LMFDB - Embedded newform 156.8.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [156,8,Mod(1,156)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("156.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 156 = 2^{2} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	156.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-81] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.7320639755$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11078x - 379248 $ x^3 - x^2 - 11078*x - 379248
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-78.5628$ of defining polynomial
Character	$\chi$	$=$	156.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} +103.038 q^{5} +1545.57 q^{7} +729.000 q^{9} +2051.35 q^{11} +2197.00 q^{13} -2782.01 q^{15} +15739.1 q^{17} +287.921 q^{19} -41730.3 q^{21} -40106.6 q^{23} -67508.3 q^{25} -19683.0 q^{27} +87360.8 q^{29} -100323. q^{31} -55386.6 q^{33} +159251. q^{35} +225233. q^{37} -59319.0 q^{39} +239941. q^{41} -488162. q^{43} +75114.4 q^{45} +1.15547e6 q^{47} +1.56523e6 q^{49} -424954. q^{51} -146943. q^{53} +211367. q^{55} -7773.87 q^{57} -543986. q^{59} +702239. q^{61} +1.12672e6 q^{63} +226374. q^{65} +3.24888e6 q^{67} +1.08288e6 q^{69} +1.07234e6 q^{71} -1.53003e6 q^{73} +1.82272e6 q^{75} +3.17050e6 q^{77} +6.54521e6 q^{79} +531441. q^{81} +4.85528e6 q^{83} +1.62171e6 q^{85} -2.35874e6 q^{87} -9.33838e6 q^{89} +3.39561e6 q^{91} +2.70871e6 q^{93} +29666.7 q^{95} -1.27026e6 q^{97} +1.49544e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 81 q^{3} + 524 q^{5} + 722 q^{7} + 2187 q^{9} + 6752 q^{11} + 6591 q^{13} - 14148 q^{15} - 15418 q^{17} + 1826 q^{19} - 19494 q^{21} + 16536 q^{23} + 10513 q^{25} - 59049 q^{27} - 151194 q^{29} + 46338 q^{31}+ \cdots + 4922208 q^{99}+O(q^{100}) $ 3 * q - 81 * q^3 + 524 * q^5 + 722 * q^7 + 2187 * q^9 + 6752 * q^11 + 6591 * q^13 - 14148 * q^15 - 15418 * q^17 + 1826 * q^19 - 19494 * q^21 + 16536 * q^23 + 10513 * q^25 - 59049 * q^27 - 151194 * q^29 + 46338 * q^31 - 182304 * q^33 - 223864 * q^35 - 112378 * q^37 - 177957 * q^39 + 324944 * q^41 + 1282940 * q^43 + 381996 * q^45 + 917072 * q^47 + 559359 * q^49 + 416286 * q^51 + 2688314 * q^53 + 4049992 * q^55 - 49302 * q^57 + 3176620 * q^59 + 2165222 * q^61 + 526338 * q^63 + 1151228 * q^65 + 6699990 * q^67 - 446472 * q^69 + 7734556 * q^71 + 5784690 * q^73 - 283851 * q^75 - 2868232 * q^77 + 8426640 * q^79 + 1594323 * q^81 + 4126284 * q^83 - 8292584 * q^85 + 4082238 * q^87 + 5207756 * q^89 + 1586234 * q^91 - 1251126 * q^93 + 18113000 * q^95 - 7634614 * q^97 + 4922208 * 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
$5$	103.038	0.368638	0.184319	−	0.982866i	$-0.440992\pi$
$5$	103.038	0.368638	0.184319	+	0.982866i	$0.440992\pi$
$6$	0	0
$7$	1545.57	1.70312	0.851558	−	0.524260i	$-0.175658\pi$
$7$	1545.57	1.70312	0.851558	+	0.524260i	$0.175658\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	2051.35	0.464693	0.232347	−	0.972633i	$-0.425360\pi$
$11$	2051.35	0.464693	0.232347	+	0.972633i	$0.425360\pi$
$12$	0	0
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	0	0
$15$	−2782.01	−0.212834
$16$	0	0
$17$	15739.1	0.776976	0.388488	−	0.921454i	$-0.372998\pi$
$17$	15739.1	0.776976	0.388488	+	0.921454i	$0.372998\pi$
$18$	0	0
$19$	287.921	0.00963022	0.00481511	−	0.999988i	$-0.498467\pi$
$19$	287.921	0.00963022	0.00481511	+	0.999988i	$0.498467\pi$
$20$	0	0
$21$	−41730.3	−0.983295
$22$	0	0
$23$	−40106.6	−0.687335	−0.343667	−	0.939091i	$-0.611669\pi$
$23$	−40106.6	−0.687335	−0.343667	+	0.939091i	$0.611669\pi$
$24$	0	0
$25$	−67508.3	−0.864106
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	87360.8	0.665156	0.332578	−	0.943076i	$-0.392082\pi$
$29$	87360.8	0.665156	0.332578	+	0.943076i	$0.392082\pi$
$30$	0	0
$31$	−100323.	−0.604829	−0.302415	−	0.953177i	$-0.597793\pi$
$31$	−100323.	−0.604829	−0.302415	+	0.953177i	$0.597793\pi$
$32$	0	0
$33$	−55386.6	−0.268291
$34$	0	0
$35$	159251.	0.627834
$36$	0	0
$37$	225233.	0.731015	0.365508	−	0.930808i	$-0.380895\pi$
$37$	225233.	0.731015	0.365508	+	0.930808i	$0.380895\pi$
$38$	0	0
$39$	−59319.0	−0.160128
$40$	0	0
$41$	239941.	0.543702	0.271851	−	0.962339i	$-0.412364\pi$
$41$	239941.	0.543702	0.271851	+	0.962339i	$0.412364\pi$
$42$	0	0
$43$	−488162.	−0.936321	−0.468160	−	0.883644i	$-0.655083\pi$
$43$	−488162.	−0.936321	−0.468160	+	0.883644i	$0.655083\pi$
$44$	0	0
$45$	75114.4	0.122879
$46$	0	0
$47$	1.15547e6	1.62337	0.811683	−	0.584098i	$-0.198552\pi$
$47$	1.15547e6	1.62337	0.811683	+	0.584098i	$0.198552\pi$
$48$	0	0
$49$	1.56523e6	1.90061
$50$	0	0
$51$	−424954.	−0.448587
$52$	0	0
$53$	−146943.	−0.135576	−0.0677882	−	0.997700i	$-0.521594\pi$
$53$	−146943.	−0.135576	−0.0677882	+	0.997700i	$0.521594\pi$
$54$	0	0
$55$	211367.	0.171304
$56$	0	0
$57$	−7773.87	−0.00556001
$58$	0	0
$59$	−543986.	−0.344831	−0.172415	−	0.985024i	$-0.555157\pi$
$59$	−543986.	−0.344831	−0.172415	+	0.985024i	$0.555157\pi$
$60$	0	0
$61$	702239.	0.396123	0.198062	−	0.980190i	$-0.436535\pi$
$61$	702239.	0.396123	0.198062	+	0.980190i	$0.436535\pi$
$62$	0	0
$63$	1.12672e6	0.567706
$64$	0	0
$65$	226374.	0.102242
$66$	0	0
$67$	3.24888e6	1.31969	0.659845	−	0.751402i	$-0.270622\pi$
$67$	3.24888e6	1.31969	0.659845	+	0.751402i	$0.270622\pi$
$68$	0	0
$69$	1.08288e6	0.396833
$70$	0	0
$71$	1.07234e6	0.355572	0.177786	−	0.984069i	$-0.443106\pi$
$71$	1.07234e6	0.355572	0.177786	+	0.984069i	$0.443106\pi$
$72$	0	0
$73$	−1.53003e6	−0.460331	−0.230165	−	0.973152i	$-0.573927\pi$
$73$	−1.53003e6	−0.460331	−0.230165	+	0.973152i	$0.573927\pi$
$74$	0	0
$75$	1.82272e6	0.498892
$76$	0	0
$77$	3.17050e6	0.791427
$78$	0	0
$79$	6.54521e6	1.49358	0.746790	−	0.665060i	$-0.231594\pi$
$79$	6.54521e6	1.49358	0.746790	+	0.665060i	$0.231594\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	4.85528e6	0.932055	0.466027	−	0.884770i	$-0.345685\pi$
$83$	4.85528e6	0.932055	0.466027	+	0.884770i	$0.345685\pi$
$84$	0	0
$85$	1.62171e6	0.286423
$86$	0	0
$87$	−2.35874e6	−0.384028
$88$	0	0
$89$	−9.33838e6	−1.40413	−0.702063	−	0.712114i	$-0.747738\pi$
$89$	−9.33838e6	−1.40413	−0.702063	+	0.712114i	$0.747738\pi$
$90$	0	0
$91$	3.39561e6	0.472360
$92$	0	0
$93$	2.70871e6	0.349198
$94$	0	0
$95$	29666.7	0.00355007
$96$	0	0
$97$	−1.27026e6	−0.141316	−0.0706580	−	0.997501i	$-0.522510\pi$
$97$	−1.27026e6	−0.141316	−0.0706580	+	0.997501i	$0.522510\pi$
$98$	0	0
$99$	1.49544e6	0.154898
$100$	0	0
$101$	1.88833e6	0.182369	0.0911847	−	0.995834i	$-0.470935\pi$
$101$	1.88833e6	0.182369	0.0911847	+	0.995834i	$0.470935\pi$
$102$	0	0
$103$	−6.75236e6	−0.608871	−0.304436	−	0.952533i	$-0.598468\pi$
$103$	−6.75236e6	−0.608871	−0.304436	+	0.952533i	$0.598468\pi$
$104$	0	0
$105$	−4.29979e6	−0.362480
$106$	0	0
$107$	2.15607e7	1.70145	0.850727	−	0.525607i	$-0.176162\pi$
$107$	2.15607e7	1.70145	0.850727	+	0.525607i	$0.176162\pi$
$108$	0	0
$109$	−4.31002e6	−0.318777	−0.159388	−	0.987216i	$-0.550952\pi$
$109$	−4.31002e6	−0.318777	−0.159388	+	0.987216i	$0.550952\pi$
$110$	0	0
$111$	−6.08130e6	−0.422052
$112$	0	0
$113$	1.32045e7	0.860888	0.430444	−	0.902617i	$-0.358357\pi$
$113$	1.32045e7	0.860888	0.430444	+	0.902617i	$0.358357\pi$
$114$	0	0
$115$	−4.13249e6	−0.253378
$116$	0	0
$117$	1.60161e6	0.0924500
$118$	0	0
$119$	2.43257e7	1.32328
$120$	0	0
$121$	−1.52791e7	−0.784060
$122$	0	0
$123$	−6.47841e6	−0.313906
$124$	0	0
$125$	−1.50057e7	−0.687181
$126$	0	0
$127$	1.22368e7	0.530095	0.265048	−	0.964235i	$-0.414612\pi$
$127$	1.22368e7	0.530095	0.265048	+	0.964235i	$0.414612\pi$
$128$	0	0
$129$	1.31804e7	0.540585
$130$	0	0
$131$	4.32414e7	1.68055	0.840273	−	0.542163i	$-0.182395\pi$
$131$	4.32414e7	1.68055	0.840273	+	0.542163i	$0.182395\pi$
$132$	0	0
$133$	445001.	0.0164014
$134$	0	0
$135$	−2.02809e6	−0.0709445
$136$	0	0
$137$	1.98598e7	0.659862	0.329931	−	0.944005i	$-0.392975\pi$
$137$	1.98598e7	0.659862	0.329931	+	0.944005i	$0.392975\pi$
$138$	0	0
$139$	−1.22852e7	−0.387998	−0.193999	−	0.981002i	$-0.562146\pi$
$139$	−1.22852e7	−0.387998	−0.193999	+	0.981002i	$0.562146\pi$
$140$	0	0
$141$	−3.11977e7	−0.937251
$142$	0	0
$143$	4.50683e6	0.128883
$144$	0	0
$145$	9.00144e6	0.245202
$146$	0	0
$147$	−4.22612e7	−1.09732
$148$	0	0
$149$	−4.27992e7	−1.05995	−0.529973	−	0.848014i	$-0.677798\pi$
$149$	−4.27992e7	−1.05995	−0.529973	+	0.848014i	$0.677798\pi$
$150$	0	0
$151$	−5.77012e7	−1.36385	−0.681924	−	0.731423i	$-0.738856\pi$
$151$	−5.77012e7	−1.36385	−0.681924	+	0.731423i	$0.738856\pi$
$152$	0	0
$153$	1.14738e7	0.258992
$154$	0	0
$155$	−1.03370e7	−0.222963
$156$	0	0
$157$	2.34313e7	0.483223	0.241612	−	0.970373i	$-0.422324\pi$
$157$	2.34313e7	0.483223	0.241612	+	0.970373i	$0.422324\pi$
$158$	0	0
$159$	3.96746e6	0.0782750
$160$	0	0
$161$	−6.19874e7	−1.17061
$162$	0	0
$163$	−1.20613e7	−0.218140	−0.109070	−	0.994034i	$-0.534787\pi$
$163$	−1.20613e7	−0.218140	−0.109070	+	0.994034i	$0.534787\pi$
$164$	0	0
$165$	−5.70690e6	−0.0989023
$166$	0	0
$167$	−9.11187e7	−1.51391	−0.756955	−	0.653467i	$-0.773314\pi$
$167$	−9.11187e7	−1.51391	−0.756955	+	0.653467i	$0.773314\pi$
$168$	0	0
$169$	4.82681e6	0.0769231
$170$	0	0
$171$	209895.	0.00321007
$172$	0	0
$173$	−4.60142e7	−0.675663	−0.337832	−	0.941207i	$-0.609693\pi$
$173$	−4.60142e7	−0.675663	−0.337832	+	0.941207i	$0.609693\pi$
$174$	0	0
$175$	−1.04338e8	−1.47167
$176$	0	0
$177$	1.46876e7	0.199088
$178$	0	0
$179$	1.24534e8	1.62294	0.811472	−	0.584391i	$-0.198667\pi$
$179$	1.24534e8	1.62294	0.811472	+	0.584391i	$0.198667\pi$
$180$	0	0
$181$	893882.	0.0112048	0.00560241	−	0.999984i	$-0.498217\pi$
$181$	893882.	0.0112048	0.00560241	+	0.999984i	$0.498217\pi$
$182$	0	0
$183$	−1.89604e7	−0.228702
$184$	0	0
$185$	2.32075e7	0.269480
$186$	0	0
$187$	3.22864e7	0.361055
$188$	0	0
$189$	−3.04214e7	−0.327765
$190$	0	0
$191$	−3.30297e7	−0.342995	−0.171497	−	0.985185i	$-0.554860\pi$
$191$	−3.30297e7	−0.342995	−0.171497	+	0.985185i	$0.554860\pi$
$192$	0	0
$193$	−9.03252e7	−0.904395	−0.452198	−	0.891918i	$-0.649360\pi$
$193$	−9.03252e7	−0.904395	−0.452198	+	0.891918i	$0.649360\pi$
$194$	0	0
$195$	−6.11209e6	−0.0590294
$196$	0	0
$197$	3.80945e7	0.355001	0.177501	−	0.984121i	$-0.443199\pi$
$197$	3.80945e7	0.355001	0.177501	+	0.984121i	$0.443199\pi$
$198$	0	0
$199$	−5.69352e7	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$199$	−5.69352e7	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$200$	0	0
$201$	−8.77198e7	−0.761924
$202$	0	0
$203$	1.35022e8	1.13284
$204$	0	0
$205$	2.47229e7	0.200429
$206$	0	0
$207$	−2.92377e7	−0.229112
$208$	0	0
$209$	590629.	0.00447510
$210$	0	0
$211$	−1.39694e8	−1.02374	−0.511870	−	0.859063i	$-0.671047\pi$
$211$	−1.39694e8	−1.02374	−0.511870	+	0.859063i	$0.671047\pi$
$212$	0	0
$213$	−2.89532e7	−0.205290
$214$	0	0
$215$	−5.02991e7	−0.345164
$216$	0	0
$217$	−1.55055e8	−1.03009
$218$	0	0
$219$	4.13108e7	0.265772
$220$	0	0
$221$	3.45787e7	0.215494
$222$	0	0
$223$	1.07307e8	0.647980	0.323990	−	0.946060i	$-0.394976\pi$
$223$	1.07307e8	0.647980	0.323990	+	0.946060i	$0.394976\pi$
$224$	0	0
$225$	−4.92135e7	−0.288035
$226$	0	0
$227$	2.17396e8	1.23356	0.616782	−	0.787134i	$-0.288436\pi$
$227$	2.17396e8	1.23356	0.616782	+	0.787134i	$0.288436\pi$
$228$	0	0
$229$	−2.60127e8	−1.43140	−0.715700	−	0.698408i	$-0.753892\pi$
$229$	−2.60127e8	−1.43140	−0.715700	+	0.698408i	$0.753892\pi$
$230$	0	0
$231$	−8.56036e7	−0.456931
$232$	0	0
$233$	1.11100e8	0.575400	0.287700	−	0.957721i	$-0.407109\pi$
$233$	1.11100e8	0.575400	0.287700	+	0.957721i	$0.407109\pi$
$234$	0	0
$235$	1.19057e8	0.598435
$236$	0	0
$237$	−1.76721e8	−0.862319
$238$	0	0
$239$	3.33752e8	1.58136	0.790681	−	0.612229i	$-0.209727\pi$
$239$	3.33752e8	1.58136	0.790681	+	0.612229i	$0.209727\pi$
$240$	0	0
$241$	−1.04044e7	−0.0478802	−0.0239401	−	0.999713i	$-0.507621\pi$
$241$	−1.04044e7	−0.0478802	−0.0239401	+	0.999713i	$0.507621\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	1.61278e8	0.700636
$246$	0	0
$247$	632563.	0.00267094
$248$	0	0
$249$	−1.31093e8	−0.538122
$250$	0	0
$251$	3.80790e8	1.51994	0.759972	−	0.649956i	$-0.225213\pi$
$251$	3.80790e8	1.51994	0.759972	+	0.649956i	$0.225213\pi$
$252$	0	0
$253$	−8.22728e7	−0.319400
$254$	0	0
$255$	−4.37863e7	−0.165366
$256$	0	0
$257$	1.61752e8	0.594406	0.297203	−	0.954814i	$-0.403946\pi$
$257$	1.61752e8	0.594406	0.297203	+	0.954814i	$0.403946\pi$
$258$	0	0
$259$	3.48113e8	1.24500
$260$	0	0
$261$	6.36860e7	0.221719
$262$	0	0
$263$	1.92014e8	0.650859	0.325430	−	0.945566i	$-0.394491\pi$
$263$	1.92014e8	0.650859	0.325430	+	0.945566i	$0.394491\pi$
$264$	0	0
$265$	−1.51407e7	−0.0499786
$266$	0	0
$267$	2.52136e8	0.810673
$268$	0	0
$269$	3.90213e8	1.22227	0.611137	−	0.791525i	$-0.290712\pi$
$269$	3.90213e8	1.22227	0.611137	+	0.791525i	$0.290712\pi$
$270$	0	0
$271$	4.80310e7	0.146598	0.0732992	−	0.997310i	$-0.476647\pi$
$271$	4.80310e7	0.146598	0.0732992	+	0.997310i	$0.476647\pi$
$272$	0	0
$273$	−9.16814e7	−0.272717
$274$	0	0
$275$	−1.38483e8	−0.401544
$276$	0	0
$277$	4.36165e8	1.23302	0.616512	−	0.787346i	$-0.288545\pi$
$277$	4.36165e8	1.23302	0.616512	+	0.787346i	$0.288545\pi$
$278$	0	0
$279$	−7.31351e7	−0.201610
$280$	0	0
$281$	4.10047e8	1.10246	0.551229	−	0.834354i	$-0.314159\pi$
$281$	4.10047e8	1.10246	0.551229	+	0.834354i	$0.314159\pi$
$282$	0	0
$283$	−2.90967e8	−0.763118	−0.381559	−	0.924345i	$-0.624613\pi$
$283$	−2.90967e8	−0.763118	−0.381559	+	0.924345i	$0.624613\pi$
$284$	0	0
$285$	−801001.	−0.00204963
$286$	0	0
$287$	3.70845e8	0.925988
$288$	0	0
$289$	−1.62621e8	−0.396309
$290$	0	0
$291$	3.42970e7	0.0815889
$292$	0	0
$293$	−5.24519e8	−1.21822	−0.609109	−	0.793087i	$-0.708473\pi$
$293$	−5.24519e8	−1.21822	−0.609109	+	0.793087i	$0.708473\pi$
$294$	0	0
$295$	−5.60510e7	−0.127118
$296$	0	0
$297$	−4.03768e7	−0.0894303
$298$	0	0
$299$	−8.81142e7	−0.190632
$300$	0	0
$301$	−7.54487e8	−1.59466
$302$	0	0
$303$	−5.09848e7	−0.105291
$304$	0	0
$305$	7.23570e7	0.146026
$306$	0	0
$307$	8.33623e8	1.64432	0.822158	−	0.569260i	$-0.192770\pi$
$307$	8.33623e8	1.64432	0.822158	+	0.569260i	$0.192770\pi$
$308$	0	0
$309$	1.82314e8	0.351532
$310$	0	0
$311$	−1.01024e9	−1.90442	−0.952210	−	0.305444i	$-0.901195\pi$
$311$	−1.01024e9	−1.90442	−0.952210	+	0.305444i	$0.901195\pi$
$312$	0	0
$313$	−2.64485e8	−0.487524	−0.243762	−	0.969835i	$-0.578382\pi$
$313$	−2.64485e8	−0.487524	−0.243762	+	0.969835i	$0.578382\pi$
$314$	0	0
$315$	1.16094e8	0.209278
$316$	0	0
$317$	−1.85507e8	−0.327079	−0.163539	−	0.986537i	$-0.552291\pi$
$317$	−1.85507e8	−0.327079	−0.163539	+	0.986537i	$0.552291\pi$
$318$	0	0
$319$	1.79208e8	0.309094
$320$	0	0
$321$	−5.82140e8	−0.982335
$322$	0	0
$323$	4.53161e6	0.00748244
$324$	0	0
$325$	−1.48316e8	−0.239660
$326$	0	0
$327$	1.16371e8	0.184046
$328$	0	0
$329$	1.78586e9	2.76478
$330$	0	0
$331$	7.44872e7	0.112897	0.0564486	−	0.998406i	$-0.482022\pi$
$331$	7.44872e7	0.112897	0.0564486	+	0.998406i	$0.482022\pi$
$332$	0	0
$333$	1.64195e8	0.243672
$334$	0	0
$335$	3.34757e8	0.486489
$336$	0	0
$337$	4.11657e7	0.0585910	0.0292955	−	0.999571i	$-0.490674\pi$
$337$	4.11657e7	0.0585910	0.0292955	+	0.999571i	$0.490674\pi$
$338$	0	0
$339$	−3.56521e8	−0.497034
$340$	0	0
$341$	−2.05797e8	−0.281060
$342$	0	0
$343$	1.14633e9	1.53384
$344$	0	0
$345$	1.11577e8	0.146288
$346$	0	0
$347$	1.99660e8	0.256530	0.128265	−	0.991740i	$-0.459059\pi$
$347$	1.99660e8	0.256530	0.128265	+	0.991740i	$0.459059\pi$
$348$	0	0
$349$	6.16798e8	0.776702	0.388351	−	0.921512i	$-0.373045\pi$
$349$	6.16798e8	0.776702	0.388351	+	0.921512i	$0.373045\pi$
$350$	0	0
$351$	−4.32436e7	−0.0533761
$352$	0	0
$353$	−1.15496e8	−0.139751	−0.0698755	−	0.997556i	$-0.522260\pi$
$353$	−1.15496e8	−0.139751	−0.0698755	+	0.997556i	$0.522260\pi$
$354$	0	0
$355$	1.10491e8	0.131078
$356$	0	0
$357$	−6.56795e8	−0.763996
$358$	0	0
$359$	−2.07789e8	−0.237024	−0.118512	−	0.992953i	$-0.537812\pi$
$359$	−2.07789e8	−0.237024	−0.118512	+	0.992953i	$0.537812\pi$
$360$	0	0
$361$	−8.93789e8	−0.999907
$362$	0	0
$363$	4.12536e8	0.452677
$364$	0	0
$365$	−1.57651e8	−0.169696
$366$	0	0
$367$	2.05412e8	0.216917	0.108459	−	0.994101i	$-0.465409\pi$
$367$	2.05412e8	0.216917	0.108459	+	0.994101i	$0.465409\pi$
$368$	0	0
$369$	1.74917e8	0.181234
$370$	0	0
$371$	−2.27110e8	−0.230902
$372$	0	0
$373$	−1.21011e9	−1.20738	−0.603689	−	0.797220i	$-0.706303\pi$
$373$	−1.21011e9	−1.20738	−0.603689	+	0.797220i	$0.706303\pi$
$374$	0	0
$375$	4.05154e8	0.396744
$376$	0	0
$377$	1.91932e8	0.184481
$378$	0	0
$379$	−1.36346e9	−1.28649	−0.643244	−	0.765661i	$-0.722412\pi$
$379$	−1.36346e9	−1.28649	−0.643244	+	0.765661i	$0.722412\pi$
$380$	0	0
$381$	−3.30393e8	−0.306051
$382$	0	0
$383$	1.63252e9	1.48479	0.742393	−	0.669964i	$-0.233691\pi$
$383$	1.63252e9	1.48479	0.742393	+	0.669964i	$0.233691\pi$
$384$	0	0
$385$	3.26681e8	0.291750
$386$	0	0
$387$	−3.55870e8	−0.312107
$388$	0	0
$389$	−1.94704e9	−1.67707	−0.838533	−	0.544851i	$-0.816586\pi$
$389$	−1.94704e9	−1.67707	−0.838533	+	0.544851i	$0.816586\pi$
$390$	0	0
$391$	−6.31240e8	−0.534042
$392$	0	0
$393$	−1.16752e9	−0.970264
$394$	0	0
$395$	6.74402e8	0.550591
$396$	0	0
$397$	−6.02131e8	−0.482975	−0.241487	−	0.970404i	$-0.577635\pi$
$397$	−6.02131e8	−0.482975	−0.241487	+	0.970404i	$0.577635\pi$
$398$	0	0
$399$	−1.20150e7	−0.00946934
$400$	0	0
$401$	−2.14476e9	−1.66102	−0.830509	−	0.557005i	$-0.811950\pi$
$401$	−2.14476e9	−1.66102	−0.830509	+	0.557005i	$0.811950\pi$
$402$	0	0
$403$	−2.20409e8	−0.167749
$404$	0	0
$405$	5.47584e7	0.0409598
$406$	0	0
$407$	4.62033e8	0.339698
$408$	0	0
$409$	−2.14085e9	−1.54723	−0.773616	−	0.633655i	$-0.781554\pi$
$409$	−2.14085e9	−1.54723	−0.773616	+	0.633655i	$0.781554\pi$
$410$	0	0
$411$	−5.36215e8	−0.380971
$412$	0	0
$413$	−8.40767e8	−0.587287
$414$	0	0
$415$	5.00277e8	0.343591
$416$	0	0
$417$	3.31700e8	0.224011
$418$	0	0
$419$	1.32017e9	0.876762	0.438381	−	0.898789i	$-0.355552\pi$
$419$	1.32017e9	0.876762	0.438381	+	0.898789i	$0.355552\pi$
$420$	0	0
$421$	2.87412e8	0.187723	0.0938616	−	0.995585i	$-0.470079\pi$
$421$	2.87412e8	0.187723	0.0938616	+	0.995585i	$0.470079\pi$
$422$	0	0
$423$	8.42338e8	0.541122
$424$	0	0
$425$	−1.06252e9	−0.671389
$426$	0	0
$427$	1.08536e9	0.674644
$428$	0	0
$429$	−1.21684e8	−0.0744105
$430$	0	0
$431$	−4.17009e8	−0.250885	−0.125443	−	0.992101i	$-0.540035\pi$
$431$	−4.17009e8	−0.250885	−0.125443	+	0.992101i	$0.540035\pi$
$432$	0	0
$433$	−2.79223e9	−1.65289	−0.826446	−	0.563017i	$-0.809641\pi$
$433$	−2.79223e9	−1.65289	−0.826446	+	0.563017i	$0.809641\pi$
$434$	0	0
$435$	−2.43039e8	−0.141567
$436$	0	0
$437$	−1.15475e7	−0.00661918
$438$	0	0
$439$	2.80847e9	1.58432	0.792161	−	0.610312i	$-0.208956\pi$
$439$	2.80847e9	1.58432	0.792161	+	0.610312i	$0.208956\pi$
$440$	0	0
$441$	1.14105e9	0.633535
$442$	0	0
$443$	−1.71891e9	−0.939379	−0.469689	−	0.882832i	$-0.655634\pi$
$443$	−1.71891e9	−0.939379	−0.469689	+	0.882832i	$0.655634\pi$
$444$	0	0
$445$	−9.62204e8	−0.517615
$446$	0	0
$447$	1.15558e9	0.611960
$448$	0	0
$449$	−2.28347e9	−1.19051	−0.595254	−	0.803538i	$-0.702949\pi$
$449$	−2.28347e9	−1.19051	−0.595254	+	0.803538i	$0.702949\pi$
$450$	0	0
$451$	4.92204e8	0.252655
$452$	0	0
$453$	1.55793e9	0.787418
$454$	0	0
$455$	3.49875e8	0.174130
$456$	0	0
$457$	1.12197e9	0.549888	0.274944	−	0.961460i	$-0.411341\pi$
$457$	1.12197e9	0.549888	0.274944	+	0.961460i	$0.411341\pi$
$458$	0	0
$459$	−3.09792e8	−0.149529
$460$	0	0
$461$	−9.98918e8	−0.474872	−0.237436	−	0.971403i	$-0.576307\pi$
$461$	−9.98918e8	−0.474872	−0.237436	+	0.971403i	$0.576307\pi$
$462$	0	0
$463$	−3.42597e9	−1.60417	−0.802085	−	0.597210i	$-0.796276\pi$
$463$	−3.42597e9	−1.60417	−0.802085	+	0.597210i	$0.796276\pi$
$464$	0	0
$465$	2.79099e8	0.128728
$466$	0	0
$467$	−3.78896e9	−1.72152	−0.860758	−	0.509015i	$-0.830010\pi$
$467$	−3.78896e9	−1.72152	−0.860758	+	0.509015i	$0.830010\pi$
$468$	0	0
$469$	5.02136e9	2.24759
$470$	0	0
$471$	−6.32646e8	−0.278989
$472$	0	0
$473$	−1.00139e9	−0.435102
$474$	0	0
$475$	−1.94371e7	−0.00832153
$476$	0	0
$477$	−1.07122e8	−0.0451921
$478$	0	0
$479$	−3.72870e9	−1.55019	−0.775093	−	0.631848i	$-0.782297\pi$
$479$	−3.72870e9	−1.55019	−0.775093	+	0.631848i	$0.782297\pi$
$480$	0	0
$481$	4.94837e8	0.202747
$482$	0	0
$483$	1.67366e9	0.675853
$484$	0	0
$485$	−1.30885e8	−0.0520945
$486$	0	0
$487$	−1.52815e9	−0.599537	−0.299768	−	0.954012i	$-0.596909\pi$
$487$	−1.52815e9	−0.599537	−0.299768	+	0.954012i	$0.596909\pi$
$488$	0	0
$489$	3.25654e8	0.125943
$490$	0	0
$491$	7.43741e7	0.0283555	0.0141777	−	0.999899i	$-0.495487\pi$
$491$	7.43741e7	0.0283555	0.0141777	+	0.999899i	$0.495487\pi$
$492$	0	0
$493$	1.37498e9	0.516810
$494$	0	0
$495$	1.54086e8	0.0571013
$496$	0	0
$497$	1.65737e9	0.605581
$498$	0	0
$499$	4.43694e9	1.59857	0.799285	−	0.600952i	$-0.205212\pi$
$499$	4.43694e9	1.59857	0.799285	+	0.600952i	$0.205212\pi$
$500$	0	0
$501$	2.46021e9	0.874056
$502$	0	0
$503$	3.92561e8	0.137537	0.0687684	−	0.997633i	$-0.478093\pi$
$503$	3.92561e8	0.137537	0.0687684	+	0.997633i	$0.478093\pi$
$504$	0	0
$505$	1.94569e8	0.0672284
$506$	0	0
$507$	−1.30324e8	−0.0444116
$508$	0	0
$509$	−2.20670e9	−0.741704	−0.370852	−	0.928692i	$-0.620934\pi$
$509$	−2.20670e9	−0.741704	−0.370852	+	0.928692i	$0.620934\pi$
$510$	0	0
$511$	−2.36476e9	−0.783997
$512$	0	0
$513$	−5.66715e6	−0.00185334
$514$	0	0
$515$	−6.95747e8	−0.224453
$516$	0	0
$517$	2.37028e9	0.754367
$518$	0	0
$519$	1.24238e9	0.390094
$520$	0	0
$521$	6.06878e9	1.88005	0.940025	−	0.341106i	$-0.110802\pi$
$521$	6.06878e9	1.88005	0.940025	+	0.341106i	$0.110802\pi$
$522$	0	0
$523$	4.87009e9	1.48861	0.744305	−	0.667840i	$-0.232781\pi$
$523$	4.87009e9	1.48861	0.744305	+	0.667840i	$0.232781\pi$
$524$	0	0
$525$	2.81714e9	0.849671
$526$	0	0
$527$	−1.57898e9	−0.469937
$528$	0	0
$529$	−1.79629e9	−0.527571
$530$	0	0
$531$	−3.96566e8	−0.114944
$532$	0	0
$533$	5.27150e8	0.150796
$534$	0	0
$535$	2.22157e9	0.627222
$536$	0	0
$537$	−3.36243e9	−0.937007
$538$	0	0
$539$	3.21084e9	0.883199
$540$	0	0
$541$	4.82787e9	1.31089	0.655443	−	0.755244i	$-0.272482\pi$
$541$	4.82787e9	1.31089	0.655443	+	0.755244i	$0.272482\pi$
$542$	0	0
$543$	−2.41348e7	−0.00646911
$544$	0	0
$545$	−4.44094e8	−0.117513
$546$	0	0
$547$	−2.28441e9	−0.596787	−0.298393	−	0.954443i	$-0.596451\pi$
$547$	−2.28441e9	−0.596787	−0.298393	+	0.954443i	$0.596451\pi$
$548$	0	0
$549$	5.11932e8	0.132041
$550$	0	0
$551$	2.51530e7	0.00640560
$552$	0	0
$553$	1.01160e10	2.54374
$554$	0	0
$555$	−6.26602e8	−0.155585
$556$	0	0
$557$	−3.69402e9	−0.905745	−0.452872	−	0.891575i	$-0.649601\pi$
$557$	−3.69402e9	−0.905745	−0.452872	+	0.891575i	$0.649601\pi$
$558$	0	0
$559$	−1.07249e9	−0.259689
$560$	0	0
$561$	−8.71732e8	−0.208455
$562$	0	0
$563$	−5.92280e9	−1.39877	−0.699387	−	0.714743i	$-0.746544\pi$
$563$	−5.92280e9	−1.39877	−0.699387	+	0.714743i	$0.746544\pi$
$564$	0	0
$565$	1.36056e9	0.317356
$566$	0	0
$567$	8.21377e8	0.189235
$568$	0	0
$569$	6.76988e9	1.54059	0.770297	−	0.637685i	$-0.220108\pi$
$569$	6.76988e9	1.54059	0.770297	+	0.637685i	$0.220108\pi$
$570$	0	0
$571$	1.44850e9	0.325605	0.162802	−	0.986659i	$-0.447947\pi$
$571$	1.44850e9	0.325605	0.162802	+	0.986659i	$0.447947\pi$
$572$	0	0
$573$	8.91801e8	0.198028
$574$	0	0
$575$	2.70753e9	0.593930
$576$	0	0
$577$	−7.28581e9	−1.57893	−0.789464	−	0.613796i	$-0.789642\pi$
$577$	−7.28581e9	−1.57893	−0.789464	+	0.613796i	$0.789642\pi$
$578$	0	0
$579$	2.43878e9	0.522153
$580$	0	0
$581$	7.50416e9	1.58740
$582$	0	0
$583$	−3.01433e8	−0.0630014
$584$	0	0
$585$	1.65026e8	0.0340806
$586$	0	0
$587$	−1.29361e9	−0.263980	−0.131990	−	0.991251i	$-0.542137\pi$
$587$	−1.29361e9	−0.263980	−0.131990	+	0.991251i	$0.542137\pi$
$588$	0	0
$589$	−2.88850e7	−0.00582464
$590$	0	0
$591$	−1.02855e9	−0.204960
$592$	0	0
$593$	−3.11678e9	−0.613783	−0.306892	−	0.951744i	$-0.599289\pi$
$593$	−3.11678e9	−0.613783	−0.306892	+	0.951744i	$0.599289\pi$
$594$	0	0
$595$	2.50647e9	0.487812
$596$	0	0
$597$	1.53725e9	0.295689
$598$	0	0
$599$	−8.17385e8	−0.155393	−0.0776967	−	0.996977i	$-0.524757\pi$
$599$	−8.17385e8	−0.155393	−0.0776967	+	0.996977i	$0.524757\pi$
$600$	0	0
$601$	−6.03603e9	−1.13420	−0.567101	−	0.823648i	$-0.691935\pi$
$601$	−6.03603e9	−1.13420	−0.567101	+	0.823648i	$0.691935\pi$
$602$	0	0
$603$	2.36843e9	0.439897
$604$	0	0
$605$	−1.57432e9	−0.289035
$606$	0	0
$607$	−4.92208e9	−0.893282	−0.446641	−	0.894713i	$-0.647380\pi$
$607$	−4.92208e9	−0.893282	−0.446641	+	0.894713i	$0.647380\pi$
$608$	0	0
$609$	−3.64559e9	−0.654044
$610$	0	0
$611$	2.53857e9	0.450241
$612$	0	0
$613$	−4.71965e9	−0.827557	−0.413778	−	0.910378i	$-0.635791\pi$
$613$	−4.71965e9	−0.827557	−0.413778	+	0.910378i	$0.635791\pi$
$614$	0	0
$615$	−6.67519e8	−0.115718
$616$	0	0
$617$	1.04505e9	0.179117	0.0895586	−	0.995982i	$-0.471454\pi$
$617$	1.04505e9	0.179117	0.0895586	+	0.995982i	$0.471454\pi$
$618$	0	0
$619$	−7.05814e8	−0.119612	−0.0598058	−	0.998210i	$-0.519048\pi$
$619$	−7.05814e8	−0.119612	−0.0598058	+	0.998210i	$0.519048\pi$
$620$	0	0
$621$	7.89418e8	0.132278
$622$	0	0
$623$	−1.44331e10	−2.39139
$624$	0	0
$625$	3.72793e9	0.610784
$626$	0	0
$627$	−1.59470e7	−0.00258370
$628$	0	0
$629$	3.54496e9	0.567981
$630$	0	0
$631$	4.33706e8	0.0687215	0.0343607	−	0.999409i	$-0.489060\pi$
$631$	4.33706e8	0.0687215	0.0343607	+	0.999409i	$0.489060\pi$
$632$	0	0
$633$	3.77174e9	0.591056
$634$	0	0
$635$	1.26085e9	0.195413
$636$	0	0
$637$	3.43881e9	0.527133
$638$	0	0
$639$	7.81735e8	0.118524
$640$	0	0
$641$	4.81614e9	0.722264	0.361132	−	0.932515i	$-0.382390\pi$
$641$	4.81614e9	0.722264	0.361132	+	0.932515i	$0.382390\pi$
$642$	0	0
$643$	1.05505e10	1.56507	0.782535	−	0.622607i	$-0.213926\pi$
$643$	1.05505e10	1.56507	0.782535	+	0.622607i	$0.213926\pi$
$644$	0	0
$645$	1.35807e9	0.199280
$646$	0	0
$647$	−1.02842e10	−1.49282	−0.746410	−	0.665486i	$-0.768224\pi$
$647$	−1.02842e10	−1.49282	−0.746410	+	0.665486i	$0.768224\pi$
$648$	0	0
$649$	−1.11591e9	−0.160241
$650$	0	0
$651$	4.18649e9	0.594725
$652$	0	0
$653$	−1.06008e10	−1.48984	−0.744922	−	0.667152i	$-0.767513\pi$
$653$	−1.06008e10	−1.48984	−0.744922	+	0.667152i	$0.767513\pi$
$654$	0	0
$655$	4.45549e9	0.619514
$656$	0	0
$657$	−1.11539e9	−0.153444
$658$	0	0
$659$	−3.01390e9	−0.410232	−0.205116	−	0.978738i	$-0.565757\pi$
$659$	−3.01390e9	−0.410232	−0.205116	+	0.978738i	$0.565757\pi$
$660$	0	0
$661$	7.70634e9	1.03787	0.518935	−	0.854814i	$-0.326329\pi$
$661$	7.70634e9	1.03787	0.518935	+	0.854814i	$0.326329\pi$
$662$	0	0
$663$	−9.33625e8	−0.124416
$664$	0	0
$665$	4.58519e7	0.00604618
$666$	0	0
$667$	−3.50374e9	−0.457185
$668$	0	0
$669$	−2.89729e9	−0.374111
$670$	0	0
$671$	1.44054e9	0.184076
$672$	0	0
$673$	−5.70284e9	−0.721171	−0.360586	−	0.932726i	$-0.617423\pi$
$673$	−5.70284e9	−0.721171	−0.360586	+	0.932726i	$0.617423\pi$
$674$	0	0
$675$	1.32877e9	0.166297
$676$	0	0
$677$	1.46189e9	0.181073	0.0905366	−	0.995893i	$-0.471142\pi$
$677$	1.46189e9	0.181073	0.0905366	+	0.995893i	$0.471142\pi$
$678$	0	0
$679$	−1.96327e9	−0.240678
$680$	0	0
$681$	−5.86970e9	−0.712199
$682$	0	0
$683$	−5.88648e9	−0.706941	−0.353470	−	0.935446i	$-0.614998\pi$
$683$	−5.88648e9	−0.706941	−0.353470	+	0.935446i	$0.614998\pi$
$684$	0	0
$685$	2.04631e9	0.243250
$686$	0	0
$687$	7.02343e9	0.826419
$688$	0	0
$689$	−3.22834e8	−0.0376021
$690$	0	0
$691$	−1.46661e10	−1.69099	−0.845495	−	0.533984i	$-0.820694\pi$
$691$	−1.46661e10	−1.69099	−0.845495	+	0.533984i	$0.820694\pi$
$692$	0	0
$693$	2.31130e9	0.263809
$694$	0	0
$695$	−1.26583e9	−0.143031
$696$	0	0
$697$	3.77644e9	0.422443
$698$	0	0
$699$	−2.99971e9	−0.332207
$700$	0	0
$701$	4.05367e9	0.444463	0.222231	−	0.974994i	$-0.428666\pi$
$701$	4.05367e9	0.444463	0.222231	+	0.974994i	$0.428666\pi$
$702$	0	0
$703$	6.48494e7	0.00703984
$704$	0	0
$705$	−3.21454e9	−0.345507
$706$	0	0
$707$	2.91853e9	0.310596
$708$	0	0
$709$	−2.59089e9	−0.273015	−0.136507	−	0.990639i	$-0.543588\pi$
$709$	−2.59089e9	−0.273015	−0.136507	+	0.990639i	$0.543588\pi$
$710$	0	0
$711$	4.77146e9	0.497860
$712$	0	0
$713$	4.02360e9	0.415720
$714$	0	0
$715$	4.64373e8	0.0475111
$716$	0	0
$717$	−9.01130e9	−0.912999
$718$	0	0
$719$	−5.11555e9	−0.513264	−0.256632	−	0.966509i	$-0.582613\pi$
$719$	−5.11555e9	−0.513264	−0.256632	+	0.966509i	$0.582613\pi$
$720$	0	0
$721$	−1.04362e10	−1.03698
$722$	0	0
$723$	2.80918e8	0.0276436
$724$	0	0
$725$	−5.89757e9	−0.574765
$726$	0	0
$727$	−1.79156e10	−1.72926	−0.864629	−	0.502410i	$-0.832447\pi$
$727$	−1.79156e10	−1.72926	−0.864629	+	0.502410i	$0.832447\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−7.68321e9	−0.727498
$732$	0	0
$733$	2.97200e8	0.0278731	0.0139365	−	0.999903i	$-0.495564\pi$
$733$	2.97200e8	0.0278731	0.0139365	+	0.999903i	$0.495564\pi$
$734$	0	0
$735$	−4.35449e9	−0.404513
$736$	0	0
$737$	6.66461e9	0.613251
$738$	0	0
$739$	−4.58030e8	−0.0417483	−0.0208741	−	0.999782i	$-0.506645\pi$
$739$	−4.58030e8	−0.0417483	−0.0208741	+	0.999782i	$0.506645\pi$
$740$	0	0
$741$	−1.70792e7	−0.00154207
$742$	0	0
$743$	−2.09647e10	−1.87512	−0.937558	−	0.347830i	$-0.886919\pi$
$743$	−2.09647e10	−1.87512	−0.937558	+	0.347830i	$0.886919\pi$
$744$	0	0
$745$	−4.40993e9	−0.390737
$746$	0	0
$747$	3.53950e9	0.310685
$748$	0	0
$749$	3.33235e10	2.89778
$750$	0	0
$751$	−2.93646e9	−0.252979	−0.126490	−	0.991968i	$-0.540371\pi$
$751$	−2.93646e9	−0.252979	−0.126490	+	0.991968i	$0.540371\pi$
$752$	0	0
$753$	−1.02813e10	−0.877540
$754$	0	0
$755$	−5.94540e9	−0.502767
$756$	0	0
$757$	−5.51634e9	−0.462184	−0.231092	−	0.972932i	$-0.574230\pi$
$757$	−5.51634e9	−0.462184	−0.231092	+	0.972932i	$0.574230\pi$
$758$	0	0
$759$	2.22137e9	0.184406
$760$	0	0
$761$	2.42782e10	1.99696	0.998482	−	0.0550775i	$-0.0175406\pi$
$761$	2.42782e10	1.99696	0.998482	+	0.0550775i	$0.0175406\pi$
$762$	0	0
$763$	−6.66142e9	−0.542914
$764$	0	0
$765$	1.18223e9	0.0954744
$766$	0	0
$767$	−1.19514e9	−0.0956389
$768$	0	0
$769$	−1.06551e10	−0.844916	−0.422458	−	0.906382i	$-0.638833\pi$
$769$	−1.06551e10	−0.844916	−0.422458	+	0.906382i	$0.638833\pi$
$770$	0	0
$771$	−4.36730e9	−0.343180
$772$	0	0
$773$	1.05446e10	0.821114	0.410557	−	0.911835i	$-0.365334\pi$
$773$	1.05446e10	0.821114	0.410557	+	0.911835i	$0.365334\pi$
$774$	0	0
$775$	6.77260e9	0.522636
$776$	0	0
$777$	−9.39905e9	−0.718804
$778$	0	0
$779$	6.90841e7	0.00523597
$780$	0	0
$781$	2.19975e9	0.165232
$782$	0	0
$783$	−1.71952e9	−0.128009
$784$	0	0
$785$	2.41431e9	0.178135
$786$	0	0
$787$	1.16175e10	0.849572	0.424786	−	0.905294i	$-0.360349\pi$
$787$	1.16175e10	0.849572	0.424786	+	0.905294i	$0.360349\pi$
$788$	0	0
$789$	−5.18437e9	−0.375774
$790$	0	0
$791$	2.04084e10	1.46619
$792$	0	0
$793$	1.54282e9	0.109865
$794$	0	0
$795$	4.08798e8	0.0288552
$796$	0	0
$797$	−3.66787e9	−0.256631	−0.128316	−	0.991733i	$-0.540957\pi$
$797$	−3.66787e9	−0.256631	−0.128316	+	0.991733i	$0.540957\pi$
$798$	0	0
$799$	1.81860e10	1.26132
$800$	0	0
$801$	−6.80768e9	−0.468042
$802$	0	0
$803$	−3.13863e9	−0.213913
$804$	0	0
$805$	−6.38703e9	−0.431532
$806$	0	0
$807$	−1.05357e10	−0.705680
$808$	0	0
$809$	−9.44725e9	−0.627315	−0.313657	−	0.949536i	$-0.601554\pi$
$809$	−9.44725e9	−0.627315	−0.313657	+	0.949536i	$0.601554\pi$
$810$	0	0
$811$	−5.20312e9	−0.342524	−0.171262	−	0.985226i	$-0.554784\pi$
$811$	−5.20312e9	−0.342524	−0.171262	+	0.985226i	$0.554784\pi$
$812$	0	0
$813$	−1.29684e9	−0.0846386
$814$	0	0
$815$	−1.24276e9	−0.0804150
$816$	0	0
$817$	−1.40552e8	−0.00901697
$818$	0	0
$819$	2.47540e9	0.157453
$820$	0	0
$821$	5.16219e9	0.325562	0.162781	−	0.986662i	$-0.447954\pi$
$821$	5.16219e9	0.325562	0.162781	+	0.986662i	$0.447954\pi$
$822$	0	0
$823$	−2.49257e9	−0.155865	−0.0779324	−	0.996959i	$-0.524832\pi$
$823$	−2.49257e9	−0.155865	−0.0779324	+	0.996959i	$0.524832\pi$
$824$	0	0
$825$	3.73905e9	0.231832
$826$	0	0
$827$	−2.54401e8	−0.0156405	−0.00782024	−	0.999969i	$-0.502489\pi$
$827$	−2.54401e8	−0.0156405	−0.00782024	+	0.999969i	$0.502489\pi$
$828$	0	0
$829$	−1.01908e10	−0.621251	−0.310626	−	0.950532i	$-0.600539\pi$
$829$	−1.01908e10	−0.621251	−0.310626	+	0.950532i	$0.600539\pi$
$830$	0	0
$831$	−1.17764e10	−0.711886
$832$	0	0
$833$	2.46352e10	1.47672
$834$	0	0
$835$	−9.38865e9	−0.558085
$836$	0	0
$837$	1.97465e9	0.116399
$838$	0	0
$839$	2.17137e10	1.26931	0.634653	−	0.772797i	$-0.281143\pi$
$839$	2.17137e10	1.26931	0.634653	+	0.772797i	$0.281143\pi$
$840$	0	0
$841$	−9.61797e9	−0.557568
$842$	0	0
$843$	−1.10713e10	−0.636504
$844$	0	0
$845$	4.97343e8	0.0283568
$846$	0	0
$847$	−2.36149e10	−1.33535
$848$	0	0
$849$	7.85612e9	0.440586
$850$	0	0
$851$	−9.03334e9	−0.502452
$852$	0	0
$853$	2.47699e10	1.36648	0.683240	−	0.730194i	$-0.260570\pi$
$853$	2.47699e10	1.36648	0.683240	+	0.730194i	$0.260570\pi$
$854$	0	0
$855$	2.16270e7	0.00118336
$856$	0	0
$857$	3.85505e9	0.209217	0.104609	−	0.994513i	$-0.466641\pi$
$857$	3.85505e9	0.209217	0.104609	+	0.994513i	$0.466641\pi$
$858$	0	0
$859$	1.32583e10	0.713694	0.356847	−	0.934163i	$-0.383852\pi$
$859$	1.32583e10	0.713694	0.356847	+	0.934163i	$0.383852\pi$
$860$	0	0
$861$	−1.00128e10	−0.534619
$862$	0	0
$863$	1.14506e10	0.606442	0.303221	−	0.952920i	$-0.401938\pi$
$863$	1.14506e10	0.606442	0.303221	+	0.952920i	$0.401938\pi$
$864$	0	0
$865$	−4.74119e9	−0.249076
$866$	0	0
$867$	4.39076e9	0.228809
$868$	0	0
$869$	1.34265e10	0.694057
$870$	0	0
$871$	7.13779e9	0.366016
$872$	0	0
$873$	−9.26020e8	−0.0471054
$874$	0	0
$875$	−2.31923e10	−1.17035
$876$	0	0
$877$	1.24703e9	0.0624279	0.0312140	−	0.999513i	$-0.490063\pi$
$877$	1.24703e9	0.0624279	0.0312140	+	0.999513i	$0.490063\pi$
$878$	0	0
$879$	1.41620e10	0.703338
$880$	0	0
$881$	−9.74816e9	−0.480294	−0.240147	−	0.970737i	$-0.577196\pi$
$881$	−9.74816e9	−0.480294	−0.240147	+	0.970737i	$0.577196\pi$
$882$	0	0
$883$	−1.31328e10	−0.641939	−0.320970	−	0.947089i	$-0.604009\pi$
$883$	−1.31328e10	−0.641939	−0.320970	+	0.947089i	$0.604009\pi$
$884$	0	0
$885$	1.51338e9	0.0733916
$886$	0	0
$887$	2.00851e10	0.966364	0.483182	−	0.875520i	$-0.339481\pi$
$887$	2.00851e10	0.966364	0.483182	+	0.875520i	$0.339481\pi$
$888$	0	0
$889$	1.89127e10	0.902814
$890$	0	0
$891$	1.09017e9	0.0516326
$892$	0	0
$893$	3.32685e8	0.0156334
$894$	0	0
$895$	1.28317e10	0.598279
$896$	0	0
$897$	2.37908e9	0.110062
$898$	0	0
$899$	−8.76426e9	−0.402306
$900$	0	0
$901$	−2.31275e9	−0.105339
$902$	0	0
$903$	2.03711e10	0.920679
$904$	0	0
$905$	9.21034e7	0.00413053
$906$	0	0
$907$	1.14140e10	0.507939	0.253970	−	0.967212i	$-0.418264\pi$
$907$	1.14140e10	0.507939	0.253970	+	0.967212i	$0.418264\pi$
$908$	0	0
$909$	1.37659e9	0.0607898
$910$	0	0
$911$	2.96830e10	1.30075	0.650375	−	0.759613i	$-0.274612\pi$
$911$	2.96830e10	1.30075	0.650375	+	0.759613i	$0.274612\pi$
$912$	0	0
$913$	9.95991e9	0.433119
$914$	0	0
$915$	−1.95364e9	−0.0843083
$916$	0	0
$917$	6.68325e10	2.86217
$918$	0	0
$919$	4.29969e10	1.82740	0.913699	−	0.406391i	$-0.133213\pi$
$919$	4.29969e10	1.82740	0.913699	+	0.406391i	$0.133213\pi$
$920$	0	0
$921$	−2.25078e10	−0.949346
$922$	0	0
$923$	2.35593e9	0.0986180
$924$	0	0
$925$	−1.52051e10	−0.631674
$926$	0	0
$927$	−4.92247e9	−0.202957
$928$	0	0
$929$	−1.89526e10	−0.775557	−0.387779	−	0.921753i	$-0.626758\pi$
$929$	−1.89526e10	−0.775557	−0.387779	+	0.921753i	$0.626758\pi$
$930$	0	0
$931$	4.50663e8	0.0183032
$932$	0	0
$933$	2.72765e10	1.09952
$934$	0	0
$935$	3.32671e9	0.133099
$936$	0	0
$937$	2.04780e10	0.813204	0.406602	−	0.913605i	$-0.366714\pi$
$937$	2.04780e10	0.813204	0.406602	+	0.913605i	$0.366714\pi$
$938$	0	0
$939$	7.14110e9	0.281472
$940$	0	0
$941$	3.81380e10	1.49209	0.746045	−	0.665896i	$-0.231951\pi$
$941$	3.81380e10	1.49209	0.746045	+	0.665896i	$0.231951\pi$
$942$	0	0
$943$	−9.62321e9	−0.373705
$944$	0	0
$945$	−3.13454e9	−0.120827
$946$	0	0
$947$	2.14389e10	0.820309	0.410154	−	0.912016i	$-0.365475\pi$
$947$	2.14389e10	0.820309	0.410154	+	0.912016i	$0.365475\pi$
$948$	0	0
$949$	−3.36148e9	−0.127673
$950$	0	0
$951$	5.00868e9	0.188839
$952$	0	0
$953$	−3.74717e10	−1.40242	−0.701210	−	0.712954i	$-0.747357\pi$
$953$	−3.74717e10	−1.40242	−0.701210	+	0.712954i	$0.747357\pi$
$954$	0	0
$955$	−3.40330e9	−0.126441
$956$	0	0
$957$	−4.83861e9	−0.178455
$958$	0	0
$959$	3.06946e10	1.12382
$960$	0	0
$961$	−1.74480e10	−0.634182
$962$	0	0
$963$	1.57178e10	0.567152
$964$	0	0
$965$	−9.30689e9	−0.333395
$966$	0	0
$967$	1.37908e10	0.490452	0.245226	−	0.969466i	$-0.421138\pi$
$967$	1.37908e10	0.490452	0.245226	+	0.969466i	$0.421138\pi$
$968$	0	0
$969$	−1.22353e8	−0.00431999
$970$	0	0
$971$	−4.01218e10	−1.40641	−0.703207	−	0.710985i	$-0.748249\pi$
$971$	−4.01218e10	−1.40641	−0.703207	+	0.710985i	$0.748249\pi$
$972$	0	0
$973$	−1.89875e10	−0.660806
$974$	0	0
$975$	4.00452e9	0.138368
$976$	0	0
$977$	−1.34176e10	−0.460304	−0.230152	−	0.973155i	$-0.573922\pi$
$977$	−1.34176e10	−0.460304	−0.230152	+	0.973155i	$0.573922\pi$
$978$	0	0
$979$	−1.91563e10	−0.652488
$980$	0	0
$981$	−3.14201e9	−0.106259
$982$	0	0
$983$	−1.61694e10	−0.542946	−0.271473	−	0.962446i	$-0.587511\pi$
$983$	−1.61694e10	−0.542946	−0.271473	+	0.962446i	$0.587511\pi$
$984$	0	0
$985$	3.92516e9	0.130867
$986$	0	0
$987$	−4.82181e10	−1.59625
$988$	0	0
$989$	1.95785e10	0.643566
$990$	0	0
$991$	−1.22415e10	−0.399556	−0.199778	−	0.979841i	$-0.564022\pi$
$991$	−1.22415e10	−0.399556	−0.199778	+	0.979841i	$0.564022\pi$
$992$	0	0
$993$	−2.01115e9	−0.0651813
$994$	0	0
$995$	−5.86647e9	−0.188797
$996$	0	0
$997$	8.83907e9	0.282471	0.141235	−	0.989976i	$-0.454893\pi$
$997$	8.83907e9	0.282471	0.141235	+	0.989976i	$0.454893\pi$
$998$	0	0
$999$	−4.43327e9	−0.140684

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.8.a.c.1.2	✓	3
3.2	odd	2		468.8.a.d.1.2		3
4.3	odd	2		624.8.a.m.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.8.a.c.1.2	✓	3	1.1	even	1	trivial
468.8.a.d.1.2		3	3.2	odd	2
624.8.a.m.1.2		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	156.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +103.038 q^{5} +1545.57 q^{7} +729.000 q^{9} +2051.35 q^{11} +2197.00 q^{13} -2782.01 q^{15} +15739.1 q^{17} +287.921 q^{19} -41730.3 q^{21} -40106.6 q^{23} -67508.3 q^{25} -19683.0 q^{27} +87360.8 q^{29} -100323. q^{31} -55386.6 q^{33} +159251. q^{35} +225233. q^{37} -59319.0 q^{39} +239941. q^{41} -488162. q^{43} +75114.4 q^{45} +1.15547e6 q^{47} +1.56523e6 q^{49} -424954. q^{51} -146943. q^{53} +211367. q^{55} -7773.87 q^{57} -543986. q^{59} +702239. q^{61} +1.12672e6 q^{63} +226374. q^{65} +3.24888e6 q^{67} +1.08288e6 q^{69} +1.07234e6 q^{71} -1.53003e6 q^{73} +1.82272e6 q^{75} +3.17050e6 q^{77} +6.54521e6 q^{79} +531441. q^{81} +4.85528e6 q^{83} +1.62171e6 q^{85} -2.35874e6 q^{87} -9.33838e6 q^{89} +3.39561e6 q^{91} +2.70871e6 q^{93} +29666.7 q^{95} -1.27026e6 q^{97} +1.49544e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 81 q^{3} + 524 q^{5} + 722 q^{7} + 2187 q^{9} + 6752 q^{11} + 6591 q^{13} - 14148 q^{15} - 15418 q^{17} + 1826 q^{19} - 19494 q^{21} + 16536 q^{23} + 10513 q^{25} - 59049 q^{27} - 151194 q^{29} + 46338 q^{31}+ \cdots + 4922208 q^{99}+O(q^{100}) \) 3 * q - 81 * q^3 + 524 * q^5 + 722 * q^7 + 2187 * q^9 + 6752 * q^11 + 6591 * q^13 - 14148 * q^15 - 15418 * q^17 + 1826 * q^19 - 19494 * q^21 + 16536 * q^23 + 10513 * q^25 - 59049 * q^27 - 151194 * q^29 + 46338 * q^31 - 182304 * q^33 - 223864 * q^35 - 112378 * q^37 - 177957 * q^39 + 324944 * q^41 + 1282940 * q^43 + 381996 * q^45 + 917072 * q^47 + 559359 * q^49 + 416286 * q^51 + 2688314 * q^53 + 4049992 * q^55 - 49302 * q^57 + 3176620 * q^59 + 2165222 * q^61 + 526338 * q^63 + 1151228 * q^65 + 6699990 * q^67 - 446472 * q^69 + 7734556 * q^71 + 5784690 * q^73 - 283851 * q^75 - 2868232 * q^77 + 8426640 * q^79 + 1594323 * q^81 + 4126284 * q^83 - 8292584 * q^85 + 4082238 * q^87 + 5207756 * q^89 + 1586234 * q^91 - 1251126 * q^93 + 18113000 * q^95 - 7634614 * q^97 + 4922208 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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