	0	0
$481$	4.64251e22	0.738788
$482$	0	0
$483$	−1.16829e22	−0.179473
$484$	0	0
$485$	−8.06852e22	−1.19671
$486$	0	0
$487$	−2.24546e22	−0.321595	−0.160798	−	0.986987i	$-0.551407\pi$
$487$	−2.24546e22	−0.321595	−0.160798	+	0.986987i	$0.551407\pi$
$488$	0	0
$489$	2.09326e23	2.89532
$490$	0	0
$491$	−5.70054e22	−0.761593	−0.380797	−	0.924659i	$-0.624350\pi$
$491$	−5.70054e22	−0.761593	−0.380797	+	0.924659i	$0.624350\pi$
$492$	0	0
$493$	−3.88333e22	−0.501194
$494$	0	0
$495$	−2.53469e22	−0.316068
$496$	0	0
$497$	1.37703e22	0.165926
$498$	0	0
$499$	−4.00593e22	−0.466497	−0.233249	−	0.972417i	$-0.574936\pi$
$499$	−4.00593e22	−0.466497	−0.233249	+	0.972417i	$0.574936\pi$
$500$	0	0
$501$	2.25078e23	2.53345
$502$	0	0
$503$	1.64428e22	0.178915	0.0894576	−	0.995991i	$-0.471487\pi$
$503$	1.64428e22	0.178915	0.0894576	+	0.995991i	$0.471487\pi$
$504$	0	0
$505$	1.48100e23	1.55804
$506$	0	0
$507$	6.59426e22	0.670809
$508$	0	0
$509$	−1.17557e23	−1.15651	−0.578254	−	0.815857i	$-0.696266\pi$
$509$	−1.17557e23	−1.15651	−0.578254	+	0.815857i	$0.696266\pi$
$510$	0	0
$511$	−1.06682e22	−0.101511
$512$	0	0
$513$	−4.56796e23	−4.20460
$514$	0	0
$515$	1.04335e23	0.929110
$516$	0	0
$517$	−1.71013e20	−0.00147353
$518$	0	0
$519$	−3.24431e23	−2.70520
$520$	0	0
$521$	1.01491e23	0.819043	0.409522	−	0.912300i	$-0.365696\pi$
$521$	1.01491e23	0.819043	0.409522	+	0.912300i	$0.365696\pi$
$522$	0	0
$523$	1.28745e23	1.00570	0.502849	−	0.864374i	$-0.332285\pi$
$523$	1.28745e23	1.00570	0.502849	+	0.864374i	$0.332285\pi$
$524$	0	0
$525$	−6.33528e21	−0.0479084
$526$	0	0
$527$	−2.66072e22	−0.194809
$528$	0	0
$529$	−3.15249e22	−0.223502
$530$	0	0
$531$	4.16607e23	2.86038
$532$	0	0
$533$	−7.86071e22	−0.522734
$534$	0	0
$535$	1.23238e23	0.793848
$536$	0	0
$537$	−9.89396e22	−0.617431
$538$	0	0
$539$	1.79850e22	0.108744
$540$	0	0
$541$	−4.58302e21	−0.0268519	−0.0134259	−	0.999910i	$-0.504274\pi$
$541$	−4.58302e21	−0.0268519	−0.0134259	+	0.999910i	$0.504274\pi$
$542$	0	0
$543$	−1.78525e23	−1.01368
$544$	0	0
$545$	−1.56068e23	−0.858903
$546$	0	0
$547$	−1.73760e23	−0.926955	−0.463477	−	0.886109i	$-0.653399\pi$
$547$	−1.73760e23	−0.926955	−0.463477	+	0.886109i	$0.653399\pi$
$548$	0	0
$549$	7.00372e23	3.62213
$550$	0	0
$551$	3.83704e23	1.92401
$552$	0	0
$553$	−2.32790e22	−0.113188
$554$	0	0
$555$	−4.10303e23	−1.93470
$556$	0	0
$557$	−3.13961e22	−0.143584	−0.0717922	−	0.997420i	$-0.522872\pi$
$557$	−3.13961e22	−0.143584	−0.0717922	+	0.997420i	$0.522872\pi$
$558$	0	0
$559$	−2.00773e23	−0.890647
$560$	0	0
$561$	−1.76669e22	−0.0760286
$562$	0	0
$563$	2.59071e23	1.08168	0.540839	−	0.841126i	$-0.318107\pi$
$563$	2.59071e23	1.08168	0.540839	+	0.841126i	$0.318107\pi$
$564$	0	0
$565$	−4.63037e23	−1.87587
$566$	0	0
$567$	8.47366e22	0.333131
$568$	0	0
$569$	−2.43221e23	−0.927998	−0.463999	−	0.885836i	$-0.653586\pi$
$569$	−2.43221e23	−0.927998	−0.463999	+	0.885836i	$0.653586\pi$
$570$	0	0
$571$	−1.84280e23	−0.682452	−0.341226	−	0.939981i	$-0.610842\pi$
$571$	−1.84280e23	−0.682452	−0.341226	+	0.939981i	$0.610842\pi$
$572$	0	0
$573$	−3.50119e23	−1.25864
$574$	0	0
$575$	5.93922e22	0.207278
$576$	0	0
$577$	2.30483e23	0.780987	0.390493	−	0.920606i	$-0.372304\pi$
$577$	2.30483e23	0.780987	0.390493	+	0.920606i	$0.372304\pi$
$578$	0	0
$579$	1.65027e23	0.542984
$580$	0	0
$581$	2.89244e22	0.0924202
$582$	0	0
$583$	1.42325e21	0.00441671
$584$	0	0
$585$	7.65885e23	2.30855
$586$	0	0
$587$	6.23301e23	1.82505	0.912524	−	0.409024i	$-0.134131\pi$
$587$	6.23301e23	1.82505	0.912524	+	0.409024i	$0.134131\pi$
$588$	0	0
$589$	2.62900e23	0.747843
$590$	0	0
$591$	−1.06292e24	−2.93769
$592$	0	0
$593$	3.24824e23	0.872334	0.436167	−	0.899866i	$-0.356336\pi$
$593$	3.24824e23	0.872334	0.436167	+	0.899866i	$0.356336\pi$
$594$	0	0
$595$	−1.67199e22	−0.0436354
$596$	0	0
$597$	1.18615e23	0.300856
$598$	0	0
$599$	−5.52096e22	−0.136109	−0.0680544	−	0.997682i	$-0.521679\pi$
$599$	−5.52096e22	−0.136109	−0.0680544	+	0.997682i	$0.521679\pi$
$600$	0	0
$601$	1.14994e23	0.275576	0.137788	−	0.990462i	$-0.456001\pi$
$601$	1.14994e23	0.275576	0.137788	+	0.990462i	$0.456001\pi$
$602$	0	0
$603$	−8.47290e23	−1.97395
$604$	0	0
$605$	−4.84736e23	−1.09795
$606$	0	0
$607$	3.28819e23	0.724190	0.362095	−	0.932141i	$-0.382062\pi$
$607$	3.28819e23	0.724190	0.362095	+	0.932141i	$0.382062\pi$
$608$	0	0
$609$	−1.30589e23	−0.279679
$610$	0	0
$611$	5.16734e21	0.0107626
$612$	0	0
$613$	8.48661e23	1.71918	0.859588	−	0.510988i	$-0.170720\pi$
$613$	8.48661e23	1.71918	0.859588	+	0.510988i	$0.170720\pi$
$614$	0	0
$615$	6.94725e23	1.36891
$616$	0	0
$617$	1.44331e22	0.0276652	0.0138326	−	0.999904i	$-0.495597\pi$
$617$	1.44331e22	0.0276652	0.0138326	+	0.999904i	$0.495597\pi$
$618$	0	0
$619$	−8.65832e23	−1.61459	−0.807296	−	0.590146i	$-0.799070\pi$
$619$	−8.65832e23	−1.61459	−0.807296	+	0.590146i	$0.799070\pi$
$620$	0	0
$621$	−1.45746e24	−2.64434
$622$	0	0
$623$	3.57338e22	0.0630857
$624$	0	0
$625$	−6.86789e23	−1.17989
$626$	0	0
$627$	1.74563e23	0.291863
$628$	0	0
$629$	−2.06209e23	−0.335566
$630$	0	0
$631$	9.17406e23	1.45316	0.726578	−	0.687084i	$-0.241110\pi$
$631$	9.17406e23	1.45316	0.726578	+	0.687084i	$0.241110\pi$
$632$	0	0
$633$	8.02709e23	1.23773
$634$	0	0
$635$	8.59093e22	0.128962
$636$	0	0
$637$	−5.43437e23	−0.794263
$638$	0	0
$639$	2.80175e24	3.98724
$640$	0	0
$641$	−1.04783e24	−1.45210	−0.726050	−	0.687642i	$-0.758646\pi$
$641$	−1.04783e24	−1.45210	−0.726050	+	0.687642i	$0.758646\pi$
$642$	0	0
$643$	9.84086e22	0.132813	0.0664064	−	0.997793i	$-0.478847\pi$
$643$	9.84086e22	0.132813	0.0664064	+	0.997793i	$0.478847\pi$
$644$	0	0
$645$	1.77442e24	2.33238
$646$	0	0
$647$	−1.49070e24	−1.90855	−0.954273	−	0.298937i	$-0.903368\pi$
$647$	−1.49070e24	−1.90855	−0.954273	+	0.298937i	$0.903368\pi$
$648$	0	0
$649$	−9.76153e22	−0.121741
$650$	0	0
$651$	−8.94748e22	−0.108708
$652$	0	0
$653$	8.28026e23	0.980126	0.490063	−	0.871687i	$-0.336974\pi$
$653$	8.28026e23	0.980126	0.490063	+	0.871687i	$0.336974\pi$
$654$	0	0
$655$	−9.33094e23	−1.07615
$656$	0	0
$657$	−2.17059e24	−2.43934
$658$	0	0
$659$	−1.75140e24	−1.91805	−0.959025	−	0.283320i	$-0.908564\pi$
$659$	−1.75140e24	−1.91805	−0.959025	+	0.283320i	$0.908564\pi$
$660$	0	0
$661$	1.25523e24	1.33971	0.669856	−	0.742491i	$-0.266356\pi$
$661$	1.25523e24	1.33971	0.669856	+	0.742491i	$0.266356\pi$
$662$	0	0
$663$	5.33825e23	0.555308
$664$	0	0
$665$	1.65206e23	0.167510
$666$	0	0
$667$	1.22425e24	1.21004
$668$	0	0
$669$	−1.65362e24	−1.59336
$670$	0	0
$671$	−1.64104e23	−0.154162
$672$	0	0
$673$	4.30567e23	0.394378	0.197189	−	0.980365i	$-0.436819\pi$
$673$	4.30567e23	0.394378	0.197189	+	0.980365i	$0.436819\pi$
$674$	0	0
$675$	−7.90335e23	−0.705877
$676$	0	0
$677$	4.23476e23	0.368829	0.184415	−	0.982849i	$-0.440961\pi$
$677$	4.23476e23	0.368829	0.184415	+	0.982849i	$0.440961\pi$
$678$	0	0
$679$	−1.36364e23	−0.115826
$680$	0	0
$681$	1.67241e24	1.38546
$682$	0	0
$683$	1.08746e23	0.0878690	0.0439345	−	0.999034i	$-0.486011\pi$
$683$	1.08746e23	0.0878690	0.0439345	+	0.999034i	$0.486011\pi$
$684$	0	0
$685$	1.87053e23	0.147433
$686$	0	0
$687$	−8.04824e23	−0.618826
$688$	0	0
$689$	−4.30051e22	−0.0322594
$690$	0	0
$691$	−1.21387e24	−0.888404	−0.444202	−	0.895927i	$-0.646513\pi$
$691$	−1.21387e24	−0.888404	−0.444202	+	0.895927i	$0.646513\pi$
$692$	0	0
$693$	−4.28382e22	−0.0305913
$694$	0	0
$695$	1.64744e24	1.14799
$696$	0	0
$697$	3.49154e23	0.237431
$698$	0	0
$699$	1.49642e24	0.993109
$700$	0	0
$701$	2.16022e24	1.39925	0.699625	−	0.714510i	$-0.253350\pi$
$701$	2.16022e24	1.39925	0.699625	+	0.714510i	$0.253350\pi$
$702$	0	0
$703$	2.03751e24	1.28819
$704$	0	0
$705$	−4.56687e22	−0.0281845
$706$	0	0
$707$	2.50300e23	0.150798
$708$	0	0
$709$	−3.06732e24	−1.80412	−0.902062	−	0.431606i	$-0.857947\pi$
$709$	−3.06732e24	−1.80412	−0.902062	+	0.431606i	$0.857947\pi$
$710$	0	0
$711$	−4.73642e24	−2.71993
$712$	0	0
$713$	8.38810e23	0.470330
$714$	0	0
$715$	−1.79455e23	−0.0982546
$716$	0	0
$717$	−2.73527e24	−1.46247
$718$	0	0
$719$	−1.12483e24	−0.587343	−0.293671	−	0.955906i	$-0.594877\pi$
$719$	−1.12483e24	−0.587343	−0.293671	+	0.955906i	$0.594877\pi$
$720$	0	0
$721$	1.76333e23	0.0899258
$722$	0	0
$723$	−8.95611e23	−0.446112
$724$	0	0
$725$	6.63874e23	0.323007
$726$	0	0
$727$	−3.37960e24	−1.60629	−0.803144	−	0.595785i	$-0.796841\pi$
$727$	−3.37960e24	−1.60629	−0.803144	+	0.595785i	$0.796841\pi$
$728$	0	0
$729$	5.00387e24	2.32339
$730$	0	0
$731$	8.91786e23	0.404541
$732$	0	0
$733$	1.47640e24	0.654365	0.327183	−	0.944961i	$-0.393901\pi$
$733$	1.47640e24	0.654365	0.327183	+	0.944961i	$0.393901\pi$
$734$	0	0
$735$	4.80286e24	2.07997
$736$	0	0
$737$	1.98529e23	0.0840136
$738$	0	0
$739$	2.15072e24	0.889419	0.444709	−	0.895675i	$-0.353307\pi$
$739$	2.15072e24	0.889419	0.444709	+	0.895675i	$0.353307\pi$
$740$	0	0
$741$	−5.27462e24	−2.13175
$742$	0	0
$743$	−1.03206e24	−0.407663	−0.203831	−	0.979006i	$-0.565339\pi$
$743$	−1.03206e24	−0.407663	−0.203831	+	0.979006i	$0.565339\pi$
$744$	0	0
$745$	−5.19923e24	−2.00729
$746$	0	0
$747$	5.88503e24	2.22087
$748$	0	0
$749$	2.08281e23	0.0768342
$750$	0	0
$751$	4.96754e23	0.179144	0.0895719	−	0.995980i	$-0.471450\pi$
$751$	4.96754e23	0.179144	0.0895719	+	0.995980i	$0.471450\pi$
$752$	0	0
$753$	1.34203e23	0.0473155
$754$	0	0
$755$	4.64168e23	0.160002
$756$	0	0
$757$	3.61087e24	1.21702	0.608509	−	0.793547i	$-0.291768\pi$
$757$	3.61087e24	1.21702	0.608509	+	0.793547i	$0.291768\pi$
$758$	0	0
$759$	5.56963e23	0.183557
$760$	0	0
$761$	−6.85509e23	−0.220924	−0.110462	−	0.993880i	$-0.535233\pi$
$761$	−6.85509e23	−0.220924	−0.110462	+	0.993880i	$0.535233\pi$
$762$	0	0
$763$	−2.63767e23	−0.0831307
$764$	0	0
$765$	−3.40187e24	−1.04857
$766$	0	0
$767$	2.94955e24	0.889191
$768$	0	0
$769$	−6.43730e24	−1.89815	−0.949074	−	0.315054i	$-0.897977\pi$
$769$	−6.43730e24	−1.89815	−0.949074	+	0.315054i	$0.897977\pi$
$770$	0	0
$771$	−2.70817e24	−0.781112
$772$	0	0
$773$	4.44679e24	1.25465	0.627323	−	0.778759i	$-0.284151\pi$
$773$	4.44679e24	1.25465	0.627323	+	0.778759i	$0.284151\pi$
$774$	0	0
$775$	4.54862e23	0.125549
$776$	0	0
$777$	−6.93441e23	−0.187254
$778$	0	0
$779$	−3.44991e24	−0.911465
$780$	0	0
$781$	−6.56478e23	−0.169702
$782$	0	0
$783$	−1.62911e25	−4.12075
$784$	0	0
$785$	−8.60765e23	−0.213055
$786$	0	0
$787$	6.94797e24	1.68296	0.841478	−	0.540292i	$-0.181686\pi$
$787$	6.94797e24	1.68296	0.841478	+	0.540292i	$0.181686\pi$
$788$	0	0
$789$	1.22231e25	2.89753
$790$	0	0
$791$	−7.82566e23	−0.181560
$792$	0	0
$793$	4.95859e24	1.12599
$794$	0	0
$795$	3.80076e23	0.0844793
$796$	0	0
$797$	5.46753e23	0.118958	0.0594792	−	0.998230i	$-0.481056\pi$
$797$	5.46753e23	0.118958	0.0594792	+	0.998230i	$0.481056\pi$
$798$	0	0
$799$	−2.29521e22	−0.00488848
$800$	0	0
$801$	7.27050e24	1.51596
$802$	0	0
$803$	5.08590e23	0.103821
$804$	0	0
$805$	5.27106e23	0.105350
$806$	0	0
$807$	3.08706e24	0.604115
$808$	0	0
$809$	−4.28024e24	−0.820174	−0.410087	−	0.912046i	$-0.634502\pi$
$809$	−4.28024e24	−0.820174	−0.410087	+	0.912046i	$0.634502\pi$
$810$	0	0
$811$	7.01384e24	1.31607	0.658034	−	0.752988i	$-0.271388\pi$
$811$	7.01384e24	1.31607	0.658034	+	0.752988i	$0.271388\pi$
$812$	0	0
$813$	−8.06333e24	−1.48165
$814$	0	0
$815$	−9.44431e24	−1.69954
$816$	0	0
$817$	−8.81156e24	−1.55298
$818$	0	0
$819$	1.29440e24	0.223437
$820$	0	0
$821$	1.96574e24	0.332360	0.166180	−	0.986095i	$-0.446857\pi$
$821$	1.96574e24	0.332360	0.166180	+	0.986095i	$0.446857\pi$
$822$	0	0
$823$	4.44335e24	0.735889	0.367944	−	0.929848i	$-0.380062\pi$
$823$	4.44335e24	0.735889	0.367944	+	0.929848i	$0.380062\pi$
$824$	0	0
$825$	3.02024e23	0.0489986
$826$	0	0
$827$	−8.17355e24	−1.29901	−0.649507	−	0.760355i	$-0.725025\pi$
$827$	−8.17355e24	−1.29901	−0.649507	+	0.760355i	$0.725025\pi$
$828$	0	0
$829$	5.09518e24	0.793316	0.396658	−	0.917966i	$-0.370170\pi$
$829$	5.09518e24	0.793316	0.396658	+	0.917966i	$0.370170\pi$
$830$	0	0
$831$	1.85828e25	2.83467
$832$	0	0
$833$	2.41381e24	0.360762
$834$	0	0
$835$	−1.01550e25	−1.48712
$836$	0	0
$837$	−1.11621e25	−1.60169
$838$	0	0
$839$	−1.09555e25	−1.54048	−0.770239	−	0.637755i	$-0.779863\pi$
$839$	−1.09555e25	−1.54048	−0.770239	+	0.637755i	$0.779863\pi$
$840$	0	0
$841$	6.42726e24	0.885646
$842$	0	0
$843$	−1.35716e25	−1.83272
$844$	0	0
$845$	−2.97519e24	−0.393760
$846$	0	0
$847$	−8.19238e23	−0.106268
$848$	0	0
$849$	1.31727e25	1.67479
$850$	0	0
$851$	6.50089e24	0.810162
$852$	0	0
$853$	−1.17613e25	−1.43677	−0.718386	−	0.695644i	$-0.755119\pi$
$853$	−1.17613e25	−1.43677	−0.718386	+	0.695644i	$0.755119\pi$
$854$	0	0
$855$	3.36132e25	4.02530
$856$	0	0
$857$	−1.55161e24	−0.182157	−0.0910787	−	0.995844i	$-0.529031\pi$
$857$	−1.55161e24	−0.182157	−0.0910787	+	0.995844i	$0.529031\pi$
$858$	0	0
$859$	−1.15282e25	−1.32685	−0.663424	−	0.748244i	$-0.730897\pi$
$859$	−1.15282e25	−1.32685	−0.663424	+	0.748244i	$0.730897\pi$
$860$	0	0
$861$	1.17414e24	0.132493
$862$	0	0
$863$	7.30526e24	0.808246	0.404123	−	0.914705i	$-0.367577\pi$
$863$	7.30526e24	0.808246	0.404123	+	0.914705i	$0.367577\pi$
$864$	0	0
$865$	1.46376e25	1.58794
$866$	0	0
$867$	1.54281e25	1.64116
$868$	0	0
$869$	1.10979e24	0.115764
$870$	0	0
$871$	−5.99876e24	−0.613631
$872$	0	0
$873$	−2.77450e25	−2.78332
$874$	0	0
$875$	−9.29320e23	−0.0914317
$876$	0	0
$877$	8.83912e24	0.852928	0.426464	−	0.904504i	$-0.359759\pi$
$877$	8.83912e24	0.852928	0.426464	+	0.904504i	$0.359759\pi$
$878$	0	0
$879$	−3.11972e25	−2.95264
$880$	0	0
$881$	5.00598e24	0.464722	0.232361	−	0.972630i	$-0.425355\pi$
$881$	5.00598e24	0.464722	0.232361	+	0.972630i	$0.425355\pi$
$882$	0	0
$883$	9.42478e24	0.858233	0.429117	−	0.903249i	$-0.358825\pi$
$883$	9.42478e24	0.858233	0.429117	+	0.903249i	$0.358825\pi$
$884$	0	0
$885$	−2.60679e25	−2.32857
$886$	0	0
$887$	1.59100e25	1.39418	0.697090	−	0.716983i	$-0.254478\pi$
$887$	1.59100e25	1.39418	0.697090	+	0.716983i	$0.254478\pi$
$888$	0	0
$889$	1.45193e23	0.0124819
$890$	0	0
$891$	−4.03968e24	−0.340711
$892$	0	0
$893$	2.26785e23	0.0187662
$894$	0	0
$895$	4.46393e24	0.362428
$896$	0	0
$897$	−1.68292e25	−1.34069
$898$	0	0
$899$	9.37605e24	0.732930
$900$	0	0
$901$	1.91018e23	0.0146526
$902$	0	0
$903$	2.99890e24	0.225744
$904$	0	0
$905$	8.05467e24	0.595024
$906$	0	0
$907$	−1.39294e25	−1.00988	−0.504942	−	0.863153i	$-0.668486\pi$
$907$	−1.39294e25	−1.00988	−0.504942	+	0.863153i	$0.668486\pi$
$908$	0	0
$909$	5.09267e25	3.62370
$910$	0	0
$911$	−4.99198e24	−0.348632	−0.174316	−	0.984690i	$-0.555771\pi$
$911$	−4.99198e24	−0.348632	−0.174316	+	0.984690i	$0.555771\pi$
$912$	0	0
$913$	−1.37892e24	−0.0945231
$914$	0	0
$915$	−4.38237e25	−2.94869
$916$	0	0
$917$	−1.57700e24	−0.104158
$918$	0	0
$919$	−3.29906e24	−0.213899	−0.106950	−	0.994264i	$-0.534108\pi$
$919$	−3.29906e24	−0.213899	−0.106950	+	0.994264i	$0.534108\pi$
$920$	0	0
$921$	4.44735e23	0.0283070
$922$	0	0
$923$	1.98362e25	1.23949
$924$	0	0
$925$	3.52524e24	0.216264
$926$	0	0
$927$	3.58772e25	2.16093
$928$	0	0
$929$	1.47943e25	0.874903	0.437451	−	0.899242i	$-0.355881\pi$
$929$	1.47943e25	0.874903	0.437451	+	0.899242i	$0.355881\pi$
$930$	0	0
$931$	−2.38504e25	−1.38492
$932$	0	0
$933$	3.78186e25	2.15631
$934$	0	0
$935$	7.97093e23	0.0446283
$936$	0	0
$937$	−8.80353e24	−0.484028	−0.242014	−	0.970273i	$-0.577808\pi$
$937$	−8.80353e24	−0.484028	−0.242014	+	0.970273i	$0.577808\pi$
$938$	0	0
$939$	−1.62730e25	−0.878640
$940$	0	0
$941$	−2.08706e25	−1.10668	−0.553342	−	0.832954i	$-0.686648\pi$
$941$	−2.08706e25	−1.10668	−0.553342	+	0.832954i	$0.686648\pi$
$942$	0	0
$943$	−1.10073e25	−0.573234
$944$	0	0
$945$	−7.01423e24	−0.358765
$946$	0	0
$947$	1.52689e24	0.0767067	0.0383533	−	0.999264i	$-0.487789\pi$
$947$	1.52689e24	0.0767067	0.0383533	+	0.999264i	$0.487789\pi$
$948$	0	0
$949$	−1.53676e25	−0.758304
$950$	0	0
$951$	−4.26482e25	−2.06712
$952$	0	0
$953$	−2.94509e25	−1.40220	−0.701099	−	0.713064i	$-0.747307\pi$
$953$	−2.94509e25	−1.40220	−0.701099	+	0.713064i	$0.747307\pi$
$954$	0	0
$955$	1.57966e25	0.738814
$956$	0	0
$957$	6.22562e24	0.286043
$958$	0	0
$959$	3.16134e23	0.0142696
$960$	0	0
$961$	−1.61260e25	−0.715118
$962$	0	0
$963$	4.23775e25	1.84634
$964$	0	0
$965$	−7.44565e24	−0.318728
$966$	0	0
$967$	3.64906e25	1.53481	0.767407	−	0.641160i	$-0.221547\pi$
$967$	3.64906e25	1.53481	0.767407	+	0.641160i	$0.221547\pi$
$968$	0	0
$969$	2.34286e25	0.968264
$970$	0	0
$971$	4.03378e25	1.63813	0.819065	−	0.573700i	$-0.194492\pi$
$971$	4.03378e25	1.63813	0.819065	+	0.573700i	$0.194492\pi$
$972$	0	0
$973$	2.78429e24	0.111110
$974$	0	0
$975$	−9.12599e24	−0.357883
$976$	0	0
$977$	−1.53014e25	−0.589694	−0.294847	−	0.955545i	$-0.595269\pi$
$977$	−1.53014e25	−0.589694	−0.294847	+	0.955545i	$0.595269\pi$
$978$	0	0
$979$	−1.70355e24	−0.0645212
$980$	0	0
$981$	−5.36667e25	−1.99764
$982$	0	0
$983$	−1.72483e25	−0.631016	−0.315508	−	0.948923i	$-0.602175\pi$
$983$	−1.72483e25	−0.631016	−0.315508	+	0.948923i	$0.602175\pi$
$984$	0	0
$985$	4.79565e25	1.72441
$986$	0	0
$987$	−7.71834e22	−0.00272790
$988$	0	0
$989$	−2.81142e25	−0.976691
$990$	0	0
$991$	4.22399e25	1.44244	0.721219	−	0.692707i	$-0.243582\pi$
$991$	4.22399e25	1.44244	0.721219	+	0.692707i	$0.243582\pi$
$992$	0	0
$993$	7.90905e25	2.65494
$994$	0	0
$995$	−5.35165e24	−0.176600
$996$	0	0
$997$	−2.14018e25	−0.694290	−0.347145	−	0.937812i	$-0.612849\pi$
$997$	−2.14018e25	−0.694290	−0.347145	+	0.937812i	$0.612849\pi$
$998$	0	0
$999$	−8.65077e25	−2.75898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.18.a.a.1.1	✓	2
3.2	odd	2		36.18.a.d.1.1		2
4.3	odd	2		16.18.a.e.1.2		2
5.2	odd	4		100.18.c.a.49.4		4
5.3	odd	4		100.18.c.a.49.1		4
5.4	even	2		100.18.a.b.1.2		2
8.3	odd	2		64.18.a.g.1.1		2
8.5	even	2		64.18.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.18.a.a.1.1	✓	2	1.1	even	1	trivial
16.18.a.e.1.2		2	4.3	odd	2
36.18.a.d.1.1		2	3.2	odd	2
64.18.a.g.1.1		2	8.3	odd	2
64.18.a.l.1.2		2	8.5	even	2
100.18.a.b.1.2		2	5.4	even	2
100.18.c.a.49.1		4	5.3	odd	4
100.18.c.a.49.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(f(q)\)	\(=\)	\(q-21516.4 q^{3} +970774. q^{5} +1.64068e6 q^{7} +3.33817e8 q^{9} +O(q^{10})\)	\(q-21516.4 q^{3} +970774. q^{5} +1.64068e6 q^{7} +3.33817e8 q^{9} -7.82166e7 q^{11} +2.36340e9 q^{13} -2.08876e10 q^{15} -1.04976e10 q^{17} +1.03725e11 q^{19} -3.53015e10 q^{21} +3.30946e11 q^{23} +1.79462e11 q^{25} -4.40391e12 q^{27} +3.69924e12 q^{29} +2.53459e12 q^{31} +1.68294e12 q^{33} +1.59273e12 q^{35} +1.96434e13 q^{37} -5.08519e13 q^{39} -3.32602e13 q^{41} -8.49511e13 q^{43} +3.24061e14 q^{45} +2.18640e12 q^{47} -2.29939e14 q^{49} +2.25872e14 q^{51} -1.81963e13 q^{53} -7.59307e13 q^{55} -2.23179e15 q^{57} +1.24801e15 q^{59} +2.09807e15 q^{61} +5.47686e14 q^{63} +2.29433e15 q^{65} -2.53819e15 q^{67} -7.12077e15 q^{69} +8.39308e15 q^{71} -6.50233e15 q^{73} -3.86138e15 q^{75} -1.28328e14 q^{77} -1.41887e16 q^{79} +5.16473e16 q^{81} +1.76295e16 q^{83} -1.01908e16 q^{85} -7.95945e16 q^{87} +2.17799e16 q^{89} +3.87758e15 q^{91} -5.45352e16 q^{93} +1.00694e17 q^{95} -8.31143e16 q^{97} -2.61100e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5880 q^{3} + 604044 q^{5} + 25350160 q^{7} + 449174682 q^{9}+O(q^{10}) \) 2 * q - 5880 * q^3 + 604044 * q^5 + 25350160 * q^7 + 449174682 * q^9	\( 2 q - 5880 q^{3} + 604044 q^{5} + 25350160 q^{7} + 449174682 q^{9} + 1259648280 q^{11} - 1320052580 q^{13} - 26621930448 q^{15} - 27498226140 q^{17} + 101133633832 q^{19} + 335430207552 q^{21} + 134767491120 q^{23} - 448986850114 q^{25} - 4619416117680 q^{27} + 2337155582652 q^{29} + 278836113472 q^{31} + 22602380058720 q^{33} - 7102242382752 q^{35} + 20929802888140 q^{37} - 108447997173648 q^{39} + 4166592315732 q^{41} - 111143148534440 q^{43} + 281755357404444 q^{45} + 196772651157600 q^{47} + 99570326741874 q^{49} - 39956666918256 q^{51} - 487965122736660 q^{53} - 566565363246960 q^{55} - 22\!\cdots\!40 q^{57}+ \cdots + 12\!\cdots\!80 q^{99}+O(q^{100}) \) 2 * q - 5880 * q^3 + 604044 * q^5 + 25350160 * q^7 + 449174682 * q^9 + 1259648280 * q^11 - 1320052580 * q^13 - 26621930448 * q^15 - 27498226140 * q^17 + 101133633832 * q^19 + 335430207552 * q^21 + 134767491120 * q^23 - 448986850114 * q^25 - 4619416117680 * q^27 + 2337155582652 * q^29 + 278836113472 * q^31 + 22602380058720 * q^33 - 7102242382752 * q^35 + 20929802888140 * q^37 - 108447997173648 * q^39 + 4166592315732 * q^41 - 111143148534440 * q^43 + 281755357404444 * q^45 + 196772651157600 * q^47 + 99570326741874 * q^49 - 39956666918256 * q^51 - 487965122736660 * q^53 - 566565363246960 * q^55 - 2272313631999840 * q^57 + 2835904197813624 * q^59 - 67544034994436 * q^61 + 3282762121966800 * q^63 + 3645157343001768 * q^65 - 995171321546360 * q^67 - 10188302100105024 * q^69 + 3882245493215376 * q^71 - 12746425881769580 * q^73 - 13688080703624712 * q^75 + 31591755846002880 * q^77 - 14984271534065504 * q^79 + 33380230287529458 * q^81 + 43899417809893800 * q^83 - 3956216991540264 * q^85 - 100892756486768400 * q^87 + 51909007958846388 * q^89 - 83455169400462112 * q^91 - 89807116632334080 * q^93 + 101643889304423664 * q^95 - 49281007789848380 * q^97 + 128223271309133880 * q^99

Introduction

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