LMFDB - Embedded newform 240.8.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.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:	$74.9724061162$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10761}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2690 $ x^2 - x - 2690
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-51.3676$ of defining polynomial
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} -125.000 q^{5} +1692.82 q^{7} +729.000 q^{9} +7555.29 q^{11} +12911.8 q^{13} +3375.00 q^{15} +17583.1 q^{17} -31062.5 q^{19} -45706.2 q^{21} -82318.0 q^{23} +15625.0 q^{25} -19683.0 q^{27} +157988. q^{29} +128193. q^{31} -203993. q^{33} -211603. q^{35} -137200. q^{37} -348618. q^{39} -177128. q^{41} -7915.59 q^{43} -91125.0 q^{45} -291482. q^{47} +2.04211e6 q^{49} -474742. q^{51} -776917. q^{53} -944411. q^{55} +838687. q^{57} +919013. q^{59} +2.08842e6 q^{61} +1.23407e6 q^{63} -1.61397e6 q^{65} +1.52915e6 q^{67} +2.22259e6 q^{69} -2.74055e6 q^{71} +3.98355e6 q^{73} -421875. q^{75} +1.27898e7 q^{77} -3.98416e6 q^{79} +531441. q^{81} -5.01862e6 q^{83} -2.19788e6 q^{85} -4.26567e6 q^{87} -1.12205e7 q^{89} +2.18573e7 q^{91} -3.46121e6 q^{93} +3.88281e6 q^{95} -2.13699e6 q^{97} +5.50781e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 54 q^{3} - 250 q^{5} + 896 q^{7} + 1458 q^{9} + 5152 q^{11} + 8396 q^{13} + 6750 q^{15} + 7780 q^{17} - 54656 q^{19} - 24192 q^{21} - 122312 q^{23} + 31250 q^{25} - 39366 q^{27} + 116804 q^{29} - 109592 q^{31}+ \cdots + 3755808 q^{99}+O(q^{100}) $ 2 * q - 54 * q^3 - 250 * q^5 + 896 * q^7 + 1458 * q^9 + 5152 * q^11 + 8396 * q^13 + 6750 * q^15 + 7780 * q^17 - 54656 * q^19 - 24192 * q^21 - 122312 * q^23 + 31250 * q^25 - 39366 * q^27 + 116804 * q^29 - 109592 * q^31 - 139104 * q^33 - 112000 * q^35 + 46764 * q^37 - 226692 * q^39 - 299484 * q^41 - 234920 * q^43 - 182250 * q^45 - 505784 * q^47 + 1853490 * q^49 - 210060 * q^51 + 791412 * q^53 - 644000 * q^55 + 1475712 * q^57 - 1179424 * q^59 + 3733676 * q^61 + 653184 * q^63 - 1049500 * q^65 - 1462888 * q^67 + 3302424 * q^69 - 2573184 * q^71 + 6000276 * q^73 - 843750 * q^75 + 14704768 * q^77 - 3399816 * q^79 + 1062882 * q^81 + 1016 * q^83 - 972500 * q^85 - 3153708 * q^87 - 7438476 * q^89 + 25455584 * q^91 + 2958984 * q^93 + 6832000 * q^95 + 10345220 * q^97 + 3755808 * q^99

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	1692.82	1.86538	0.932692	−	0.360673i	$-0.117453\pi$
$7$	1692.82	1.86538	0.932692	+	0.360673i	$0.117453\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	7555.29	1.71150	0.855750	−	0.517390i	$-0.173096\pi$
$11$	7555.29	1.71150	0.855750	+	0.517390i	$0.173096\pi$
$12$	0	0
$13$	12911.8	1.62999	0.814993	−	0.579471i	$-0.196741\pi$
$13$	12911.8	1.62999	0.814993	+	0.579471i	$0.196741\pi$
$14$	0	0
$15$	3375.00	0.258199
$16$	0	0
$17$	17583.1	0.868007	0.434003	−	0.900911i	$-0.357101\pi$
$17$	17583.1	0.868007	0.434003	+	0.900911i	$0.357101\pi$
$18$	0	0
$19$	−31062.5	−1.03896	−0.519479	−	0.854483i	$-0.673874\pi$
$19$	−31062.5	−1.03896	−0.519479	+	0.854483i	$0.673874\pi$
$20$	0	0
$21$	−45706.2	−1.07698
$22$	0	0
$23$	−82318.0	−1.41074	−0.705371	−	0.708839i	$-0.749219\pi$
$23$	−82318.0	−1.41074	−0.705371	+	0.708839i	$0.749219\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	157988.	1.20290	0.601452	−	0.798909i	$-0.294589\pi$
$29$	157988.	1.20290	0.601452	+	0.798909i	$0.294589\pi$
$30$	0	0
$31$	128193.	0.772855	0.386428	−	0.922320i	$-0.373709\pi$
$31$	128193.	0.772855	0.386428	+	0.922320i	$0.373709\pi$
$32$	0	0
$33$	−203993.	−0.988135
$34$	0	0
$35$	−211603.	−0.834225
$36$	0	0
$37$	−137200.	−0.445296	−0.222648	−	0.974899i	$-0.571470\pi$
$37$	−137200.	−0.445296	−0.222648	+	0.974899i	$0.571470\pi$
$38$	0	0
$39$	−348618.	−0.941073
$40$	0	0
$41$	−177128.	−0.401369	−0.200685	−	0.979656i	$-0.564317\pi$
$41$	−177128.	−0.401369	−0.200685	+	0.979656i	$0.564317\pi$
$42$	0	0
$43$	−7915.59	−0.0151825	−0.00759125	−	0.999971i	$-0.502416\pi$
$43$	−7915.59	−0.0151825	−0.00759125	+	0.999971i	$0.502416\pi$
$44$	0	0
$45$	−91125.0	−0.149071
$46$	0	0
$47$	−291482.	−0.409514	−0.204757	−	0.978813i	$-0.565640\pi$
$47$	−291482.	−0.409514	−0.204757	+	0.978813i	$0.565640\pi$
$48$	0	0
$49$	2.04211e6	2.47966
$50$	0	0
$51$	−474742.	−0.501144
$52$	0	0
$53$	−776917.	−0.716818	−0.358409	−	0.933565i	$-0.616681\pi$
$53$	−776917.	−0.716818	−0.358409	+	0.933565i	$0.616681\pi$
$54$	0	0
$55$	−944411.	−0.765406
$56$	0	0
$57$	838687.	0.599843
$58$	0	0
$59$	919013.	0.582559	0.291279	−	0.956638i	$-0.405919\pi$
$59$	919013.	0.582559	0.291279	+	0.956638i	$0.405919\pi$
$60$	0	0
$61$	2.08842e6	1.17805	0.589024	−	0.808116i	$-0.299512\pi$
$61$	2.08842e6	1.17805	0.589024	+	0.808116i	$0.299512\pi$
$62$	0	0
$63$	1.23407e6	0.621795
$64$	0	0
$65$	−1.61397e6	−0.728952
$66$	0	0
$67$	1.52915e6	0.621140	0.310570	−	0.950551i	$-0.399480\pi$
$67$	1.52915e6	0.621140	0.310570	+	0.950551i	$0.399480\pi$
$68$	0	0
$69$	2.22259e6	0.814492
$70$	0	0
$71$	−2.74055e6	−0.908726	−0.454363	−	0.890817i	$-0.650133\pi$
$71$	−2.74055e6	−0.908726	−0.454363	+	0.890817i	$0.650133\pi$
$72$	0	0
$73$	3.98355e6	1.19851	0.599253	−	0.800560i	$-0.295464\pi$
$73$	3.98355e6	1.19851	0.599253	+	0.800560i	$0.295464\pi$
$74$	0	0
$75$	−421875.	−0.115470
$76$	0	0
$77$	1.27898e7	3.19261
$78$	0	0
$79$	−3.98416e6	−0.909163	−0.454581	−	0.890705i	$-0.650211\pi$
$79$	−3.98416e6	−0.909163	−0.454581	+	0.890705i	$0.650211\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−5.01862e6	−0.963409	−0.481705	−	0.876334i	$-0.659982\pi$
$83$	−5.01862e6	−0.963409	−0.481705	+	0.876334i	$0.659982\pi$
$84$	0	0
$85$	−2.19788e6	−0.388184
$86$	0	0
$87$	−4.26567e6	−0.694496
$88$	0	0
$89$	−1.12205e7	−1.68713	−0.843565	−	0.537026i	$-0.819548\pi$
$89$	−1.12205e7	−1.68713	−0.843565	+	0.537026i	$0.819548\pi$
$90$	0	0
$91$	2.18573e7	3.04055
$92$	0	0
$93$	−3.46121e6	−0.446208
$94$	0	0
$95$	3.88281e6	0.464637
$96$	0	0
$97$	−2.13699e6	−0.237740	−0.118870	−	0.992910i	$-0.537927\pi$
$97$	−2.13699e6	−0.237740	−0.118870	+	0.992910i	$0.537927\pi$
$98$	0	0
$99$	5.50781e6	0.570500
$100$	0	0
$101$	3.50951e6	0.338939	0.169470	−	0.985535i	$-0.445795\pi$
$101$	3.50951e6	0.338939	0.169470	+	0.985535i	$0.445795\pi$
$102$	0	0
$103$	−2.02866e7	−1.82927	−0.914636	−	0.404278i	$-0.867523\pi$
$103$	−2.02866e7	−1.82927	−0.914636	+	0.404278i	$0.867523\pi$
$104$	0	0
$105$	5.71328e6	0.481640
$106$	0	0
$107$	2.10480e7	1.66099	0.830495	−	0.557026i	$-0.188057\pi$
$107$	2.10480e7	1.66099	0.830495	+	0.557026i	$0.188057\pi$
$108$	0	0
$109$	9.59898e6	0.709958	0.354979	−	0.934874i	$-0.384488\pi$
$109$	9.59898e6	0.709958	0.354979	+	0.934874i	$0.384488\pi$
$110$	0	0
$111$	3.70440e6	0.257092
$112$	0	0
$113$	873393.	0.0569423	0.0284712	−	0.999595i	$-0.490936\pi$
$113$	873393.	0.0569423	0.0284712	+	0.999595i	$0.490936\pi$
$114$	0	0
$115$	1.02897e7	0.630903
$116$	0	0
$117$	9.41267e6	0.543328
$118$	0	0
$119$	2.97650e7	1.61917
$120$	0	0
$121$	3.75953e7	1.92923
$122$	0	0
$123$	4.78246e6	0.231731
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	2.23717e7	0.969138	0.484569	−	0.874753i	$-0.338977\pi$
$127$	2.23717e7	0.969138	0.484569	+	0.874753i	$0.338977\pi$
$128$	0	0
$129$	213721.	0.00876562
$130$	0	0
$131$	3.10149e7	1.20537	0.602686	−	0.797979i	$-0.294097\pi$
$131$	3.10149e7	1.20537	0.602686	+	0.797979i	$0.294097\pi$
$132$	0	0
$133$	−5.25833e7	−1.93806
$134$	0	0
$135$	2.46038e6	0.0860663
$136$	0	0
$137$	4.86660e6	0.161698	0.0808488	−	0.996726i	$-0.474237\pi$
$137$	4.86660e6	0.161698	0.0808488	+	0.996726i	$0.474237\pi$
$138$	0	0
$139$	−4.84162e7	−1.52911	−0.764555	−	0.644558i	$-0.777041\pi$
$139$	−4.84162e7	−1.52911	−0.764555	+	0.644558i	$0.777041\pi$
$140$	0	0
$141$	7.87000e6	0.236433
$142$	0	0
$143$	9.75521e7	2.78972
$144$	0	0
$145$	−1.97485e7	−0.537955
$146$	0	0
$147$	−5.51369e7	−1.43163
$148$	0	0
$149$	−1.58127e6	−0.0391610	−0.0195805	−	0.999808i	$-0.506233\pi$
$149$	−1.58127e6	−0.0391610	−0.0195805	+	0.999808i	$0.506233\pi$
$150$	0	0
$151$	−1.68785e7	−0.398946	−0.199473	−	0.979903i	$-0.563923\pi$
$151$	−1.68785e7	−0.398946	−0.199473	+	0.979903i	$0.563923\pi$
$152$	0	0
$153$	1.28180e7	0.289336
$154$	0	0
$155$	−1.60241e7	−0.345631
$156$	0	0
$157$	1.64799e7	0.339864	0.169932	−	0.985456i	$-0.445645\pi$
$157$	1.64799e7	0.339864	0.169932	+	0.985456i	$0.445645\pi$
$158$	0	0
$159$	2.09768e7	0.413855
$160$	0	0
$161$	−1.39350e8	−2.63157
$162$	0	0
$163$	5.60123e7	1.01304	0.506520	−	0.862228i	$-0.330931\pi$
$163$	5.60123e7	1.01304	0.506520	+	0.862228i	$0.330931\pi$
$164$	0	0
$165$	2.54991e7	0.441907
$166$	0	0
$167$	−1.14884e7	−0.190877	−0.0954385	−	0.995435i	$-0.530425\pi$
$167$	−1.14884e7	−0.190877	−0.0954385	+	0.995435i	$0.530425\pi$
$168$	0	0
$169$	1.03965e8	1.65685
$170$	0	0
$171$	−2.26445e7	−0.346320
$172$	0	0
$173$	−1.15198e7	−0.169155	−0.0845776	−	0.996417i	$-0.526954\pi$
$173$	−1.15198e7	−0.169155	−0.0845776	+	0.996417i	$0.526954\pi$
$174$	0	0
$175$	2.64504e7	0.373077
$176$	0	0
$177$	−2.48134e7	−0.336340
$178$	0	0
$179$	−8.58637e7	−1.11898	−0.559492	−	0.828835i	$-0.689004\pi$
$179$	−8.58637e7	−1.11898	−0.559492	+	0.828835i	$0.689004\pi$
$180$	0	0
$181$	8.81910e7	1.10548	0.552738	−	0.833355i	$-0.313583\pi$
$181$	8.81910e7	1.10548	0.552738	+	0.833355i	$0.313583\pi$
$182$	0	0
$183$	−5.63872e7	−0.680146
$184$	0	0
$185$	1.71500e7	0.199142
$186$	0	0
$187$	1.32845e8	1.48559
$188$	0	0
$189$	−3.33198e7	−0.358993
$190$	0	0
$191$	−1.48367e8	−1.54071	−0.770355	−	0.637615i	$-0.779921\pi$
$191$	−1.48367e8	−1.54071	−0.770355	+	0.637615i	$0.779921\pi$
$192$	0	0
$193$	1.15408e8	1.15554	0.577769	−	0.816200i	$-0.303923\pi$
$193$	1.15408e8	1.15554	0.577769	+	0.816200i	$0.303923\pi$
$194$	0	0
$195$	4.35772e7	0.420860
$196$	0	0
$197$	5.25418e7	0.489636	0.244818	−	0.969569i	$-0.421272\pi$
$197$	5.25418e7	0.489636	0.244818	+	0.969569i	$0.421272\pi$
$198$	0	0
$199$	1.89448e8	1.70414	0.852068	−	0.523431i	$-0.175348\pi$
$199$	1.89448e8	1.70414	0.852068	+	0.523431i	$0.175348\pi$
$200$	0	0
$201$	−4.12872e7	−0.358615
$202$	0	0
$203$	2.67445e8	2.24388
$204$	0	0
$205$	2.21410e7	0.179498
$206$	0	0
$207$	−6.00098e7	−0.470247
$208$	0	0
$209$	−2.34686e8	−1.77818
$210$	0	0
$211$	−2.55914e8	−1.87545	−0.937725	−	0.347379i	$-0.887072\pi$
$211$	−2.55914e8	−1.87545	−0.937725	+	0.347379i	$0.887072\pi$
$212$	0	0
$213$	7.39947e7	0.524653
$214$	0	0
$215$	989448.	0.00678982
$216$	0	0
$217$	2.17008e8	1.44167
$218$	0	0
$219$	−1.07556e8	−0.691958
$220$	0	0
$221$	2.27028e8	1.41484
$222$	0	0
$223$	−2.85375e8	−1.72325	−0.861625	−	0.507546i	$-0.830553\pi$
$223$	−2.85375e8	−1.72325	−0.861625	+	0.507546i	$0.830553\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	1.64896e8	0.935663	0.467831	−	0.883818i	$-0.345035\pi$
$227$	1.64896e8	0.935663	0.467831	+	0.883818i	$0.345035\pi$
$228$	0	0
$229$	2.46143e7	0.135445	0.0677226	−	0.997704i	$-0.478427\pi$
$229$	2.46143e7	0.135445	0.0677226	+	0.997704i	$0.478427\pi$
$230$	0	0
$231$	−3.45324e8	−1.84325
$232$	0	0
$233$	−4.02168e7	−0.208287	−0.104143	−	0.994562i	$-0.533210\pi$
$233$	−4.02168e7	−0.208287	−0.104143	+	0.994562i	$0.533210\pi$
$234$	0	0
$235$	3.64352e7	0.183140
$236$	0	0
$237$	1.07572e8	0.524905
$238$	0	0
$239$	−1.65900e8	−0.786058	−0.393029	−	0.919526i	$-0.628573\pi$
$239$	−1.65900e8	−0.786058	−0.393029	+	0.919526i	$0.628573\pi$
$240$	0	0
$241$	4.44599e7	0.204602	0.102301	−	0.994754i	$-0.467380\pi$
$241$	4.44599e7	0.204602	0.102301	+	0.994754i	$0.467380\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−2.55263e8	−1.10894
$246$	0	0
$247$	−4.01071e8	−1.69349
$248$	0	0
$249$	1.35503e8	0.556225
$250$	0	0
$251$	−2.17092e8	−0.866534	−0.433267	−	0.901266i	$-0.642639\pi$
$251$	−2.17092e8	−0.866534	−0.433267	+	0.901266i	$0.642639\pi$
$252$	0	0
$253$	−6.21936e8	−2.41448
$254$	0	0
$255$	5.93428e7	0.224118
$256$	0	0
$257$	−2.62078e8	−0.963084	−0.481542	−	0.876423i	$-0.659923\pi$
$257$	−2.62078e8	−0.963084	−0.481542	+	0.876423i	$0.659923\pi$
$258$	0	0
$259$	−2.32256e8	−0.830648
$260$	0	0
$261$	1.15173e8	0.400968
$262$	0	0
$263$	1.61193e8	0.546388	0.273194	−	0.961959i	$-0.411920\pi$
$263$	1.61193e8	0.546388	0.273194	+	0.961959i	$0.411920\pi$
$264$	0	0
$265$	9.71146e7	0.320571
$266$	0	0
$267$	3.02955e8	0.974065
$268$	0	0
$269$	−4.83488e8	−1.51444	−0.757221	−	0.653159i	$-0.773444\pi$
$269$	−4.83488e8	−1.51444	−0.757221	+	0.653159i	$0.773444\pi$
$270$	0	0
$271$	1.72618e8	0.526858	0.263429	−	0.964679i	$-0.415147\pi$
$271$	1.72618e8	0.526858	0.263429	+	0.964679i	$0.415147\pi$
$272$	0	0
$273$	−5.90148e8	−1.75546
$274$	0	0
$275$	1.18051e8	0.342300
$276$	0	0
$277$	3.74034e8	1.05738	0.528690	−	0.848815i	$-0.322683\pi$
$277$	3.74034e8	1.05738	0.528690	+	0.848815i	$0.322683\pi$
$278$	0	0
$279$	9.34527e7	0.257618
$280$	0	0
$281$	2.01716e8	0.542335	0.271168	−	0.962532i	$-0.412590\pi$
$281$	2.01716e8	0.542335	0.271168	+	0.962532i	$0.412590\pi$
$282$	0	0
$283$	5.12679e8	1.34460	0.672300	−	0.740279i	$-0.265307\pi$
$283$	5.12679e8	1.34460	0.672300	+	0.740279i	$0.265307\pi$
$284$	0	0
$285$	−1.04836e8	−0.268258
$286$	0	0
$287$	−2.99847e8	−0.748708
$288$	0	0
$289$	−1.01175e8	−0.246565
$290$	0	0
$291$	5.76987e7	0.137259
$292$	0	0
$293$	−2.32116e8	−0.539098	−0.269549	−	0.962987i	$-0.586875\pi$
$293$	−2.32116e8	−0.539098	−0.269549	+	0.962987i	$0.586875\pi$
$294$	0	0
$295$	−1.14877e8	−0.260528
$296$	0	0
$297$	−1.48711e8	−0.329378
$298$	0	0
$299$	−1.06287e9	−2.29949
$300$	0	0
$301$	−1.33997e7	−0.0283212
$302$	0	0
$303$	−9.47568e7	−0.195687
$304$	0	0
$305$	−2.61052e8	−0.526839
$306$	0	0
$307$	2.50803e8	0.494708	0.247354	−	0.968925i	$-0.420439\pi$
$307$	2.50803e8	0.494708	0.247354	+	0.968925i	$0.420439\pi$
$308$	0	0
$309$	5.47737e8	1.05613
$310$	0	0
$311$	4.63371e8	0.873508	0.436754	−	0.899581i	$-0.356128\pi$
$311$	4.63371e8	0.873508	0.436754	+	0.899581i	$0.356128\pi$
$312$	0	0
$313$	−9.96023e8	−1.83596	−0.917982	−	0.396622i	$-0.870182\pi$
$313$	−9.96023e8	−1.83596	−0.917982	+	0.396622i	$0.870182\pi$
$314$	0	0
$315$	−1.54258e8	−0.278075
$316$	0	0
$317$	−1.28892e8	−0.227258	−0.113629	−	0.993523i	$-0.536247\pi$
$317$	−1.28892e8	−0.227258	−0.113629	+	0.993523i	$0.536247\pi$
$318$	0	0
$319$	1.19364e9	2.05877
$320$	0	0
$321$	−5.68295e8	−0.958973
$322$	0	0
$323$	−5.46173e8	−0.901823
$324$	0	0
$325$	2.01746e8	0.325997
$326$	0	0
$327$	−2.59172e8	−0.409894
$328$	0	0
$329$	−4.93427e8	−0.763901
$330$	0	0
$331$	−4.34739e8	−0.658917	−0.329458	−	0.944170i	$-0.606866\pi$
$331$	−4.34739e8	−0.658917	−0.329458	+	0.944170i	$0.606866\pi$
$332$	0	0
$333$	−1.00019e8	−0.148432
$334$	0	0
$335$	−1.91144e8	−0.277782
$336$	0	0
$337$	3.00957e8	0.428350	0.214175	−	0.976795i	$-0.431294\pi$
$337$	3.00957e8	0.428350	0.214175	+	0.976795i	$0.431294\pi$
$338$	0	0
$339$	−2.35816e7	−0.0328757
$340$	0	0
$341$	9.68535e8	1.32274
$342$	0	0
$343$	2.06281e9	2.76013
$344$	0	0
$345$	−2.77823e8	−0.364252
$346$	0	0
$347$	2.45979e7	0.0316043	0.0158021	−	0.999875i	$-0.494970\pi$
$347$	2.45979e7	0.0316043	0.0158021	+	0.999875i	$0.494970\pi$
$348$	0	0
$349$	1.48910e9	1.87515	0.937575	−	0.347784i	$-0.113066\pi$
$349$	1.48910e9	1.87515	0.937575	+	0.347784i	$0.113066\pi$
$350$	0	0
$351$	−2.54142e8	−0.313691
$352$	0	0
$353$	−9.72005e8	−1.17613	−0.588067	−	0.808812i	$-0.700111\pi$
$353$	−9.72005e8	−1.17613	−0.588067	+	0.808812i	$0.700111\pi$
$354$	0	0
$355$	3.42568e8	0.406394
$356$	0	0
$357$	−8.03655e8	−0.934826
$358$	0	0
$359$	6.72359e8	0.766956	0.383478	−	0.923550i	$-0.374726\pi$
$359$	6.72359e8	0.766956	0.383478	+	0.923550i	$0.374726\pi$
$360$	0	0
$361$	7.10052e7	0.0794356
$362$	0	0
$363$	−1.01507e9	−1.11384
$364$	0	0
$365$	−4.97944e8	−0.535988
$366$	0	0
$367$	−1.87253e8	−0.197741	−0.0988706	−	0.995100i	$-0.531523\pi$
$367$	−1.87253e8	−0.197741	−0.0988706	+	0.995100i	$0.531523\pi$
$368$	0	0
$369$	−1.29126e8	−0.133790
$370$	0	0
$371$	−1.31518e9	−1.33714
$372$	0	0
$373$	1.43161e9	1.42838	0.714192	−	0.699950i	$-0.246794\pi$
$373$	1.43161e9	1.42838	0.714192	+	0.699950i	$0.246794\pi$
$374$	0	0
$375$	5.27344e7	0.0516398
$376$	0	0
$377$	2.03990e9	1.96071
$378$	0	0
$379$	3.52865e8	0.332944	0.166472	−	0.986046i	$-0.446762\pi$
$379$	3.52865e8	0.332944	0.166472	+	0.986046i	$0.446762\pi$
$380$	0	0
$381$	−6.04035e8	−0.559532
$382$	0	0
$383$	9.41925e8	0.856684	0.428342	−	0.903617i	$-0.359098\pi$
$383$	9.41925e8	0.856684	0.428342	+	0.903617i	$0.359098\pi$
$384$	0	0
$385$	−1.59872e9	−1.42778
$386$	0	0
$387$	−5.77046e6	−0.00506084
$388$	0	0
$389$	1.70748e9	1.47073	0.735363	−	0.677674i	$-0.237012\pi$
$389$	1.70748e9	1.47073	0.735363	+	0.677674i	$0.237012\pi$
$390$	0	0
$391$	−1.44740e9	−1.22453
$392$	0	0
$393$	−8.37402e8	−0.695922
$394$	0	0
$395$	4.98020e8	0.406590
$396$	0	0
$397$	2.43384e9	1.95220	0.976101	−	0.217317i	$-0.0697305\pi$
$397$	2.43384e9	1.95220	0.976101	+	0.217317i	$0.0697305\pi$
$398$	0	0
$399$	1.41975e9	1.11894
$400$	0	0
$401$	−3.90896e8	−0.302731	−0.151365	−	0.988478i	$-0.548367\pi$
$401$	−3.90896e8	−0.302731	−0.151365	+	0.988478i	$0.548367\pi$
$402$	0	0
$403$	1.65520e9	1.25974
$404$	0	0
$405$	−6.64301e7	−0.0496904
$406$	0	0
$407$	−1.03659e9	−0.762124
$408$	0	0
$409$	5.70045e8	0.411981	0.205991	−	0.978554i	$-0.433958\pi$
$409$	5.70045e8	0.411981	0.205991	+	0.978554i	$0.433958\pi$
$410$	0	0
$411$	−1.31398e8	−0.0933562
$412$	0	0
$413$	1.55573e9	1.08670
$414$	0	0
$415$	6.27327e8	0.430850
$416$	0	0
$417$	1.30724e9	0.882833
$418$	0	0
$419$	−1.13885e8	−0.0756338	−0.0378169	−	0.999285i	$-0.512040\pi$
$419$	−1.13885e8	−0.0756338	−0.0378169	+	0.999285i	$0.512040\pi$
$420$	0	0
$421$	−7.26241e8	−0.474344	−0.237172	−	0.971468i	$-0.576221\pi$
$421$	−7.26241e8	−0.474344	−0.237172	+	0.971468i	$0.576221\pi$
$422$	0	0
$423$	−2.12490e8	−0.136505
$424$	0	0
$425$	2.74735e8	0.173601
$426$	0	0
$427$	3.53532e9	2.19751
$428$	0	0
$429$	−2.63391e9	−1.61065
$430$	0	0
$431$	−1.98831e9	−1.19623	−0.598113	−	0.801412i	$-0.704082\pi$
$431$	−1.98831e9	−1.19623	−0.598113	+	0.801412i	$0.704082\pi$
$432$	0	0
$433$	1.76834e9	1.04679	0.523395	−	0.852090i	$-0.324665\pi$
$433$	1.76834e9	1.04679	0.523395	+	0.852090i	$0.324665\pi$
$434$	0	0
$435$	5.33209e8	0.310588
$436$	0	0
$437$	2.55700e9	1.46570
$438$	0	0
$439$	−5.84269e8	−0.329600	−0.164800	−	0.986327i	$-0.552698\pi$
$439$	−5.84269e8	−0.329600	−0.164800	+	0.986327i	$0.552698\pi$
$440$	0	0
$441$	1.48870e9	0.826553
$442$	0	0
$443$	−1.41943e9	−0.775714	−0.387857	−	0.921720i	$-0.626785\pi$
$443$	−1.41943e9	−0.775714	−0.387857	+	0.921720i	$0.626785\pi$
$444$	0	0
$445$	1.40257e9	0.754508
$446$	0	0
$447$	4.26943e7	0.0226096
$448$	0	0
$449$	7.27226e8	0.379147	0.189573	−	0.981867i	$-0.439290\pi$
$449$	7.27226e8	0.379147	0.189573	+	0.981867i	$0.439290\pi$
$450$	0	0
$451$	−1.33825e9	−0.686943
$452$	0	0
$453$	4.55720e8	0.230332
$454$	0	0
$455$	−2.73217e9	−1.35978
$456$	0	0
$457$	−3.13175e9	−1.53490	−0.767451	−	0.641107i	$-0.778475\pi$
$457$	−3.13175e9	−1.53490	−0.767451	+	0.641107i	$0.778475\pi$
$458$	0	0
$459$	−3.46087e8	−0.167048
$460$	0	0
$461$	−9.30032e8	−0.442125	−0.221062	−	0.975260i	$-0.570952\pi$
$461$	−9.30032e8	−0.442125	−0.221062	+	0.975260i	$0.570952\pi$
$462$	0	0
$463$	1.22920e9	0.575557	0.287778	−	0.957697i	$-0.407083\pi$
$463$	1.22920e9	0.575557	0.287778	+	0.957697i	$0.407083\pi$
$464$	0	0
$465$	4.32651e8	0.199550
$466$	0	0
$467$	−4.96824e8	−0.225732	−0.112866	−	0.993610i	$-0.536003\pi$
$467$	−4.96824e8	−0.225732	−0.112866	+	0.993610i	$0.536003\pi$
$468$	0	0
$469$	2.58859e9	1.15867
$470$	0	0
$471$	−4.44957e8	−0.196221
$472$	0	0
$473$	−5.98046e7	−0.0259849
$474$	0	0
$475$	−4.85351e8	−0.207792
$476$	0	0
$477$	−5.66373e8	−0.238939
$478$	0	0
$479$	−1.70976e9	−0.710820	−0.355410	−	0.934711i	$-0.615659\pi$
$479$	−1.70976e9	−0.710820	−0.355410	+	0.934711i	$0.615659\pi$
$480$	0	0
$481$	−1.77150e9	−0.725826
$482$	0	0
$483$	3.76244e9	1.51934
$484$	0	0
$485$	2.67124e8	0.106320
$486$	0	0
$487$	−3.41415e8	−0.133946	−0.0669732	−	0.997755i	$-0.521334\pi$
$487$	−3.41415e8	−0.133946	−0.0669732	+	0.997755i	$0.521334\pi$
$488$	0	0
$489$	−1.51233e9	−0.584879
$490$	0	0
$491$	−2.65319e9	−1.01154	−0.505770	−	0.862669i	$-0.668791\pi$
$491$	−2.65319e9	−1.01154	−0.505770	+	0.862669i	$0.668791\pi$
$492$	0	0
$493$	2.77791e9	1.04413
$494$	0	0
$495$	−6.88476e8	−0.255135
$496$	0	0
$497$	−4.63926e9	−1.69512
$498$	0	0
$499$	−1.83957e9	−0.662772	−0.331386	−	0.943495i	$-0.607516\pi$
$499$	−1.83957e9	−0.662772	−0.331386	+	0.943495i	$0.607516\pi$
$500$	0	0
$501$	3.10188e8	0.110203
$502$	0	0
$503$	4.81741e9	1.68782	0.843909	−	0.536487i	$-0.180249\pi$
$503$	4.81741e9	1.68782	0.843909	+	0.536487i	$0.180249\pi$
$504$	0	0
$505$	−4.38689e8	−0.151578
$506$	0	0
$507$	−2.80706e9	−0.956584
$508$	0	0
$509$	3.51073e9	1.18001	0.590004	−	0.807400i	$-0.299126\pi$
$509$	3.51073e9	1.18001	0.590004	+	0.807400i	$0.299126\pi$
$510$	0	0
$511$	6.74344e9	2.23567
$512$	0	0
$513$	6.11403e8	0.199948
$514$	0	0
$515$	2.53582e9	0.818075
$516$	0	0
$517$	−2.20223e9	−0.700883
$518$	0	0
$519$	3.11036e8	0.0976617
$520$	0	0
$521$	−3.61701e9	−1.12051	−0.560257	−	0.828319i	$-0.689298\pi$
$521$	−3.61701e9	−1.12051	−0.560257	+	0.828319i	$0.689298\pi$
$522$	0	0
$523$	1.38692e9	0.423930	0.211965	−	0.977277i	$-0.432014\pi$
$523$	1.38692e9	0.423930	0.211965	+	0.977277i	$0.432014\pi$
$524$	0	0
$525$	−7.14160e8	−0.215396
$526$	0	0
$527$	2.25402e9	0.670844
$528$	0	0
$529$	3.37143e9	0.990191
$530$	0	0
$531$	6.69961e8	0.194186
$532$	0	0
$533$	−2.28704e9	−0.654226
$534$	0	0
$535$	−2.63100e9	−0.742818
$536$	0	0
$537$	2.31832e9	0.646046
$538$	0	0
$539$	1.54287e10	4.24394
$540$	0	0
$541$	3.53611e9	0.960143	0.480071	−	0.877229i	$-0.340611\pi$
$541$	3.53611e9	0.960143	0.480071	+	0.877229i	$0.340611\pi$
$542$	0	0
$543$	−2.38116e9	−0.638247
$544$	0	0
$545$	−1.19987e9	−0.317503
$546$	0	0
$547$	3.34133e9	0.872897	0.436449	−	0.899729i	$-0.356236\pi$
$547$	3.34133e9	0.872897	0.436449	+	0.899729i	$0.356236\pi$
$548$	0	0
$549$	1.52246e9	0.392682
$550$	0	0
$551$	−4.90749e9	−1.24977
$552$	0	0
$553$	−6.74447e9	−1.69594
$554$	0	0
$555$	−4.63051e8	−0.114975
$556$	0	0
$557$	5.70595e9	1.39906	0.699528	−	0.714606i	$-0.253394\pi$
$557$	5.70595e9	1.39906	0.699528	+	0.714606i	$0.253394\pi$
$558$	0	0
$559$	−1.02204e8	−0.0247473
$560$	0	0
$561$	−3.58682e9	−0.857708
$562$	0	0
$563$	−7.54884e9	−1.78279	−0.891397	−	0.453224i	$-0.850274\pi$
$563$	−7.54884e9	−1.78279	−0.891397	+	0.453224i	$0.850274\pi$
$564$	0	0
$565$	−1.09174e8	−0.0254654
$566$	0	0
$567$	8.99635e8	0.207265
$568$	0	0
$569$	−3.70964e9	−0.844186	−0.422093	−	0.906552i	$-0.638704\pi$
$569$	−3.70964e9	−0.844186	−0.422093	+	0.906552i	$0.638704\pi$
$570$	0	0
$571$	−9.03703e8	−0.203142	−0.101571	−	0.994828i	$-0.532387\pi$
$571$	−9.03703e8	−0.203142	−0.101571	+	0.994828i	$0.532387\pi$
$572$	0	0
$573$	4.00591e9	0.889530
$574$	0	0
$575$	−1.28622e9	−0.282148
$576$	0	0
$577$	−8.61561e8	−0.186711	−0.0933557	−	0.995633i	$-0.529759\pi$
$577$	−8.61561e8	−0.186711	−0.0933557	+	0.995633i	$0.529759\pi$
$578$	0	0
$579$	−3.11601e9	−0.667150
$580$	0	0
$581$	−8.49563e9	−1.79713
$582$	0	0
$583$	−5.86984e9	−1.22683
$584$	0	0
$585$	−1.17658e9	−0.242984
$586$	0	0
$587$	3.89214e9	0.794246	0.397123	−	0.917765i	$-0.370009\pi$
$587$	3.89214e9	0.794246	0.397123	+	0.917765i	$0.370009\pi$
$588$	0	0
$589$	−3.98199e9	−0.802965
$590$	0	0
$591$	−1.41863e9	−0.282691
$592$	0	0
$593$	2.08455e9	0.410507	0.205254	−	0.978709i	$-0.434198\pi$
$593$	2.08455e9	0.410507	0.205254	+	0.978709i	$0.434198\pi$
$594$	0	0
$595$	−3.72062e9	−0.724113
$596$	0	0
$597$	−5.11510e9	−0.983884
$598$	0	0
$599$	2.98088e9	0.566696	0.283348	−	0.959017i	$-0.408555\pi$
$599$	2.98088e9	0.566696	0.283348	+	0.959017i	$0.408555\pi$
$600$	0	0
$601$	−5.37665e9	−1.01030	−0.505151	−	0.863031i	$-0.668563\pi$
$601$	−5.37665e9	−1.01030	−0.505151	+	0.863031i	$0.668563\pi$
$602$	0	0
$603$	1.11475e9	0.207047
$604$	0	0
$605$	−4.69941e9	−0.862778
$606$	0	0
$607$	1.08127e10	1.96233	0.981165	−	0.193173i	$-0.0618780\pi$
$607$	1.08127e10	1.96233	0.981165	+	0.193173i	$0.0618780\pi$
$608$	0	0
$609$	−7.22103e9	−1.29550
$610$	0	0
$611$	−3.76354e9	−0.667501
$612$	0	0
$613$	−6.17191e9	−1.08220	−0.541100	−	0.840958i	$-0.681992\pi$
$613$	−6.17191e9	−1.08220	−0.541100	+	0.840958i	$0.681992\pi$
$614$	0	0
$615$	−5.97807e8	−0.103633
$616$	0	0
$617$	−2.82724e9	−0.484579	−0.242289	−	0.970204i	$-0.577898\pi$
$617$	−2.82724e9	−0.484579	−0.242289	+	0.970204i	$0.577898\pi$
$618$	0	0
$619$	−4.42911e9	−0.750584	−0.375292	−	0.926907i	$-0.622458\pi$
$619$	−4.42911e9	−0.750584	−0.375292	+	0.926907i	$0.622458\pi$
$620$	0	0
$621$	1.62026e9	0.271497
$622$	0	0
$623$	−1.89944e10	−3.14715
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	6.33652e9	1.02663
$628$	0	0
$629$	−2.41240e9	−0.386520
$630$	0	0
$631$	−2.16710e9	−0.343381	−0.171690	−	0.985151i	$-0.554923\pi$
$631$	−2.16710e9	−0.343381	−0.171690	+	0.985151i	$0.554923\pi$
$632$	0	0
$633$	6.90968e9	1.08279
$634$	0	0
$635$	−2.79646e9	−0.433411
$636$	0	0
$637$	2.63672e10	4.04181
$638$	0	0
$639$	−1.99786e9	−0.302909
$640$	0	0
$641$	−5.53630e9	−0.830264	−0.415132	−	0.909761i	$-0.636265\pi$
$641$	−5.53630e9	−0.830264	−0.415132	+	0.909761i	$0.636265\pi$
$642$	0	0
$643$	−4.42601e9	−0.656559	−0.328280	−	0.944581i	$-0.606469\pi$
$643$	−4.42601e9	−0.656559	−0.328280	+	0.944581i	$0.606469\pi$
$644$	0	0
$645$	−2.67151e7	−0.00392011
$646$	0	0
$647$	1.09455e10	1.58881	0.794404	−	0.607389i	$-0.207783\pi$
$647$	1.09455e10	1.58881	0.794404	+	0.607389i	$0.207783\pi$
$648$	0	0
$649$	6.94341e9	0.997049
$650$	0	0
$651$	−5.85922e9	−0.832350
$652$	0	0
$653$	7.33718e9	1.03118	0.515588	−	0.856836i	$-0.327573\pi$
$653$	7.33718e9	1.03118	0.515588	+	0.856836i	$0.327573\pi$
$654$	0	0
$655$	−3.87686e9	−0.539059
$656$	0	0
$657$	2.90401e9	0.399502
$658$	0	0
$659$	3.76188e9	0.512042	0.256021	−	0.966671i	$-0.417588\pi$
$659$	3.76188e9	0.512042	0.256021	+	0.966671i	$0.417588\pi$
$660$	0	0
$661$	−5.43289e9	−0.731688	−0.365844	−	0.930676i	$-0.619220\pi$
$661$	−5.43289e9	−0.731688	−0.365844	+	0.930676i	$0.619220\pi$
$662$	0	0
$663$	−6.12976e9	−0.816857
$664$	0	0
$665$	6.57291e9	0.866726
$666$	0	0
$667$	−1.30052e10	−1.69698
$668$	0	0
$669$	7.70511e9	0.994919
$670$	0	0
$671$	1.57786e10	2.01623
$672$	0	0
$673$	4.38179e9	0.554113	0.277057	−	0.960854i	$-0.410641\pi$
$673$	4.38179e9	0.554113	0.277057	+	0.960854i	$0.410641\pi$
$674$	0	0
$675$	−3.07547e8	−0.0384900
$676$	0	0
$677$	−3.78434e9	−0.468737	−0.234369	−	0.972148i	$-0.575302\pi$
$677$	−3.78434e9	−0.468737	−0.234369	+	0.972148i	$0.575302\pi$
$678$	0	0
$679$	−3.61755e9	−0.443476
$680$	0	0
$681$	−4.45219e9	−0.540205
$682$	0	0
$683$	6.76592e8	0.0812558	0.0406279	−	0.999174i	$-0.487064\pi$
$683$	6.76592e8	0.0812558	0.0406279	+	0.999174i	$0.487064\pi$
$684$	0	0
$685$	−6.08325e8	−0.0723134
$686$	0	0
$687$	−6.64587e8	−0.0781994
$688$	0	0
$689$	−1.00314e10	−1.16840
$690$	0	0
$691$	−3.51846e9	−0.405676	−0.202838	−	0.979212i	$-0.565017\pi$
$691$	−3.51846e9	−0.405676	−0.202838	+	0.979212i	$0.565017\pi$
$692$	0	0
$693$	9.32374e9	1.06420
$694$	0	0
$695$	6.05203e9	0.683839
$696$	0	0
$697$	−3.11445e9	−0.348391
$698$	0	0
$699$	1.08585e9	0.120255
$700$	0	0
$701$	−1.05800e10	−1.16004	−0.580019	−	0.814603i	$-0.696955\pi$
$701$	−1.05800e10	−1.16004	−0.580019	+	0.814603i	$0.696955\pi$
$702$	0	0
$703$	4.26178e9	0.462644
$704$	0	0
$705$	−9.83750e8	−0.105736
$706$	0	0
$707$	5.94098e9	0.632252
$708$	0	0
$709$	−5.70513e9	−0.601179	−0.300590	−	0.953754i	$-0.597183\pi$
$709$	−5.70513e9	−0.601179	−0.300590	+	0.953754i	$0.597183\pi$
$710$	0	0
$711$	−2.90445e9	−0.303054
$712$	0	0
$713$	−1.05526e10	−1.09030
$714$	0	0
$715$	−1.21940e10	−1.24760
$716$	0	0
$717$	4.47931e9	0.453831
$718$	0	0
$719$	−1.62812e10	−1.63356	−0.816780	−	0.576950i	$-0.804243\pi$
$719$	−1.62812e10	−1.63356	−0.816780	+	0.576950i	$0.804243\pi$
$720$	0	0
$721$	−3.43416e10	−3.41230
$722$	0	0
$723$	−1.20042e9	−0.118127
$724$	0	0
$725$	2.46856e9	0.240581
$726$	0	0
$727$	−1.13792e10	−1.09836	−0.549178	−	0.835706i	$-0.685059\pi$
$727$	−1.13792e10	−1.09836	−0.549178	+	0.835706i	$0.685059\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.39180e8	−0.0131785
$732$	0	0
$733$	−7.15194e9	−0.670749	−0.335374	−	0.942085i	$-0.608863\pi$
$733$	−7.15194e9	−0.670749	−0.335374	+	0.942085i	$0.608863\pi$
$734$	0	0
$735$	6.89211e9	0.640245
$736$	0	0
$737$	1.15532e10	1.06308
$738$	0	0
$739$	−1.49488e10	−1.36255	−0.681273	−	0.732029i	$-0.738573\pi$
$739$	−1.49488e10	−1.36255	−0.681273	+	0.732029i	$0.738573\pi$
$740$	0	0
$741$	1.08289e10	0.977736
$742$	0	0
$743$	−3.53086e9	−0.315805	−0.157903	−	0.987455i	$-0.550473\pi$
$743$	−3.53086e9	−0.315805	−0.157903	+	0.987455i	$0.550473\pi$
$744$	0	0
$745$	1.97659e8	0.0175133
$746$	0	0
$747$	−3.65857e9	−0.321136
$748$	0	0
$749$	3.56305e10	3.09839
$750$	0	0
$751$	−1.74317e10	−1.50176	−0.750880	−	0.660439i	$-0.770370\pi$
$751$	−1.74317e10	−1.50176	−0.750880	+	0.660439i	$0.770370\pi$
$752$	0	0
$753$	5.86148e9	0.500293
$754$	0	0
$755$	2.10981e9	0.178414
$756$	0	0
$757$	−1.57913e10	−1.32307	−0.661535	−	0.749915i	$-0.730095\pi$
$757$	−1.57913e10	−1.32307	−0.661535	+	0.749915i	$0.730095\pi$
$758$	0	0
$759$	1.67923e10	1.39400
$760$	0	0
$761$	1.59902e10	1.31524	0.657622	−	0.753348i	$-0.271562\pi$
$761$	1.59902e10	1.31524	0.657622	+	0.753348i	$0.271562\pi$
$762$	0	0
$763$	1.62494e10	1.32434
$764$	0	0
$765$	−1.60226e9	−0.129395
$766$	0	0
$767$	1.18661e10	0.949562
$768$	0	0
$769$	1.67093e10	1.32500	0.662499	−	0.749063i	$-0.269496\pi$
$769$	1.67093e10	1.32500	0.662499	+	0.749063i	$0.269496\pi$
$770$	0	0
$771$	7.07610e9	0.556037
$772$	0	0
$773$	7.70940e9	0.600334	0.300167	−	0.953887i	$-0.402958\pi$
$773$	7.70940e9	0.600334	0.300167	+	0.953887i	$0.402958\pi$
$774$	0	0
$775$	2.00302e9	0.154571
$776$	0	0
$777$	6.27090e9	0.479575
$778$	0	0
$779$	5.50204e9	0.417006
$780$	0	0
$781$	−2.07056e10	−1.55528
$782$	0	0
$783$	−3.10967e9	−0.231499
$784$	0	0
$785$	−2.05999e9	−0.151992
$786$	0	0
$787$	6.10258e9	0.446275	0.223137	−	0.974787i	$-0.428370\pi$
$787$	6.10258e9	0.446275	0.223137	+	0.974787i	$0.428370\pi$
$788$	0	0
$789$	−4.35221e9	−0.315457
$790$	0	0
$791$	1.47850e9	0.106219
$792$	0	0
$793$	2.69651e10	1.92020
$794$	0	0
$795$	−2.62210e9	−0.185082
$796$	0	0
$797$	−1.42109e9	−0.0994300	−0.0497150	−	0.998763i	$-0.515831\pi$
$797$	−1.42109e9	−0.0994300	−0.0497150	+	0.998763i	$0.515831\pi$
$798$	0	0
$799$	−5.12513e9	−0.355461
$800$	0	0
$801$	−8.17977e9	−0.562377
$802$	0	0
$803$	3.00969e10	2.05124
$804$	0	0
$805$	1.74187e10	1.17688
$806$	0	0
$807$	1.30542e10	0.874364
$808$	0	0
$809$	−2.57373e10	−1.70900	−0.854502	−	0.519448i	$-0.826138\pi$
$809$	−2.57373e10	−1.70900	−0.854502	+	0.519448i	$0.826138\pi$
$810$	0	0
$811$	9.67313e9	0.636787	0.318393	−	0.947959i	$-0.396857\pi$
$811$	9.67313e9	0.636787	0.318393	+	0.947959i	$0.396857\pi$
$812$	0	0
$813$	−4.66069e9	−0.304182
$814$	0	0
$815$	−7.00154e9	−0.453046
$816$	0	0
$817$	2.45878e8	0.0157740
$818$	0	0
$819$	1.59340e10	1.01352
$820$	0	0
$821$	3.58162e9	0.225880	0.112940	−	0.993602i	$-0.463973\pi$
$821$	3.58162e9	0.225880	0.112940	+	0.993602i	$0.463973\pi$
$822$	0	0
$823$	−1.33606e10	−0.835461	−0.417730	−	0.908571i	$-0.637174\pi$
$823$	−1.33606e10	−0.835461	−0.417730	+	0.908571i	$0.637174\pi$
$824$	0	0
$825$	−3.18739e9	−0.197627
$826$	0	0
$827$	−1.84221e10	−1.13258	−0.566290	−	0.824206i	$-0.691622\pi$
$827$	−1.84221e10	−1.13258	−0.566290	+	0.824206i	$0.691622\pi$
$828$	0	0
$829$	−1.59999e10	−0.975387	−0.487693	−	0.873015i	$-0.662162\pi$
$829$	−1.59999e10	−0.975387	−0.487693	+	0.873015i	$0.662162\pi$
$830$	0	0
$831$	−1.00989e10	−0.610479
$832$	0	0
$833$	3.59065e10	2.15236
$834$	0	0
$835$	1.43606e9	0.0853628
$836$	0	0
$837$	−2.52322e9	−0.148736
$838$	0	0
$839$	2.54806e10	1.48951	0.744754	−	0.667339i	$-0.232567\pi$
$839$	2.54806e10	1.48951	0.744754	+	0.667339i	$0.232567\pi$
$840$	0	0
$841$	7.71028e9	0.446976
$842$	0	0
$843$	−5.44633e9	−0.313118
$844$	0	0
$845$	−1.29956e10	−0.740967
$846$	0	0
$847$	6.36421e10	3.59876
$848$	0	0
$849$	−1.38423e10	−0.776305
$850$	0	0
$851$	1.12940e10	0.628197
$852$	0	0
$853$	−1.17789e9	−0.0649803	−0.0324902	−	0.999472i	$-0.510344\pi$
$853$	−1.17789e9	−0.0649803	−0.0324902	+	0.999472i	$0.510344\pi$
$854$	0	0
$855$	2.83057e9	0.154879
$856$	0	0
$857$	3.84024e8	0.0208413	0.0104207	−	0.999946i	$-0.496683\pi$
$857$	3.84024e8	0.0208413	0.0104207	+	0.999946i	$0.496683\pi$
$858$	0	0
$859$	5.43846e9	0.292752	0.146376	−	0.989229i	$-0.453239\pi$
$859$	5.43846e9	0.292752	0.146376	+	0.989229i	$0.453239\pi$
$860$	0	0
$861$	8.09586e9	0.432267
$862$	0	0
$863$	−1.79788e10	−0.952187	−0.476093	−	0.879395i	$-0.657948\pi$
$863$	−1.79788e10	−0.952187	−0.476093	+	0.879395i	$0.657948\pi$
$864$	0	0
$865$	1.43998e9	0.0756485
$866$	0	0
$867$	2.73172e9	0.142354
$868$	0	0
$869$	−3.01015e10	−1.55603
$870$	0	0
$871$	1.97441e10	1.01245
$872$	0	0
$873$	−1.55787e9	−0.0792465
$874$	0	0
$875$	−3.30629e9	−0.166845
$876$	0	0
$877$	−1.88242e10	−0.942362	−0.471181	−	0.882037i	$-0.656172\pi$
$877$	−1.88242e10	−0.942362	−0.471181	+	0.882037i	$0.656172\pi$
$878$	0	0
$879$	6.26712e9	0.311248
$880$	0	0
$881$	1.52225e10	0.750018	0.375009	−	0.927021i	$-0.377640\pi$
$881$	1.52225e10	0.750018	0.375009	+	0.927021i	$0.377640\pi$
$882$	0	0
$883$	3.92105e10	1.91664	0.958318	−	0.285703i	$-0.0922270\pi$
$883$	3.92105e10	1.91664	0.958318	+	0.285703i	$0.0922270\pi$
$884$	0	0
$885$	3.10167e9	0.150416
$886$	0	0
$887$	1.69878e9	0.0817345	0.0408672	−	0.999165i	$-0.486988\pi$
$887$	1.69878e9	0.0817345	0.0408672	+	0.999165i	$0.486988\pi$
$888$	0	0
$889$	3.78713e10	1.80781
$890$	0	0
$891$	4.01519e9	0.190167
$892$	0	0
$893$	9.05414e9	0.425468
$894$	0	0
$895$	1.07330e10	0.500425
$896$	0	0
$897$	2.86975e10	1.32761
$898$	0	0
$899$	2.02529e10	0.929670
$900$	0	0
$901$	−1.36606e10	−0.622203
$902$	0	0
$903$	3.61792e8	0.0163513
$904$	0	0
$905$	−1.10239e10	−0.494384
$906$	0	0
$907$	−2.91325e10	−1.29644	−0.648220	−	0.761453i	$-0.724486\pi$
$907$	−2.91325e10	−1.29644	−0.648220	+	0.761453i	$0.724486\pi$
$908$	0	0
$909$	2.55843e9	0.112980
$910$	0	0
$911$	−2.33098e10	−1.02147	−0.510733	−	0.859739i	$-0.670626\pi$
$911$	−2.33098e10	−1.02147	−0.510733	+	0.859739i	$0.670626\pi$
$912$	0	0
$913$	−3.79171e10	−1.64887
$914$	0	0
$915$	7.04841e9	0.304170
$916$	0	0
$917$	5.25027e10	2.24848
$918$	0	0
$919$	−2.11867e10	−0.900449	−0.450225	−	0.892915i	$-0.648656\pi$
$919$	−2.11867e10	−0.900449	−0.450225	+	0.892915i	$0.648656\pi$
$920$	0	0
$921$	−6.77169e9	−0.285620
$922$	0	0
$923$	−3.53853e10	−1.48121
$924$	0	0
$925$	−2.14375e9	−0.0890592
$926$	0	0
$927$	−1.47889e10	−0.609757
$928$	0	0
$929$	−9.04776e9	−0.370242	−0.185121	−	0.982716i	$-0.559268\pi$
$929$	−9.04776e9	−0.370242	−0.185121	+	0.982716i	$0.559268\pi$
$930$	0	0
$931$	−6.34329e10	−2.57626
$932$	0	0
$933$	−1.25110e10	−0.504320
$934$	0	0
$935$	−1.66056e10	−0.664377
$936$	0	0
$937$	−7.58666e9	−0.301274	−0.150637	−	0.988589i	$-0.548133\pi$
$937$	−7.58666e9	−0.301274	−0.150637	+	0.988589i	$0.548133\pi$
$938$	0	0
$939$	2.68926e10	1.05999
$940$	0	0
$941$	−2.50996e10	−0.981981	−0.490991	−	0.871165i	$-0.663365\pi$
$941$	−2.50996e10	−0.981981	−0.490991	+	0.871165i	$0.663365\pi$
$942$	0	0
$943$	1.45808e10	0.566228
$944$	0	0
$945$	4.16498e9	0.160547
$946$	0	0
$947$	1.59653e10	0.610875	0.305438	−	0.952212i	$-0.401197\pi$
$947$	1.59653e10	0.610875	0.305438	+	0.952212i	$0.401197\pi$
$948$	0	0
$949$	5.14346e10	1.95355
$950$	0	0
$951$	3.48008e9	0.131207
$952$	0	0
$953$	−3.79999e10	−1.42219	−0.711094	−	0.703097i	$-0.751800\pi$
$953$	−3.79999e10	−1.42219	−0.711094	+	0.703097i	$0.751800\pi$
$954$	0	0
$955$	1.85459e10	0.689027
$956$	0	0
$957$	−3.22284e10	−1.18863
$958$	0	0
$959$	8.23829e9	0.301628
$960$	0	0
$961$	−1.10792e10	−0.402694
$962$	0	0
$963$	1.53440e10	0.553664
$964$	0	0
$965$	−1.44260e10	−0.516772
$966$	0	0
$967$	4.67649e9	0.166313	0.0831567	−	0.996536i	$-0.473500\pi$
$967$	4.67649e9	0.166313	0.0831567	+	0.996536i	$0.473500\pi$
$968$	0	0
$969$	1.47467e10	0.520668
$970$	0	0
$971$	−4.09960e10	−1.43706	−0.718529	−	0.695497i	$-0.755184\pi$
$971$	−4.09960e10	−1.43706	−0.718529	+	0.695497i	$0.755184\pi$
$972$	0	0
$973$	−8.19601e10	−2.85238
$974$	0	0
$975$	−5.44715e9	−0.188215
$976$	0	0
$977$	3.28959e10	1.12852	0.564262	−	0.825596i	$-0.309161\pi$
$977$	3.28959e10	1.12852	0.564262	+	0.825596i	$0.309161\pi$
$978$	0	0
$979$	−8.47745e10	−2.88752
$980$	0	0
$981$	6.99766e9	0.236653
$982$	0	0
$983$	1.32118e10	0.443635	0.221817	−	0.975088i	$-0.428801\pi$
$983$	1.32118e10	0.443635	0.221817	+	0.975088i	$0.428801\pi$
$984$	0	0
$985$	−6.56772e9	−0.218972
$986$	0	0
$987$	1.33225e10	0.441038
$988$	0	0
$989$	6.51595e8	0.0214186
$990$	0	0
$991$	1.78336e10	0.582077	0.291038	−	0.956711i	$-0.405999\pi$
$991$	1.78336e10	0.582077	0.291038	+	0.956711i	$0.405999\pi$
$992$	0	0
$993$	1.17379e10	0.380426
$994$	0	0
$995$	−2.36810e10	−0.762113
$996$	0	0
$997$	−2.87838e10	−0.919845	−0.459923	−	0.887959i	$-0.652123\pi$
$997$	−2.87838e10	−0.919845	−0.459923	+	0.887959i	$0.652123\pi$
$998$	0	0
$999$	2.70051e9	0.0856972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.o.1.2		2
4.3	odd	2		120.8.a.g.1.1	✓	2
12.11	even	2		360.8.a.h.1.1		2
20.3	even	4		600.8.f.f.49.4		4
20.7	even	4		600.8.f.f.49.1		4
20.19	odd	2		600.8.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.8.a.g.1.1	✓	2	4.3	odd	2
240.8.a.o.1.2		2	1.1	even	1	trivial
360.8.a.h.1.1		2	12.11	even	2
600.8.a.g.1.2		2	20.19	odd	2
600.8.f.f.49.1		4	20.7	even	4
600.8.f.f.49.4		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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} -125.000 q^{5} +1692.82 q^{7} +729.000 q^{9} +7555.29 q^{11} +12911.8 q^{13} +3375.00 q^{15} +17583.1 q^{17} -31062.5 q^{19} -45706.2 q^{21} -82318.0 q^{23} +15625.0 q^{25} -19683.0 q^{27} +157988. q^{29} +128193. q^{31} -203993. q^{33} -211603. q^{35} -137200. q^{37} -348618. q^{39} -177128. q^{41} -7915.59 q^{43} -91125.0 q^{45} -291482. q^{47} +2.04211e6 q^{49} -474742. q^{51} -776917. q^{53} -944411. q^{55} +838687. q^{57} +919013. q^{59} +2.08842e6 q^{61} +1.23407e6 q^{63} -1.61397e6 q^{65} +1.52915e6 q^{67} +2.22259e6 q^{69} -2.74055e6 q^{71} +3.98355e6 q^{73} -421875. q^{75} +1.27898e7 q^{77} -3.98416e6 q^{79} +531441. q^{81} -5.01862e6 q^{83} -2.19788e6 q^{85} -4.26567e6 q^{87} -1.12205e7 q^{89} +2.18573e7 q^{91} -3.46121e6 q^{93} +3.88281e6 q^{95} -2.13699e6 q^{97} +5.50781e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} - 250 q^{5} + 896 q^{7} + 1458 q^{9} + 5152 q^{11} + 8396 q^{13} + 6750 q^{15} + 7780 q^{17} - 54656 q^{19} - 24192 q^{21} - 122312 q^{23} + 31250 q^{25} - 39366 q^{27} + 116804 q^{29} - 109592 q^{31}+ \cdots + 3755808 q^{99}+O(q^{100}) \) 2 * q - 54 * q^3 - 250 * q^5 + 896 * q^7 + 1458 * q^9 + 5152 * q^11 + 8396 * q^13 + 6750 * q^15 + 7780 * q^17 - 54656 * q^19 - 24192 * q^21 - 122312 * q^23 + 31250 * q^25 - 39366 * q^27 + 116804 * q^29 - 109592 * q^31 - 139104 * q^33 - 112000 * q^35 + 46764 * q^37 - 226692 * q^39 - 299484 * q^41 - 234920 * q^43 - 182250 * q^45 - 505784 * q^47 + 1853490 * q^49 - 210060 * q^51 + 791412 * q^53 - 644000 * q^55 + 1475712 * q^57 - 1179424 * q^59 + 3733676 * q^61 + 653184 * q^63 - 1049500 * q^65 - 1462888 * q^67 + 3302424 * q^69 - 2573184 * q^71 + 6000276 * q^73 - 843750 * q^75 + 14704768 * q^77 - 3399816 * q^79 + 1062882 * q^81 + 1016 * q^83 - 972500 * q^85 - 3153708 * q^87 - 7438476 * q^89 + 25455584 * q^91 + 2958984 * q^93 + 6832000 * q^95 + 10345220 * q^97 + 3755808 * q^99

Introduction

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