LMFDB - Embedded newform 48.18.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,18,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	48.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:	$87.9466019254$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14569}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3642 $ x^2 - x - 3642
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-59.8511$ of defining polynomial
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6561.00 q^{3} +1.46604e6 q^{5} -2.28730e7 q^{7} +4.30467e7 q^{9} +O(q^{10})$	$q-6561.00 q^{3} +1.46604e6 q^{5} -2.28730e7 q^{7} +4.30467e7 q^{9} +5.40173e8 q^{11} -3.45398e7 q^{13} -9.61872e9 q^{15} -9.43866e9 q^{17} +2.25061e9 q^{19} +1.50070e11 q^{21} +3.45854e11 q^{23} +1.38635e12 q^{25} -2.82430e11 q^{27} +5.11838e11 q^{29} -1.14436e11 q^{31} -3.54407e12 q^{33} -3.35329e13 q^{35} -1.56972e13 q^{37} +2.26616e11 q^{39} -8.00761e13 q^{41} +3.66737e13 q^{43} +6.31084e13 q^{45} +1.17577e14 q^{47} +2.90545e14 q^{49} +6.19270e13 q^{51} -3.90043e14 q^{53} +7.91917e14 q^{55} -1.47663e13 q^{57} -1.82562e15 q^{59} +1.41053e15 q^{61} -9.84609e14 q^{63} -5.06369e13 q^{65} +1.47114e15 q^{67} -2.26915e15 q^{69} +7.31441e15 q^{71} +1.34580e16 q^{73} -9.09582e15 q^{75} -1.23554e16 q^{77} +8.37779e15 q^{79} +1.85302e15 q^{81} +2.55978e16 q^{83} -1.38375e16 q^{85} -3.35817e15 q^{87} +4.47540e16 q^{89} +7.90031e14 q^{91} +7.50815e14 q^{93} +3.29949e15 q^{95} +7.41658e16 q^{97} +2.32527e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 13122 q^{3} + 382860 q^{5} - 24471568 q^{7} + 86093442 q^{9}+O(q^{10}) $ 2 * q - 13122 * q^3 + 382860 * q^5 - 24471568 * q^7 + 86093442 * q^9	$ 2 q - 13122 q^{3} + 382860 q^{5} - 24471568 q^{7} + 86093442 q^{9} + 987553512 q^{11} - 2519398244 q^{13} - 2511944460 q^{15} - 34313126364 q^{17} - 80053542184 q^{19} + 160557957648 q^{21} - 297228742704 q^{23} + 1796695260350 q^{25} - 564859072962 q^{27} - 470374069572 q^{29} - 3400754454592 q^{31} - 6479338592232 q^{33} - 31801338154080 q^{35} + 10652012180428 q^{37} + 16529771878884 q^{39} - 113376799448748 q^{41} - 61637031489880 q^{43} + 16480867602060 q^{45} + 279645641926560 q^{47} + 60469362475890 q^{49} + 225128422074204 q^{51} - 530964038611476 q^{53} + 307321280077680 q^{55} + 525231290269224 q^{57} - 17\!\cdots\!56 q^{59}+ \cdots + 42\!\cdots\!52 q^{99}+O(q^{100}) $ 2 * q - 13122 * q^3 + 382860 * q^5 - 24471568 * q^7 + 86093442 * q^9 + 987553512 * q^11 - 2519398244 * q^13 - 2511944460 * q^15 - 34313126364 * q^17 - 80053542184 * q^19 + 160557957648 * q^21 - 297228742704 * q^23 + 1796695260350 * q^25 - 564859072962 * q^27 - 470374069572 * q^29 - 3400754454592 * q^31 - 6479338592232 * q^33 - 31801338154080 * q^35 + 10652012180428 * q^37 + 16529771878884 * q^39 - 113376799448748 * q^41 - 61637031489880 * q^43 + 16480867602060 * q^45 + 279645641926560 * q^47 + 60469362475890 * q^49 + 225128422074204 * q^51 - 530964038611476 * q^53 + 307321280077680 * q^55 + 525231290269224 * q^57 - 1727524231086456 * q^59 + 2784287656027900 * q^61 - 1053420760128528 * q^63 + 2640922679811240 * q^65 + 3329301676696184 * q^67 + 1950117780880944 * q^69 + 13489402206504816 * q^71 + 436589918136724 * q^73 - 11788117603156350 * q^75 - 13070541011736384 * q^77 - 4376041565214880 * q^79 + 3706040377703682 * q^81 + 39886442265612888 * q^83 + 13106143850552280 * q^85 + 3086124270461892 * q^87 + 6972184096107444 * q^89 + 4762196572326688 * q^91 + 22312349976578112 * q^93 + 92450059415485200 * q^95 + 183569555712460996 * q^97 + 42510940503634152 * 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
$3$	−6561.00	−0.577350
$3$	−6561.00	−0.577350
$4$	0	0
$5$	1.46604e6	1.67843	0.839213	−	0.543803i	$-0.183016\pi$
$5$	1.46604e6	1.67843	0.839213	+	0.543803i	$0.183016\pi$
$6$	0	0
$7$	−2.28730e7	−1.49965	−0.749825	−	0.661636i	$-0.769863\pi$
$7$	−2.28730e7	−1.49965	−0.749825	+	0.661636i	$0.769863\pi$
$8$	0	0
$9$	4.30467e7	0.333333
$10$	0	0
$11$	5.40173e8	0.759792	0.379896	−	0.925029i	$-0.375960\pi$
$11$	5.40173e8	0.759792	0.379896	+	0.925029i	$0.375960\pi$
$12$	0	0
$13$	−3.45398e7	−0.0117436	−0.00587181	−	0.999983i	$-0.501869\pi$
$13$	−3.45398e7	−0.0117436	−0.00587181	+	0.999983i	$0.501869\pi$
$14$	0	0
$15$	−9.61872e9	−0.969039
$16$	0	0
$17$	−9.43866e9	−0.328167	−0.164083	−	0.986446i	$-0.552467\pi$
$17$	−9.43866e9	−0.328167	−0.164083	+	0.986446i	$0.552467\pi$
$18$	0	0
$19$	2.25061e9	0.0304015	0.0152007	−	0.999884i	$-0.495161\pi$
$19$	2.25061e9	0.0304015	0.0152007	+	0.999884i	$0.495161\pi$
$20$	0	0
$21$	1.50070e11	0.865824
$22$	0	0
$23$	3.45854e11	0.920886	0.460443	−	0.887689i	$-0.347690\pi$
$23$	3.45854e11	0.920886	0.460443	+	0.887689i	$0.347690\pi$
$24$	0	0
$25$	1.38635e12	1.81711
$26$	0	0
$27$	−2.82430e11	−0.192450
$28$	0	0
$29$	5.11838e11	0.189998	0.0949992	−	0.995477i	$-0.469715\pi$
$29$	5.11838e11	0.189998	0.0949992	+	0.995477i	$0.469715\pi$
$30$	0	0
$31$	−1.14436e11	−0.0240984	−0.0120492	−	0.999927i	$-0.503835\pi$
$31$	−1.14436e11	−0.0240984	−0.0120492	+	0.999927i	$0.503835\pi$
$32$	0	0
$33$	−3.54407e12	−0.438666
$34$	0	0
$35$	−3.35329e13	−2.51705
$36$	0	0
$37$	−1.56972e13	−0.734697	−0.367349	−	0.930083i	$-0.619734\pi$
$37$	−1.56972e13	−0.734697	−0.367349	+	0.930083i	$0.619734\pi$
$38$	0	0
$39$	2.26616e11	0.00678018
$40$	0	0
$41$	−8.00761e13	−1.56617	−0.783087	−	0.621912i	$-0.786356\pi$
$41$	−8.00761e13	−1.56617	−0.783087	+	0.621912i	$0.786356\pi$
$42$	0	0
$43$	3.66737e13	0.478489	0.239245	−	0.970959i	$-0.423100\pi$
$43$	3.66737e13	0.478489	0.239245	+	0.970959i	$0.423100\pi$
$44$	0	0
$45$	6.31084e13	0.559475
$46$	0	0
$47$	1.17577e14	0.720263	0.360132	−	0.932901i	$-0.382732\pi$
$47$	1.17577e14	0.720263	0.360132	+	0.932901i	$0.382732\pi$
$48$	0	0
$49$	2.90545e14	1.24895
$50$	0	0
$51$	6.19270e13	0.189467
$52$	0	0
$53$	−3.90043e14	−0.860534	−0.430267	−	0.902702i	$-0.641581\pi$
$53$	−3.90043e14	−0.860534	−0.430267	+	0.902702i	$0.641581\pi$
$54$	0	0
$55$	7.91917e14	1.27525
$56$	0	0
$57$	−1.47663e13	−0.0175523
$58$	0	0
$59$	−1.82562e15	−1.61870	−0.809352	−	0.587323i	$-0.800182\pi$
$59$	−1.82562e15	−1.61870	−0.809352	+	0.587323i	$0.800182\pi$
$60$	0	0
$61$	1.41053e15	0.942059	0.471029	−	0.882118i	$-0.343883\pi$
$61$	1.41053e15	0.942059	0.471029	+	0.882118i	$0.343883\pi$
$62$	0	0
$63$	−9.84609e14	−0.499884
$64$	0	0
$65$	−5.06369e13	−0.0197108
$66$	0	0
$67$	1.47114e15	0.442607	0.221304	−	0.975205i	$-0.428969\pi$
$67$	1.47114e15	0.442607	0.221304	+	0.975205i	$0.428969\pi$
$68$	0	0
$69$	−2.26915e15	−0.531674
$70$	0	0
$71$	7.31441e15	1.34426	0.672130	−	0.740433i	$-0.265380\pi$
$71$	7.31441e15	1.34426	0.672130	+	0.740433i	$0.265380\pi$
$72$	0	0
$73$	1.34580e16	1.95315	0.976577	−	0.215169i	$-0.0690302\pi$
$73$	1.34580e16	1.95315	0.976577	+	0.215169i	$0.0690302\pi$
$74$	0	0
$75$	−9.09582e15	−1.04911
$76$	0	0
$77$	−1.23554e16	−1.13942
$78$	0	0
$79$	8.37779e15	0.621297	0.310648	−	0.950525i	$-0.399454\pi$
$79$	8.37779e15	0.621297	0.310648	+	0.950525i	$0.399454\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	0	0
$83$	2.55978e16	1.24750	0.623748	−	0.781625i	$-0.285609\pi$
$83$	2.55978e16	1.24750	0.623748	+	0.781625i	$0.285609\pi$
$84$	0	0
$85$	−1.38375e16	−0.550803
$86$	0	0
$87$	−3.35817e15	−0.109696
$88$	0	0
$89$	4.47540e16	1.20508	0.602541	−	0.798088i	$-0.294155\pi$
$89$	4.47540e16	1.20508	0.602541	+	0.798088i	$0.294155\pi$
$90$	0	0
$91$	7.90031e14	0.0176113
$92$	0	0
$93$	7.50815e14	0.0139132
$94$	0	0
$95$	3.29949e15	0.0510266
$96$	0	0
$97$	7.41658e16	0.960824	0.480412	−	0.877043i	$-0.340487\pi$
$97$	7.41658e16	0.960824	0.480412	+	0.877043i	$0.340487\pi$
$98$	0	0
$99$	2.32527e16	0.253264
$100$	0	0
$101$	−7.87706e16	−0.723823	−0.361912	−	0.932212i	$-0.617876\pi$
$101$	−7.87706e16	−0.723823	−0.361912	+	0.932212i	$0.617876\pi$
$102$	0	0
$103$	−6.67667e16	−0.519330	−0.259665	−	0.965699i	$-0.583612\pi$
$103$	−6.67667e16	−0.519330	−0.259665	+	0.965699i	$0.583612\pi$
$104$	0	0
$105$	2.20009e17	1.45322
$106$	0	0
$107$	2.57732e17	1.45013	0.725064	−	0.688681i	$-0.241810\pi$
$107$	2.57732e17	1.45013	0.725064	+	0.688681i	$0.241810\pi$
$108$	0	0
$109$	1.67981e17	0.807488	0.403744	−	0.914872i	$-0.367709\pi$
$109$	1.67981e17	0.807488	0.403744	+	0.914872i	$0.367709\pi$
$110$	0	0
$111$	1.02990e17	0.424178
$112$	0	0
$113$	3.01755e17	1.06779	0.533897	−	0.845550i	$-0.320727\pi$
$113$	3.01755e17	1.06779	0.533897	+	0.845550i	$0.320727\pi$
$114$	0	0
$115$	5.07037e17	1.54564
$116$	0	0
$117$	−1.48683e15	−0.00391454
$118$	0	0
$119$	2.15891e17	0.492135
$120$	0	0
$121$	−2.13660e17	−0.422716
$122$	0	0
$123$	5.25379e17	0.904231
$124$	0	0
$125$	9.13942e17	1.37146
$126$	0	0
$127$	−1.84942e17	−0.242496	−0.121248	−	0.992622i	$-0.538690\pi$
$127$	−1.84942e17	−0.242496	−0.121248	+	0.992622i	$0.538690\pi$
$128$	0	0
$129$	−2.40616e17	−0.276256
$130$	0	0
$131$	−9.78482e16	−0.0985705	−0.0492852	−	0.998785i	$-0.515694\pi$
$131$	−9.78482e16	−0.0985705	−0.0492852	+	0.998785i	$0.515694\pi$
$132$	0	0
$133$	−5.14782e16	−0.0455916
$134$	0	0
$135$	−4.14054e17	−0.323013
$136$	0	0
$137$	−3.72170e17	−0.256222	−0.128111	−	0.991760i	$-0.540891\pi$
$137$	−3.72170e17	−0.256222	−0.128111	+	0.991760i	$0.540891\pi$
$138$	0	0
$139$	−4.12327e17	−0.250967	−0.125483	−	0.992096i	$-0.540048\pi$
$139$	−4.12327e17	−0.250967	−0.125483	+	0.992096i	$0.540048\pi$
$140$	0	0
$141$	−7.71424e17	−0.415844
$142$	0	0
$143$	−1.86575e16	−0.00892271
$144$	0	0
$145$	7.50378e17	0.318898
$146$	0	0
$147$	−1.90626e18	−0.721083
$148$	0	0
$149$	4.80416e18	1.62007	0.810035	−	0.586382i	$-0.199448\pi$
$149$	4.80416e18	1.62007	0.810035	+	0.586382i	$0.199448\pi$
$150$	0	0
$151$	4.91996e18	1.48135	0.740674	−	0.671864i	$-0.234506\pi$
$151$	4.91996e18	1.48135	0.740674	+	0.671864i	$0.234506\pi$
$152$	0	0
$153$	−4.06303e17	−0.109389
$154$	0	0
$155$	−1.67768e17	−0.0404474
$156$	0	0
$157$	6.42241e18	1.38852	0.694258	−	0.719726i	$-0.255733\pi$
$157$	6.42241e18	1.38852	0.694258	+	0.719726i	$0.255733\pi$
$158$	0	0
$159$	2.55907e18	0.496829
$160$	0	0
$161$	−7.91072e18	−1.38101
$162$	0	0
$163$	7.83455e18	1.23146	0.615729	−	0.787958i	$-0.288862\pi$
$163$	7.83455e18	1.23146	0.615729	+	0.787958i	$0.288862\pi$
$164$	0	0
$165$	−5.19577e18	−0.736269
$166$	0	0
$167$	−1.03503e19	−1.32392	−0.661962	−	0.749537i	$-0.730276\pi$
$167$	−1.03503e19	−1.32392	−0.661962	+	0.749537i	$0.730276\pi$
$168$	0	0
$169$	−8.64922e18	−0.999862
$170$	0	0
$171$	9.68814e16	0.0101338
$172$	0	0
$173$	−8.18965e18	−0.776021	−0.388011	−	0.921655i	$-0.626838\pi$
$173$	−8.18965e18	−0.776021	−0.388011	+	0.921655i	$0.626838\pi$
$174$	0	0
$175$	−3.17099e19	−2.72503
$176$	0	0
$177$	1.19779e19	0.934560
$178$	0	0
$179$	2.91944e18	0.207038	0.103519	−	0.994627i	$-0.466990\pi$
$179$	2.91944e18	0.207038	0.103519	+	0.994627i	$0.466990\pi$
$180$	0	0
$181$	−2.50385e19	−1.61562	−0.807812	−	0.589440i	$-0.799349\pi$
$181$	−2.50385e19	−1.61562	−0.807812	+	0.589440i	$0.799349\pi$
$182$	0	0
$183$	−9.25447e18	−0.543898
$184$	0	0
$185$	−2.30128e19	−1.23313
$186$	0	0
$187$	−5.09851e18	−0.249338
$188$	0	0
$189$	6.46002e18	0.288608
$190$	0	0
$191$	2.01953e19	0.825025	0.412513	−	0.910952i	$-0.364651\pi$
$191$	2.01953e19	0.825025	0.412513	+	0.910952i	$0.364651\pi$
$192$	0	0
$193$	4.28113e19	1.60074	0.800372	−	0.599504i	$-0.204635\pi$
$193$	4.28113e19	1.60074	0.800372	+	0.599504i	$0.204635\pi$
$194$	0	0
$195$	3.32229e17	0.0113800
$196$	0	0
$197$	4.01977e19	1.26252	0.631261	−	0.775571i	$-0.282538\pi$
$197$	4.01977e19	1.26252	0.631261	+	0.775571i	$0.282538\pi$
$198$	0	0
$199$	−4.29123e19	−1.23689	−0.618444	−	0.785829i	$-0.712237\pi$
$199$	−4.29123e19	−1.23689	−0.618444	+	0.785829i	$0.712237\pi$
$200$	0	0
$201$	−9.65215e18	−0.255539
$202$	0	0
$203$	−1.17073e19	−0.284931
$204$	0	0
$205$	−1.17395e20	−2.62871
$206$	0	0
$207$	1.48879e19	0.306962
$208$	0	0
$209$	1.21572e18	0.0230988
$210$	0	0
$211$	−1.34631e19	−0.235909	−0.117955	−	0.993019i	$-0.537634\pi$
$211$	−1.34631e19	−0.235909	−0.117955	+	0.993019i	$0.537634\pi$
$212$	0	0
$213$	−4.79898e19	−0.776109
$214$	0	0
$215$	5.37652e19	0.803109
$216$	0	0
$217$	2.61750e18	0.0361392
$218$	0	0
$219$	−8.82980e19	−1.12765
$220$	0	0
$221$	3.26010e17	0.00385386
$222$	0	0
$223$	5.53968e19	0.606587	0.303293	−	0.952897i	$-0.401914\pi$
$223$	5.53968e19	0.606587	0.303293	+	0.952897i	$0.401914\pi$
$224$	0	0
$225$	5.96777e19	0.605704
$226$	0	0
$227$	−1.02921e19	−0.0968915	−0.0484457	−	0.998826i	$-0.515427\pi$
$227$	−1.02921e19	−0.0968915	−0.0484457	+	0.998826i	$0.515427\pi$
$228$	0	0
$229$	1.55018e20	1.35451	0.677254	−	0.735750i	$-0.263170\pi$
$229$	1.55018e20	1.35451	0.677254	+	0.735750i	$0.263170\pi$
$230$	0	0
$231$	8.10637e19	0.657846
$232$	0	0
$233$	−1.64581e19	−0.124124	−0.0620618	−	0.998072i	$-0.519768\pi$
$233$	−1.64581e19	−0.124124	−0.0620618	+	0.998072i	$0.519768\pi$
$234$	0	0
$235$	1.72373e20	1.20891
$236$	0	0
$237$	−5.49667e19	−0.358706
$238$	0	0
$239$	7.36122e19	0.447268	0.223634	−	0.974673i	$-0.428208\pi$
$239$	7.36122e19	0.447268	0.223634	+	0.974673i	$0.428208\pi$
$240$	0	0
$241$	1.00349e20	0.568028	0.284014	−	0.958820i	$-0.408334\pi$
$241$	1.00349e20	0.568028	0.284014	+	0.958820i	$0.408334\pi$
$242$	0	0
$243$	−1.21577e19	−0.0641500
$244$	0	0
$245$	4.25951e20	2.09627
$246$	0	0
$247$	−7.77357e16	−0.000357023 0
$248$	0	0
$249$	−1.67947e20	−0.720243
$250$	0	0
$251$	−3.34304e20	−1.33942	−0.669708	−	0.742624i	$-0.733581\pi$
$251$	−3.34304e20	−1.33942	−0.669708	+	0.742624i	$0.733581\pi$
$252$	0	0
$253$	1.86821e20	0.699682
$254$	0	0
$255$	9.07878e19	0.318006
$256$	0	0
$257$	−6.36862e19	−0.208744	−0.104372	−	0.994538i	$-0.533283\pi$
$257$	−6.36862e19	−0.208744	−0.104372	+	0.994538i	$0.533283\pi$
$258$	0	0
$259$	3.59043e20	1.10179
$260$	0	0
$261$	2.20330e19	0.0633328
$262$	0	0
$263$	3.67784e20	0.990761	0.495381	−	0.868676i	$-0.335029\pi$
$263$	3.67784e20	0.990761	0.495381	+	0.868676i	$0.335029\pi$
$264$	0	0
$265$	−5.71821e20	−1.44434
$266$	0	0
$267$	−2.93631e20	−0.695754
$268$	0	0
$269$	1.44158e20	0.320586	0.160293	−	0.987069i	$-0.448756\pi$
$269$	1.44158e20	0.320586	0.160293	+	0.987069i	$0.448756\pi$
$270$	0	0
$271$	−6.03451e20	−1.26009	−0.630047	−	0.776557i	$-0.716964\pi$
$271$	−6.03451e20	−1.26009	−0.630047	+	0.776557i	$0.716964\pi$
$272$	0	0
$273$	−5.18339e18	−0.0101679
$274$	0	0
$275$	7.48867e20	1.38063
$276$	0	0
$277$	3.10539e20	0.538318	0.269159	−	0.963096i	$-0.413254\pi$
$277$	3.10539e20	0.538318	0.269159	+	0.963096i	$0.413254\pi$
$278$	0	0
$279$	−4.92610e18	−0.00803281
$280$	0	0
$281$	6.43730e20	0.987871	0.493935	−	0.869499i	$-0.335558\pi$
$281$	6.43730e20	0.987871	0.493935	+	0.869499i	$0.335558\pi$
$282$	0	0
$283$	3.43859e20	0.496817	0.248408	−	0.968655i	$-0.420093\pi$
$283$	3.43859e20	0.496817	0.248408	+	0.968655i	$0.420093\pi$
$284$	0	0
$285$	−2.16480e19	−0.0294602
$286$	0	0
$287$	1.83158e21	2.34872
$288$	0	0
$289$	−7.38152e20	−0.892307
$290$	0	0
$291$	−4.86602e20	−0.554732
$292$	0	0
$293$	4.41631e20	0.474990	0.237495	−	0.971389i	$-0.423674\pi$
$293$	4.41631e20	0.474990	0.237495	+	0.971389i	$0.423674\pi$
$294$	0	0
$295$	−2.67644e21	−2.71688
$296$	0	0
$297$	−1.52561e20	−0.146222
$298$	0	0
$299$	−1.19457e19	−0.0108145
$300$	0	0
$301$	−8.38837e20	−0.717567
$302$	0	0
$303$	5.16814e20	0.417899
$304$	0	0
$305$	2.06790e21	1.58118
$306$	0	0
$307$	2.55581e20	0.184864	0.0924319	−	0.995719i	$-0.470536\pi$
$307$	2.55581e20	0.184864	0.0924319	+	0.995719i	$0.470536\pi$
$308$	0	0
$309$	4.38056e20	0.299835
$310$	0	0
$311$	−1.33035e21	−0.861988	−0.430994	−	0.902355i	$-0.641837\pi$
$311$	−1.33035e21	−0.861988	−0.430994	+	0.902355i	$0.641837\pi$
$312$	0	0
$313$	−8.09712e20	−0.496825	−0.248413	−	0.968654i	$-0.579909\pi$
$313$	−8.09712e20	−0.496825	−0.248413	+	0.968654i	$0.579909\pi$
$314$	0	0
$315$	−1.44348e21	−0.839017
$316$	0	0
$317$	−2.88959e21	−1.59159	−0.795797	−	0.605564i	$-0.792948\pi$
$317$	−2.88959e21	−1.59159	−0.795797	+	0.605564i	$0.792948\pi$
$318$	0	0
$319$	2.76481e20	0.144359
$320$	0	0
$321$	−1.69098e21	−0.837232
$322$	0	0
$323$	−2.12427e19	−0.00997675
$324$	0	0
$325$	−4.78842e19	−0.0213395
$326$	0	0
$327$	−1.10213e21	−0.466203
$328$	0	0
$329$	−2.68935e21	−1.08014
$330$	0	0
$331$	−2.93645e21	−1.12017	−0.560085	−	0.828435i	$-0.689232\pi$
$331$	−2.93645e21	−1.12017	−0.560085	+	0.828435i	$0.689232\pi$
$332$	0	0
$333$	−6.75714e20	−0.244899
$334$	0	0
$335$	2.15676e21	0.742883
$336$	0	0
$337$	−3.01200e20	−0.0986282	−0.0493141	−	0.998783i	$-0.515704\pi$
$337$	−3.01200e20	−0.0986282	−0.0493141	+	0.998783i	$0.515704\pi$
$338$	0	0
$339$	−1.97981e21	−0.616491
$340$	0	0
$341$	−6.18152e19	−0.0183098
$342$	0	0
$343$	−1.32467e21	−0.373342
$344$	0	0
$345$	−3.32667e21	−0.892375
$346$	0	0
$347$	−1.24669e21	−0.318390	−0.159195	−	0.987247i	$-0.550890\pi$
$347$	−1.24669e21	−0.318390	−0.159195	+	0.987247i	$0.550890\pi$
$348$	0	0
$349$	−9.93535e18	−0.00241639	−0.00120820	−	0.999999i	$-0.500385\pi$
$349$	−9.93535e18	−0.00241639	−0.00120820	+	0.999999i	$0.500385\pi$
$350$	0	0
$351$	9.75507e18	0.00226006
$352$	0	0
$353$	6.30999e21	1.39298	0.696488	−	0.717569i	$-0.254745\pi$
$353$	6.30999e21	1.39298	0.696488	+	0.717569i	$0.254745\pi$
$354$	0	0
$355$	1.07232e22	2.25624
$356$	0	0
$357$	−1.41646e21	−0.284134
$358$	0	0
$359$	−1.07474e21	−0.205589	−0.102795	−	0.994703i	$-0.532778\pi$
$359$	−1.07474e21	−0.205589	−0.102795	+	0.994703i	$0.532778\pi$
$360$	0	0
$361$	−5.47532e21	−0.999076
$362$	0	0
$363$	1.40183e21	0.244055
$364$	0	0
$365$	1.97300e22	3.27822
$366$	0	0
$367$	−4.61598e21	−0.732153	−0.366077	−	0.930585i	$-0.619299\pi$
$367$	−4.61598e21	−0.732153	−0.366077	+	0.930585i	$0.619299\pi$
$368$	0	0
$369$	−3.44701e21	−0.522058
$370$	0	0
$371$	8.92147e21	1.29050
$372$	0	0
$373$	6.59694e21	0.911628	0.455814	−	0.890075i	$-0.349348\pi$
$373$	6.59694e21	0.911628	0.455814	+	0.890075i	$0.349348\pi$
$374$	0	0
$375$	−5.99638e21	−0.791814
$376$	0	0
$377$	−1.76788e19	−0.00223127
$378$	0	0
$379$	−1.41137e22	−1.70297	−0.851485	−	0.524379i	$-0.824298\pi$
$379$	−1.41137e22	−1.70297	−0.851485	+	0.524379i	$0.824298\pi$
$380$	0	0
$381$	1.21340e21	0.140005
$382$	0	0
$383$	1.26426e22	1.39523	0.697617	−	0.716471i	$-0.254244\pi$
$383$	1.26426e22	1.39523	0.697617	+	0.716471i	$0.254244\pi$
$384$	0	0
$385$	−1.81135e22	−1.91244
$386$	0	0
$387$	1.57868e21	0.159496
$388$	0	0
$389$	5.65745e21	0.547078	0.273539	−	0.961861i	$-0.411806\pi$
$389$	5.65745e21	0.547078	0.273539	+	0.961861i	$0.411806\pi$
$390$	0	0
$391$	−3.26440e21	−0.302204
$392$	0	0
$393$	6.41982e20	0.0569097
$394$	0	0
$395$	1.22822e22	1.04280
$396$	0	0
$397$	5.61679e21	0.456845	0.228422	−	0.973562i	$-0.426643\pi$
$397$	5.61679e21	0.456845	0.228422	+	0.973562i	$0.426643\pi$
$398$	0	0
$399$	3.37749e20	0.0263223
$400$	0	0
$401$	−1.90221e22	−1.42079	−0.710396	−	0.703802i	$-0.751484\pi$
$401$	−1.90221e22	−1.42079	−0.710396	+	0.703802i	$0.751484\pi$
$402$	0	0
$403$	3.95260e18	0.000283003 0
$404$	0	0
$405$	2.71661e21	0.186492
$406$	0	0
$407$	−8.47922e21	−0.558217
$408$	0	0
$409$	1.22737e22	0.775045	0.387522	−	0.921860i	$-0.373331\pi$
$409$	1.22737e22	0.775045	0.387522	+	0.921860i	$0.373331\pi$
$410$	0	0
$411$	2.44181e21	0.147930
$412$	0	0
$413$	4.17574e22	2.42749
$414$	0	0
$415$	3.75276e22	2.09383
$416$	0	0
$417$	2.70528e21	0.144896
$418$	0	0
$419$	2.12528e22	1.09294	0.546470	−	0.837479i	$-0.315971\pi$
$419$	2.12528e22	1.09294	0.546470	+	0.837479i	$0.315971\pi$
$420$	0	0
$421$	−6.94281e21	−0.342876	−0.171438	−	0.985195i	$-0.554841\pi$
$421$	−6.94281e21	−0.342876	−0.171438	+	0.985195i	$0.554841\pi$
$422$	0	0
$423$	5.06131e21	0.240088
$424$	0	0
$425$	−1.30852e22	−0.596316
$426$	0	0
$427$	−3.22630e22	−1.41276
$428$	0	0
$429$	1.22412e20	0.00515153
$430$	0	0
$431$	2.09760e22	0.848526	0.424263	−	0.905539i	$-0.360533\pi$
$431$	2.09760e22	0.848526	0.424263	+	0.905539i	$0.360533\pi$
$432$	0	0
$433$	−7.77111e21	−0.302229	−0.151114	−	0.988516i	$-0.548286\pi$
$433$	−7.77111e21	−0.302229	−0.151114	+	0.988516i	$0.548286\pi$
$434$	0	0
$435$	−4.92323e21	−0.184116
$436$	0	0
$437$	7.78382e20	0.0279963
$438$	0	0
$439$	3.62044e21	0.125260	0.0626301	−	0.998037i	$-0.480051\pi$
$439$	3.62044e21	0.125260	0.0626301	+	0.998037i	$0.480051\pi$
$440$	0	0
$441$	1.25070e22	0.416318
$442$	0	0
$443$	−4.20160e22	−1.34581	−0.672905	−	0.739729i	$-0.734954\pi$
$443$	−4.20160e22	−1.34581	−0.672905	+	0.739729i	$0.734954\pi$
$444$	0	0
$445$	6.56113e22	2.02264
$446$	0	0
$447$	−3.15201e22	−0.935348
$448$	0	0
$449$	−3.62047e22	−1.03436	−0.517180	−	0.855877i	$-0.673018\pi$
$449$	−3.62047e22	−1.03436	−0.517180	+	0.855877i	$0.673018\pi$
$450$	0	0
$451$	−4.32549e22	−1.18997
$452$	0	0
$453$	−3.22798e22	−0.855257
$454$	0	0
$455$	1.15822e21	0.0295593
$456$	0	0
$457$	−3.85735e22	−0.948422	−0.474211	−	0.880411i	$-0.657267\pi$
$457$	−3.85735e22	−0.948422	−0.474211	+	0.880411i	$0.657267\pi$
$458$	0	0
$459$	2.66576e21	0.0631557
$460$	0	0
$461$	−3.51617e22	−0.802809	−0.401405	−	0.915901i	$-0.631478\pi$
$461$	−3.51617e22	−0.802809	−0.401405	+	0.915901i	$0.631478\pi$
$462$	0	0
$463$	5.56328e22	1.22431	0.612157	−	0.790737i	$-0.290302\pi$
$463$	5.56328e22	1.22431	0.612157	+	0.790737i	$0.290302\pi$
$464$	0	0
$465$	1.10073e21	0.0233523
$466$	0	0
$467$	2.54999e22	0.521610	0.260805	−	0.965392i	$-0.416012\pi$
$467$	2.54999e22	0.521610	0.260805	+	0.965392i	$0.416012\pi$
$468$	0	0
$469$	−3.36494e22	−0.663756
$470$	0	0
$471$	−4.21374e22	−0.801661
$472$	0	0
$473$	1.98101e22	0.363553
$474$	0	0
$475$	3.12012e21	0.0552429
$476$	0	0
$477$	−1.67901e22	−0.286845
$478$	0	0
$479$	−7.71958e22	−1.27275	−0.636373	−	0.771381i	$-0.719566\pi$
$479$	−7.71958e22	−1.27275	−0.636373	+	0.771381i	$0.719566\pi$
$480$	0	0
$481$	5.42180e20	0.00862800
$482$	0	0
$483$	5.19023e22	0.797325
$484$	0	0
$485$	1.08730e23	1.61267
$486$	0	0
$487$	3.67556e22	0.526413	0.263207	−	0.964739i	$-0.415220\pi$
$487$	3.67556e22	0.526413	0.263207	+	0.964739i	$0.415220\pi$
$488$	0	0
$489$	−5.14025e22	−0.710982
$490$	0	0
$491$	−8.85815e21	−0.118345	−0.0591726	−	0.998248i	$-0.518846\pi$
$491$	−8.85815e21	−0.118345	−0.0591726	+	0.998248i	$0.518846\pi$
$492$	0	0
$493$	−4.83107e21	−0.0623511
$494$	0	0
$495$	3.40894e22	0.425085
$496$	0	0
$497$	−1.67303e23	−2.01592
$498$	0	0
$499$	3.19438e22	0.371991	0.185995	−	0.982551i	$-0.440449\pi$
$499$	3.19438e22	0.371991	0.185995	+	0.982551i	$0.440449\pi$
$500$	0	0
$501$	6.79084e22	0.764368
$502$	0	0
$503$	−1.46736e23	−1.59664	−0.798322	−	0.602230i	$-0.794279\pi$
$503$	−1.46736e23	−1.59664	−0.798322	+	0.602230i	$0.794279\pi$
$504$	0	0
$505$	−1.15481e23	−1.21488
$506$	0	0
$507$	5.67476e22	0.577271
$508$	0	0
$509$	−1.22599e22	−0.120611	−0.0603054	−	0.998180i	$-0.519207\pi$
$509$	−1.22599e22	−0.120611	−0.0603054	+	0.998180i	$0.519207\pi$
$510$	0	0
$511$	−3.07825e23	−2.92905
$512$	0	0
$513$	−6.35639e20	−0.00585077
$514$	0	0
$515$	−9.78830e22	−0.871657
$516$	0	0
$517$	6.35120e22	0.547250
$518$	0	0
$519$	5.37323e22	0.448036
$520$	0	0
$521$	−1.70219e23	−1.37368	−0.686842	−	0.726807i	$-0.741004\pi$
$521$	−1.70219e23	−1.37368	−0.686842	+	0.726807i	$0.741004\pi$
$522$	0	0
$523$	−7.34078e21	−0.0573427	−0.0286713	−	0.999589i	$-0.509128\pi$
$523$	−7.34078e21	−0.0573427	−0.0286713	+	0.999589i	$0.509128\pi$
$524$	0	0
$525$	2.08049e23	1.57330
$526$	0	0
$527$	1.08012e21	0.00790830
$528$	0	0
$529$	−2.14351e22	−0.151968
$530$	0	0
$531$	−7.85868e22	−0.539568
$532$	0	0
$533$	2.76582e21	0.0183926
$534$	0	0
$535$	3.77847e23	2.43393
$536$	0	0
$537$	−1.91545e22	−0.119533
$538$	0	0
$539$	1.56944e23	0.948945
$540$	0	0
$541$	2.69319e23	1.57794	0.788970	−	0.614431i	$-0.210614\pi$
$541$	2.69319e23	1.57794	0.788970	+	0.614431i	$0.210614\pi$
$542$	0	0
$543$	1.64278e23	0.932781
$544$	0	0
$545$	2.46268e23	1.35531
$546$	0	0
$547$	5.71353e22	0.304798	0.152399	−	0.988319i	$-0.451300\pi$
$547$	5.71353e22	0.304798	0.152399	+	0.988319i	$0.451300\pi$
$548$	0	0
$549$	6.07186e22	0.314020
$550$	0	0
$551$	1.15195e21	0.00577623
$552$	0	0
$553$	−1.91625e23	−0.931728
$554$	0	0
$555$	1.50987e23	0.711951
$556$	0	0
$557$	8.13046e22	0.371831	0.185916	−	0.982566i	$-0.440475\pi$
$557$	8.13046e22	0.371831	0.185916	+	0.982566i	$0.440475\pi$
$558$	0	0
$559$	−1.26670e21	−0.00561920
$560$	0	0
$561$	3.34513e22	0.143956
$562$	0	0
$563$	−2.45069e23	−1.02322	−0.511608	−	0.859219i	$-0.670950\pi$
$563$	−2.45069e23	−1.02322	−0.511608	+	0.859219i	$0.670950\pi$
$564$	0	0
$565$	4.42386e23	1.79221
$566$	0	0
$567$	−4.23842e22	−0.166628
$568$	0	0
$569$	−4.48105e23	−1.70972	−0.854862	−	0.518855i	$-0.826358\pi$
$569$	−4.48105e23	−1.70972	−0.854862	+	0.518855i	$0.826358\pi$
$570$	0	0
$571$	4.10855e23	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$571$	4.10855e23	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$572$	0	0
$573$	−1.32502e23	−0.476329
$574$	0	0
$575$	4.79473e23	1.67335
$576$	0	0
$577$	1.87387e23	0.634959	0.317480	−	0.948265i	$-0.397163\pi$
$577$	1.87387e23	0.634959	0.317480	+	0.948265i	$0.397163\pi$
$578$	0	0
$579$	−2.80885e23	−0.924190
$580$	0	0
$581$	−5.85500e23	−1.87081
$582$	0	0
$583$	−2.10691e23	−0.653827
$584$	0	0
$585$	−2.17975e21	−0.00657026
$586$	0	0
$587$	−3.17863e23	−0.930714	−0.465357	−	0.885123i	$-0.654074\pi$
$587$	−3.17863e23	−0.930714	−0.465357	+	0.885123i	$0.654074\pi$
$588$	0	0
$589$	−2.57551e20	−0.000732627 0
$590$	0	0
$591$	−2.63737e23	−0.728917
$592$	0	0
$593$	5.64469e23	1.51592	0.757958	−	0.652304i	$-0.226197\pi$
$593$	5.64469e23	1.51592	0.757958	+	0.652304i	$0.226197\pi$
$594$	0	0
$595$	3.16505e23	0.826013
$596$	0	0
$597$	2.81547e23	0.714117
$598$	0	0
$599$	−2.56471e22	−0.0632282	−0.0316141	−	0.999500i	$-0.510065\pi$
$599$	−2.56471e22	−0.0632282	−0.0316141	+	0.999500i	$0.510065\pi$
$600$	0	0
$601$	5.16554e23	1.23789	0.618946	−	0.785433i	$-0.287560\pi$
$601$	5.16554e23	1.23789	0.618946	+	0.785433i	$0.287560\pi$
$602$	0	0
$603$	6.33278e22	0.147536
$604$	0	0
$605$	−3.13236e23	−0.709497
$606$	0	0
$607$	−6.45213e23	−1.42102	−0.710508	−	0.703689i	$-0.751535\pi$
$607$	−6.45213e23	−1.42102	−0.710508	+	0.703689i	$0.751535\pi$
$608$	0	0
$609$	7.68115e22	0.164505
$610$	0	0
$611$	−4.06110e21	−0.00845850
$612$	0	0
$613$	−5.11815e23	−1.03681	−0.518405	−	0.855135i	$-0.673474\pi$
$613$	−5.11815e23	−1.03681	−0.518405	+	0.855135i	$0.673474\pi$
$614$	0	0
$615$	7.70229e23	1.51768
$616$	0	0
$617$	−3.00782e22	−0.0576537	−0.0288269	−	0.999584i	$-0.509177\pi$
$617$	−3.00782e22	−0.0576537	−0.0288269	+	0.999584i	$0.509177\pi$
$618$	0	0
$619$	6.97971e23	1.30157	0.650784	−	0.759263i	$-0.274440\pi$
$619$	6.97971e23	1.30157	0.650784	+	0.759263i	$0.274440\pi$
$620$	0	0
$621$	−9.76794e22	−0.177225
$622$	0	0
$623$	−1.02366e24	−1.80720
$624$	0	0
$625$	2.82181e23	0.484784
$626$	0	0
$627$	−7.97633e21	−0.0133361
$628$	0	0
$629$	1.48161e23	0.241103
$630$	0	0
$631$	5.56309e23	0.881183	0.440592	−	0.897708i	$-0.354769\pi$
$631$	5.56309e23	0.881183	0.440592	+	0.897708i	$0.354769\pi$
$632$	0	0
$633$	8.83316e22	0.136202
$634$	0	0
$635$	−2.71133e23	−0.407011
$636$	0	0
$637$	−1.00354e22	−0.0146672
$638$	0	0
$639$	3.14861e23	0.448086
$640$	0	0
$641$	−3.15692e23	−0.437492	−0.218746	−	0.975782i	$-0.570197\pi$
$641$	−3.15692e23	−0.437492	−0.218746	+	0.975782i	$0.570197\pi$
$642$	0	0
$643$	−1.03616e24	−1.39841	−0.699203	−	0.714924i	$-0.746461\pi$
$643$	−1.03616e24	−1.39841	−0.699203	+	0.714924i	$0.746461\pi$
$644$	0	0
$645$	−3.52754e23	−0.463675
$646$	0	0
$647$	5.82417e23	0.745671	0.372836	−	0.927897i	$-0.378386\pi$
$647$	5.82417e23	0.745671	0.372836	+	0.927897i	$0.378386\pi$
$648$	0	0
$649$	−9.86149e23	−1.22988
$650$	0	0
$651$	−1.71734e22	−0.0208650
$652$	0	0
$653$	1.52832e23	0.180905	0.0904526	−	0.995901i	$-0.471169\pi$
$653$	1.52832e23	0.180905	0.0904526	+	0.995901i	$0.471169\pi$
$654$	0	0
$655$	−1.43450e23	−0.165443
$656$	0	0
$657$	5.79323e23	0.651051
$658$	0	0
$659$	−6.03187e23	−0.660581	−0.330290	−	0.943879i	$-0.607147\pi$
$659$	−6.03187e23	−0.660581	−0.330290	+	0.943879i	$0.607147\pi$
$660$	0	0
$661$	−1.09124e24	−1.16468	−0.582340	−	0.812945i	$-0.697863\pi$
$661$	−1.09124e24	−1.16468	−0.582340	+	0.812945i	$0.697863\pi$
$662$	0	0
$663$	−2.13895e21	−0.00222503
$664$	0	0
$665$	−7.54694e22	−0.0765221
$666$	0	0
$667$	1.77021e23	0.174967
$668$	0	0
$669$	−3.63458e23	−0.350213
$670$	0	0
$671$	7.61928e23	0.715769
$672$	0	0
$673$	−3.64938e23	−0.334265	−0.167133	−	0.985934i	$-0.553451\pi$
$673$	−3.64938e23	−0.334265	−0.167133	+	0.985934i	$0.553451\pi$
$674$	0	0
$675$	−3.91545e23	−0.349703
$676$	0	0
$677$	−3.91131e23	−0.340658	−0.170329	−	0.985387i	$-0.554483\pi$
$677$	−3.91131e23	−0.340658	−0.170329	+	0.985387i	$0.554483\pi$
$678$	0	0
$679$	−1.69640e24	−1.44090
$680$	0	0
$681$	6.75267e22	0.0559403
$682$	0	0
$683$	−8.44250e23	−0.682174	−0.341087	−	0.940032i	$-0.610795\pi$
$683$	−8.44250e23	−0.682174	−0.341087	+	0.940032i	$0.610795\pi$
$684$	0	0
$685$	−5.45617e23	−0.430049
$686$	0	0
$687$	−1.01708e24	−0.782025
$688$	0	0
$689$	1.34720e22	0.0101058
$690$	0	0
$691$	−1.21416e24	−0.888610	−0.444305	−	0.895876i	$-0.646549\pi$
$691$	−1.21416e24	−0.888610	−0.444305	+	0.895876i	$0.646549\pi$
$692$	0	0
$693$	−5.31859e23	−0.379808
$694$	0	0
$695$	−6.04490e23	−0.421229
$696$	0	0
$697$	7.55811e23	0.513966
$698$	0	0
$699$	1.07982e23	0.0716628
$700$	0	0
$701$	2.69078e24	1.74291	0.871456	−	0.490473i	$-0.163176\pi$
$701$	2.69078e24	1.74291	0.871456	+	0.490473i	$0.163176\pi$
$702$	0	0
$703$	−3.53283e22	−0.0223359
$704$	0	0
$705$	−1.13094e24	−0.697964
$706$	0	0
$707$	1.80172e24	1.08548
$708$	0	0
$709$	−1.01451e24	−0.596712	−0.298356	−	0.954455i	$-0.596438\pi$
$709$	−1.01451e24	−0.596712	−0.298356	+	0.954455i	$0.596438\pi$
$710$	0	0
$711$	3.60636e23	0.207099
$712$	0	0
$713$	−3.95782e22	−0.0221919
$714$	0	0
$715$	−2.73527e22	−0.0149761
$716$	0	0
$717$	−4.82970e23	−0.258230
$718$	0	0
$719$	3.62460e24	1.89263	0.946313	−	0.323251i	$-0.104776\pi$
$719$	3.62460e24	1.89263	0.946313	+	0.323251i	$0.104776\pi$
$720$	0	0
$721$	1.52716e24	0.778814
$722$	0	0
$723$	−6.58392e23	−0.327951
$724$	0	0
$725$	7.09585e23	0.345248
$726$	0	0
$727$	2.60528e24	1.23826	0.619129	−	0.785289i	$-0.287486\pi$
$727$	2.60528e24	1.23826	0.619129	+	0.785289i	$0.287486\pi$
$728$	0	0
$729$	7.97664e22	0.0370370
$730$	0	0
$731$	−3.46150e23	−0.157024
$732$	0	0
$733$	−2.82031e23	−0.125001	−0.0625004	−	0.998045i	$-0.519907\pi$
$733$	−2.82031e23	−0.125001	−0.0625004	+	0.998045i	$0.519907\pi$
$734$	0	0
$735$	−2.79467e24	−1.21028
$736$	0	0
$737$	7.94670e23	0.336289
$738$	0	0
$739$	2.13421e24	0.882592	0.441296	−	0.897362i	$-0.354519\pi$
$739$	2.13421e24	0.882592	0.441296	+	0.897362i	$0.354519\pi$
$740$	0	0
$741$	5.10024e20	0.000206127 0
$742$	0	0
$743$	−1.65375e24	−0.653229	−0.326614	−	0.945158i	$-0.605908\pi$
$743$	−1.65375e24	−0.653229	−0.326614	+	0.945158i	$0.605908\pi$
$744$	0	0
$745$	7.04311e24	2.71917
$746$	0	0
$747$	1.10190e24	0.415832
$748$	0	0
$749$	−5.89511e24	−2.17469
$750$	0	0
$751$	1.59968e24	0.576890	0.288445	−	0.957496i	$-0.406862\pi$
$751$	1.59968e24	0.576890	0.288445	+	0.957496i	$0.406862\pi$
$752$	0	0
$753$	2.19337e24	0.773312
$754$	0	0
$755$	7.21288e24	2.48633
$756$	0	0
$757$	−3.10841e24	−1.04767	−0.523834	−	0.851820i	$-0.675499\pi$
$757$	−3.10841e24	−1.04767	−0.523834	+	0.851820i	$0.675499\pi$
$758$	0	0
$759$	−1.22573e24	−0.403962
$760$	0	0
$761$	3.83430e24	1.23571	0.617856	−	0.786292i	$-0.288002\pi$
$761$	3.83430e24	1.23571	0.617856	+	0.786292i	$0.288002\pi$
$762$	0	0
$763$	−3.84224e24	−1.21095
$764$	0	0
$765$	−5.95659e23	−0.183601
$766$	0	0
$767$	6.30565e22	0.0190094
$768$	0	0
$769$	−3.27383e24	−0.965343	−0.482672	−	0.875801i	$-0.660334\pi$
$769$	−3.27383e24	−0.965343	−0.482672	+	0.875801i	$0.660334\pi$
$770$	0	0
$771$	4.17845e23	0.120518
$772$	0	0
$773$	−7.30757e23	−0.206180	−0.103090	−	0.994672i	$-0.532873\pi$
$773$	−7.30757e23	−0.206180	−0.103090	+	0.994672i	$0.532873\pi$
$774$	0	0
$775$	−1.58648e23	−0.0437895
$776$	0	0
$777$	−2.35568e24	−0.636118
$778$	0	0
$779$	−1.80220e23	−0.0476140
$780$	0	0
$781$	3.95104e24	1.02136
$782$	0	0
$783$	−1.44558e23	−0.0365652
$784$	0	0
$785$	9.41554e24	2.33052
$786$	0	0
$787$	−6.01570e23	−0.145714	−0.0728570	−	0.997342i	$-0.523212\pi$
$787$	−6.01570e23	−0.145714	−0.0728570	+	0.997342i	$0.523212\pi$
$788$	0	0
$789$	−2.41303e24	−0.572016
$790$	0	0
$791$	−6.90204e24	−1.60132
$792$	0	0
$793$	−4.87194e22	−0.0110632
$794$	0	0
$795$	3.75172e24	0.833891
$796$	0	0
$797$	5.16820e24	1.12446	0.562230	−	0.826981i	$-0.309944\pi$
$797$	5.16820e24	1.12446	0.562230	+	0.826981i	$0.309944\pi$
$798$	0	0
$799$	−1.10977e24	−0.236366
$800$	0	0
$801$	1.92651e24	0.401694
$802$	0	0
$803$	7.26965e24	1.48399
$804$	0	0
$805$	−1.15975e25	−2.31792
$806$	0	0
$807$	−9.45821e23	−0.185090
$808$	0	0
$809$	8.29099e24	1.58871	0.794354	−	0.607455i	$-0.207810\pi$
$809$	8.29099e24	1.58871	0.794354	+	0.607455i	$0.207810\pi$
$810$	0	0
$811$	6.74898e23	0.126637	0.0633185	−	0.997993i	$-0.479832\pi$
$811$	6.74898e23	0.126637	0.0633185	+	0.997993i	$0.479832\pi$
$812$	0	0
$813$	3.95924e24	0.727515
$814$	0	0
$815$	1.14858e25	2.06691
$816$	0	0
$817$	8.25381e22	0.0145468
$818$	0	0
$819$	3.40082e22	0.00587044
$820$	0	0
$821$	−3.52716e24	−0.596361	−0.298180	−	0.954510i	$-0.596380\pi$
$821$	−3.52716e24	−0.596361	−0.298180	+	0.954510i	$0.596380\pi$
$822$	0	0
$823$	−2.10509e24	−0.348636	−0.174318	−	0.984689i	$-0.555772\pi$
$823$	−2.10509e24	−0.348636	−0.174318	+	0.984689i	$0.555772\pi$
$824$	0	0
$825$	−4.91331e24	−0.797106
$826$	0	0
$827$	−7.78739e24	−1.23764	−0.618821	−	0.785532i	$-0.712390\pi$
$827$	−7.78739e24	−1.23764	−0.618821	+	0.785532i	$0.712390\pi$
$828$	0	0
$829$	−3.67550e24	−0.572273	−0.286137	−	0.958189i	$-0.592371\pi$
$829$	−3.67550e24	−0.572273	−0.286137	+	0.958189i	$0.592371\pi$
$830$	0	0
$831$	−2.03745e24	−0.310798
$832$	0	0
$833$	−2.74235e24	−0.409865
$834$	0	0
$835$	−1.51740e25	−2.22211
$836$	0	0
$837$	3.23201e22	0.00463774
$838$	0	0
$839$	7.12132e23	0.100135	0.0500673	−	0.998746i	$-0.484056\pi$
$839$	7.12132e23	0.100135	0.0500673	+	0.998746i	$0.484056\pi$
$840$	0	0
$841$	−6.99517e24	−0.963901
$842$	0	0
$843$	−4.22351e24	−0.570348
$844$	0	0
$845$	−1.26801e25	−1.67819
$846$	0	0
$847$	4.88706e24	0.633926
$848$	0	0
$849$	−2.25606e24	−0.286837
$850$	0	0
$851$	−5.42895e24	−0.676573
$852$	0	0
$853$	9.94213e24	1.21454	0.607271	−	0.794495i	$-0.292264\pi$
$853$	9.94213e24	1.21454	0.607271	+	0.794495i	$0.292264\pi$
$854$	0	0
$855$	1.42032e23	0.0170089
$856$	0	0
$857$	−7.12623e23	−0.0836610	−0.0418305	−	0.999125i	$-0.513319\pi$
$857$	−7.12623e23	−0.0836610	−0.0418305	+	0.999125i	$0.513319\pi$
$858$	0	0
$859$	6.42042e24	0.738961	0.369480	−	0.929239i	$-0.379536\pi$
$859$	6.42042e24	0.738961	0.369480	+	0.929239i	$0.379536\pi$
$860$	0	0
$861$	−1.20170e25	−1.35603
$862$	0	0
$863$	3.45207e24	0.381934	0.190967	−	0.981596i	$-0.438838\pi$
$863$	3.45207e24	0.381934	0.190967	+	0.981596i	$0.438838\pi$
$864$	0	0
$865$	−1.20064e25	−1.30249
$866$	0	0
$867$	4.84302e24	0.515173
$868$	0	0
$869$	4.52545e24	0.472056
$870$	0	0
$871$	−5.08130e22	−0.00519781
$872$	0	0
$873$	3.19260e24	0.320275
$874$	0	0
$875$	−2.09046e25	−2.05671
$876$	0	0
$877$	1.98750e25	1.91783	0.958915	−	0.283692i	$-0.0915595\pi$
$877$	1.98750e25	1.91783	0.958915	+	0.283692i	$0.0915595\pi$
$878$	0	0
$879$	−2.89754e24	−0.274236
$880$	0	0
$881$	−1.53861e24	−0.142835	−0.0714175	−	0.997447i	$-0.522752\pi$
$881$	−1.53861e24	−0.142835	−0.0714175	+	0.997447i	$0.522752\pi$
$882$	0	0
$883$	8.67401e24	0.789867	0.394934	−	0.918710i	$-0.370768\pi$
$883$	8.67401e24	0.789867	0.394934	+	0.918710i	$0.370768\pi$
$884$	0	0
$885$	1.75601e25	1.56859
$886$	0	0
$887$	−7.74587e24	−0.678765	−0.339383	−	0.940648i	$-0.610218\pi$
$887$	−7.74587e24	−0.678765	−0.339383	+	0.940648i	$0.610218\pi$
$888$	0	0
$889$	4.23018e24	0.363659
$890$	0	0
$891$	1.00095e24	0.0844214
$892$	0	0
$893$	2.64620e23	0.0218971
$894$	0	0
$895$	4.28003e24	0.347497
$896$	0	0
$897$	7.83760e22	0.00624378
$898$	0	0
$899$	−5.85728e22	−0.00457866
$900$	0	0
$901$	3.68149e24	0.282399
$902$	0	0
$903$	5.50361e24	0.414288
$904$	0	0
$905$	−3.67076e25	−2.71171
$906$	0	0
$907$	9.51449e24	0.689801	0.344900	−	0.938639i	$-0.387913\pi$
$907$	9.51449e24	0.689801	0.344900	+	0.938639i	$0.387913\pi$
$908$	0	0
$909$	−3.39082e24	−0.241274
$910$	0	0
$911$	5.87611e24	0.410378	0.205189	−	0.978722i	$-0.434219\pi$
$911$	5.87611e24	0.410378	0.205189	+	0.978722i	$0.434219\pi$
$912$	0	0
$913$	1.38272e25	0.947838
$914$	0	0
$915$	−1.35675e25	−0.912892
$916$	0	0
$917$	2.23808e24	0.147821
$918$	0	0
$919$	−1.95287e25	−1.26617	−0.633084	−	0.774083i	$-0.718211\pi$
$919$	−1.95287e25	−1.26617	−0.633084	+	0.774083i	$0.718211\pi$
$920$	0	0
$921$	−1.67686e24	−0.106731
$922$	0	0
$923$	−2.52638e23	−0.0157865
$924$	0	0
$925$	−2.17618e25	−1.33503
$926$	0	0
$927$	−2.87409e24	−0.173110
$928$	0	0
$929$	−4.51245e24	−0.266857	−0.133428	−	0.991058i	$-0.542599\pi$
$929$	−4.51245e24	−0.266857	−0.133428	+	0.991058i	$0.542599\pi$
$930$	0	0
$931$	6.53902e23	0.0379700
$932$	0	0
$933$	8.72840e24	0.497669
$934$	0	0
$935$	−7.47463e24	−0.418496
$936$	0	0
$937$	9.95126e24	0.547131	0.273566	−	0.961853i	$-0.411797\pi$
$937$	9.95126e24	0.547131	0.273566	+	0.961853i	$0.411797\pi$
$938$	0	0
$939$	5.31252e24	0.286842
$940$	0	0
$941$	−1.60734e25	−0.852309	−0.426155	−	0.904650i	$-0.640132\pi$
$941$	−1.60734e25	−0.852309	−0.426155	+	0.904650i	$0.640132\pi$
$942$	0	0
$943$	−2.76946e25	−1.44227
$944$	0	0
$945$	9.47067e24	0.484407
$946$	0	0
$947$	1.75270e25	0.880505	0.440253	−	0.897874i	$-0.354889\pi$
$947$	1.75270e25	0.880505	0.440253	+	0.897874i	$0.354889\pi$
$948$	0	0
$949$	−4.64838e23	−0.0229371
$950$	0	0
$951$	1.89586e25	0.918907
$952$	0	0
$953$	−2.36935e25	−1.12808	−0.564040	−	0.825747i	$-0.690754\pi$
$953$	−2.36935e25	−1.12808	−0.564040	+	0.825747i	$0.690754\pi$
$954$	0	0
$955$	2.96072e25	1.38474
$956$	0	0
$957$	−1.81399e24	−0.0833459
$958$	0	0
$959$	8.51265e24	0.384243
$960$	0	0
$961$	−2.25370e25	−0.999419
$962$	0	0
$963$	1.10945e25	0.483376
$964$	0	0
$965$	6.27633e25	2.68673
$966$	0	0
$967$	2.02177e25	0.850369	0.425185	−	0.905107i	$-0.360209\pi$
$967$	2.02177e25	0.850369	0.425185	+	0.905107i	$0.360209\pi$
$968$	0	0
$969$	1.39374e23	0.00576008
$970$	0	0
$971$	−2.75403e25	−1.11842	−0.559210	−	0.829026i	$-0.688895\pi$
$971$	−2.75403e25	−1.11842	−0.559210	+	0.829026i	$0.688895\pi$
$972$	0	0
$973$	9.43117e24	0.376362
$974$	0	0
$975$	3.14168e23	0.0123203
$976$	0	0
$977$	3.78814e25	1.45990	0.729948	−	0.683502i	$-0.239544\pi$
$977$	3.78814e25	1.45990	0.729948	+	0.683502i	$0.239544\pi$
$978$	0	0
$979$	2.41749e25	0.915612
$980$	0	0
$981$	7.23105e24	0.269163
$982$	0	0
$983$	5.25889e25	1.92393	0.961965	−	0.273171i	$-0.0880725\pi$
$983$	5.25889e25	1.92393	0.961965	+	0.273171i	$0.0880725\pi$
$984$	0	0
$985$	5.89317e25	2.11905
$986$	0	0
$987$	1.76448e25	0.623621
$988$	0	0
$989$	1.26837e25	0.440634
$990$	0	0
$991$	−6.05240e24	−0.206682	−0.103341	−	0.994646i	$-0.532953\pi$
$991$	−6.05240e24	−0.206682	−0.103341	+	0.994646i	$0.532953\pi$
$992$	0	0
$993$	1.92660e25	0.646731
$994$	0	0
$995$	−6.29113e25	−2.07602
$996$	0	0
$997$	−4.03816e24	−0.131001	−0.0655005	−	0.997853i	$-0.520864\pi$
$997$	−4.03816e24	−0.131001	−0.0655005	+	0.997853i	$0.520864\pi$
$998$	0	0
$999$	4.43336e24	0.141393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.18.a.h.1.2		2
4.3	odd	2		3.18.a.b.1.1	✓	2
12.11	even	2		9.18.a.c.1.2		2
20.3	even	4		75.18.b.c.49.3		4
20.7	even	4		75.18.b.c.49.2		4
20.19	odd	2		75.18.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.18.a.b.1.1	✓	2	4.3	odd	2
9.18.a.c.1.2		2	12.11	even	2
48.18.a.h.1.2		2	1.1	even	1	trivial
75.18.a.b.1.2		2	20.19	odd	2
75.18.b.c.49.2		4	20.7	even	4
75.18.b.c.49.3		4	20.3	even	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q-6561.00 q^{3} +1.46604e6 q^{5} -2.28730e7 q^{7} +4.30467e7 q^{9} +O(q^{10})\)	\(q-6561.00 q^{3} +1.46604e6 q^{5} -2.28730e7 q^{7} +4.30467e7 q^{9} +5.40173e8 q^{11} -3.45398e7 q^{13} -9.61872e9 q^{15} -9.43866e9 q^{17} +2.25061e9 q^{19} +1.50070e11 q^{21} +3.45854e11 q^{23} +1.38635e12 q^{25} -2.82430e11 q^{27} +5.11838e11 q^{29} -1.14436e11 q^{31} -3.54407e12 q^{33} -3.35329e13 q^{35} -1.56972e13 q^{37} +2.26616e11 q^{39} -8.00761e13 q^{41} +3.66737e13 q^{43} +6.31084e13 q^{45} +1.17577e14 q^{47} +2.90545e14 q^{49} +6.19270e13 q^{51} -3.90043e14 q^{53} +7.91917e14 q^{55} -1.47663e13 q^{57} -1.82562e15 q^{59} +1.41053e15 q^{61} -9.84609e14 q^{63} -5.06369e13 q^{65} +1.47114e15 q^{67} -2.26915e15 q^{69} +7.31441e15 q^{71} +1.34580e16 q^{73} -9.09582e15 q^{75} -1.23554e16 q^{77} +8.37779e15 q^{79} +1.85302e15 q^{81} +2.55978e16 q^{83} -1.38375e16 q^{85} -3.35817e15 q^{87} +4.47540e16 q^{89} +7.90031e14 q^{91} +7.50815e14 q^{93} +3.29949e15 q^{95} +7.41658e16 q^{97} +2.32527e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13122 q^{3} + 382860 q^{5} - 24471568 q^{7} + 86093442 q^{9}+O(q^{10}) \) 2 * q - 13122 * q^3 + 382860 * q^5 - 24471568 * q^7 + 86093442 * q^9	\( 2 q - 13122 q^{3} + 382860 q^{5} - 24471568 q^{7} + 86093442 q^{9} + 987553512 q^{11} - 2519398244 q^{13} - 2511944460 q^{15} - 34313126364 q^{17} - 80053542184 q^{19} + 160557957648 q^{21} - 297228742704 q^{23} + 1796695260350 q^{25} - 564859072962 q^{27} - 470374069572 q^{29} - 3400754454592 q^{31} - 6479338592232 q^{33} - 31801338154080 q^{35} + 10652012180428 q^{37} + 16529771878884 q^{39} - 113376799448748 q^{41} - 61637031489880 q^{43} + 16480867602060 q^{45} + 279645641926560 q^{47} + 60469362475890 q^{49} + 225128422074204 q^{51} - 530964038611476 q^{53} + 307321280077680 q^{55} + 525231290269224 q^{57} - 17\!\cdots\!56 q^{59}+ \cdots + 42\!\cdots\!52 q^{99}+O(q^{100}) \) 2 * q - 13122 * q^3 + 382860 * q^5 - 24471568 * q^7 + 86093442 * q^9 + 987553512 * q^11 - 2519398244 * q^13 - 2511944460 * q^15 - 34313126364 * q^17 - 80053542184 * q^19 + 160557957648 * q^21 - 297228742704 * q^23 + 1796695260350 * q^25 - 564859072962 * q^27 - 470374069572 * q^29 - 3400754454592 * q^31 - 6479338592232 * q^33 - 31801338154080 * q^35 + 10652012180428 * q^37 + 16529771878884 * q^39 - 113376799448748 * q^41 - 61637031489880 * q^43 + 16480867602060 * q^45 + 279645641926560 * q^47 + 60469362475890 * q^49 + 225128422074204 * q^51 - 530964038611476 * q^53 + 307321280077680 * q^55 + 525231290269224 * q^57 - 1727524231086456 * q^59 + 2784287656027900 * q^61 - 1053420760128528 * q^63 + 2640922679811240 * q^65 + 3329301676696184 * q^67 + 1950117780880944 * q^69 + 13489402206504816 * q^71 + 436589918136724 * q^73 - 11788117603156350 * q^75 - 13070541011736384 * q^77 - 4376041565214880 * q^79 + 3706040377703682 * q^81 + 39886442265612888 * q^83 + 13106143850552280 * q^85 + 3086124270461892 * q^87 + 6972184096107444 * q^89 + 4762196572326688 * q^91 + 22312349976578112 * q^93 + 92450059415485200 * q^95 + 183569555712460996 * q^97 + 42510940503634152 * q^99

Introduction

L-functions

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