LMFDB - Embedded newform 104.10.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [104,10,Mod(1,104)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("104.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	104.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,141] 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:	$53.5637269610$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 $ x^8 - 3x^7 - 135356x^6 - 24398x^5 + 5213582205x^4 + 598076469x^3 - 44131402568826x^2 + 60107523849756*x + 1558753465556664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$106.986$ of defining polynomial
Character	$\chi$	$=$	104.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-88.9864 q^{3} -1188.35 q^{5} +3819.67 q^{7} -11764.4 q^{9} -44508.8 q^{11} +28561.0 q^{13} +105747. q^{15} -333157. q^{17} +62901.0 q^{19} -339899. q^{21} -2.12366e6 q^{23} -540958. q^{25} +2.79839e6 q^{27} -7.05902e6 q^{29} +8.97951e6 q^{31} +3.96068e6 q^{33} -4.53909e6 q^{35} +1.61465e7 q^{37} -2.54154e6 q^{39} +3.64237e6 q^{41} -3.21697e7 q^{43} +1.39802e7 q^{45} +1.39318e7 q^{47} -2.57637e7 q^{49} +2.96465e7 q^{51} +7.19727e7 q^{53} +5.28918e7 q^{55} -5.59733e6 q^{57} -7.52616e7 q^{59} +2.09407e8 q^{61} -4.49362e7 q^{63} -3.39404e7 q^{65} +2.66714e8 q^{67} +1.88977e8 q^{69} +1.43915e8 q^{71} +3.49049e8 q^{73} +4.81380e7 q^{75} -1.70009e8 q^{77} -5.57499e8 q^{79} -1.74600e7 q^{81} -1.21402e8 q^{83} +3.95906e8 q^{85} +6.28157e8 q^{87} +4.63605e8 q^{89} +1.09094e8 q^{91} -7.99054e8 q^{93} -7.47481e7 q^{95} -3.61385e8 q^{97} +5.23620e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 141 q^{3} + 2051 q^{5} - 2417 q^{7} + 115741 q^{9} - 53118 q^{11} + 228488 q^{13} - 464555 q^{15} + 433095 q^{17} - 434954 q^{19} + 906875 q^{21} - 1124296 q^{23} + 5966065 q^{25} + 7820643 q^{27}+ \cdots - 641626736 q^{99}+O(q^{100}) $ 8 * q + 141 * q^3 + 2051 * q^5 - 2417 * q^7 + 115741 * q^9 - 53118 * q^11 + 228488 * q^13 - 464555 * q^15 + 433095 * q^17 - 434954 * q^19 + 906875 * q^21 - 1124296 * q^23 + 5966065 * q^25 + 7820643 * q^27 - 1913432 * q^29 + 5220148 * q^31 - 1626882 * q^33 + 950707 * q^35 - 419409 * q^37 + 4027101 * q^39 + 12148046 * q^41 + 32143187 * q^43 + 18893544 * q^45 + 57898411 * q^47 + 104302637 * q^49 + 198521477 * q^51 + 273106866 * q^53 + 141214662 * q^55 + 253032058 * q^57 + 50734502 * q^59 + 139520018 * q^61 - 20344158 * q^63 + 58578611 * q^65 + 528144786 * q^67 + 483506576 * q^69 + 334814973 * q^71 + 630864344 * q^73 + 2265796264 * q^75 + 939664146 * q^77 + 865037608 * q^79 + 3194281040 * q^81 + 1520229952 * q^83 + 2130158385 * q^85 + 1167867006 * q^87 + 894700240 * q^89 - 69031937 * q^91 + 3803209548 * q^93 - 2695926542 * q^95 + 980232164 * q^97 - 641626736 * 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
$3$	−88.9864	−0.634276	−0.317138	−	0.948379i	$-0.602722\pi$
$3$	−88.9864	−0.634276	−0.317138	+	0.948379i	$0.602722\pi$
$4$	0	0
$5$	−1188.35	−0.850311	−0.425156	−	0.905120i	$-0.639781\pi$
$5$	−1188.35	−0.850311	−0.425156	+	0.905120i	$0.639781\pi$
$6$	0	0
$7$	3819.67	0.601291	0.300645	−	0.953736i	$-0.402798\pi$
$7$	3819.67	0.601291	0.300645	+	0.953736i	$0.402798\pi$
$8$	0	0
$9$	−11764.4	−0.597694
$10$	0	0
$11$	−44508.8	−0.916597	−0.458299	−	0.888798i	$-0.651541\pi$
$11$	−44508.8	−0.916597	−0.458299	+	0.888798i	$0.651541\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	105747.	0.539332
$16$	0	0
$17$	−333157.	−0.967451	−0.483726	−	0.875220i	$-0.660717\pi$
$17$	−333157.	−0.967451	−0.483726	+	0.875220i	$0.660717\pi$
$18$	0	0
$19$	62901.0	0.110730	0.0553651	−	0.998466i	$-0.482368\pi$
$19$	62901.0	0.110730	0.0553651	+	0.998466i	$0.482368\pi$
$20$	0	0
$21$	−339899.	−0.381384
$22$	0	0
$23$	−2.12366e6	−1.58238	−0.791190	−	0.611571i	$-0.790538\pi$
$23$	−2.12366e6	−1.58238	−0.791190	+	0.611571i	$0.790538\pi$
$24$	0	0
$25$	−540958.	−0.276971
$26$	0	0
$27$	2.79839e6	1.01338
$28$	0	0
$29$	−7.05902e6	−1.85333	−0.926667	−	0.375884i	$-0.877339\pi$
$29$	−7.05902e6	−1.85333	−0.926667	+	0.375884i	$0.877339\pi$
$30$	0	0
$31$	8.97951e6	1.74632	0.873162	−	0.487430i	$-0.162066\pi$
$31$	8.97951e6	1.74632	0.873162	+	0.487430i	$0.162066\pi$
$32$	0	0
$33$	3.96068e6	0.581375
$34$	0	0
$35$	−4.53909e6	−0.511284
$36$	0	0
$37$	1.61465e7	1.41635	0.708173	−	0.706039i	$-0.249520\pi$
$37$	1.61465e7	1.41635	0.708173	+	0.706039i	$0.249520\pi$
$38$	0	0
$39$	−2.54154e6	−0.175916
$40$	0	0
$41$	3.64237e6	0.201306	0.100653	−	0.994922i	$-0.467907\pi$
$41$	3.64237e6	0.201306	0.100653	+	0.994922i	$0.467907\pi$
$42$	0	0
$43$	−3.21697e7	−1.43496	−0.717479	−	0.696580i	$-0.754704\pi$
$43$	−3.21697e7	−1.43496	−0.717479	+	0.696580i	$0.754704\pi$
$44$	0	0
$45$	1.39802e7	0.508226
$46$	0	0
$47$	1.39318e7	0.416453	0.208226	−	0.978081i	$-0.433231\pi$
$47$	1.39318e7	0.416453	0.208226	+	0.978081i	$0.433231\pi$
$48$	0	0
$49$	−2.57637e7	−0.638449
$50$	0	0
$51$	2.96465e7	0.613631
$52$	0	0
$53$	7.19727e7	1.25293	0.626464	−	0.779450i	$-0.284501\pi$
$53$	7.19727e7	1.25293	0.626464	+	0.779450i	$0.284501\pi$
$54$	0	0
$55$	5.28918e7	0.779393
$56$	0	0
$57$	−5.59733e6	−0.0702335
$58$	0	0
$59$	−7.52616e7	−0.808610	−0.404305	−	0.914624i	$-0.632487\pi$
$59$	−7.52616e7	−0.808610	−0.404305	+	0.914624i	$0.632487\pi$
$60$	0	0
$61$	2.09407e8	1.93646	0.968228	−	0.250068i	$-0.0804529\pi$
$61$	2.09407e8	1.93646	0.968228	+	0.250068i	$0.0804529\pi$
$62$	0	0
$63$	−4.49362e7	−0.359388
$64$	0	0
$65$	−3.39404e7	−0.235834
$66$	0	0
$67$	2.66714e8	1.61700	0.808500	−	0.588497i	$-0.200280\pi$
$67$	2.66714e8	1.61700	0.808500	+	0.588497i	$0.200280\pi$
$68$	0	0
$69$	1.88977e8	1.00366
$70$	0	0
$71$	1.43915e8	0.672117	0.336058	−	0.941841i	$-0.390906\pi$
$71$	1.43915e8	0.672117	0.336058	+	0.941841i	$0.390906\pi$
$72$	0	0
$73$	3.49049e8	1.43858	0.719290	−	0.694710i	$-0.244467\pi$
$73$	3.49049e8	1.43858	0.719290	+	0.694710i	$0.244467\pi$
$74$	0	0
$75$	4.81380e7	0.175676
$76$	0	0
$77$	−1.70009e8	−0.551142
$78$	0	0
$79$	−5.57499e8	−1.61036	−0.805178	−	0.593033i	$-0.797930\pi$
$79$	−5.57499e8	−1.61036	−0.805178	+	0.593033i	$0.797930\pi$
$80$	0	0
$81$	−1.74600e7	−0.0450673
$82$	0	0
$83$	−1.21402e8	−0.280786	−0.140393	−	0.990096i	$-0.544837\pi$
$83$	−1.21402e8	−0.280786	−0.140393	+	0.990096i	$0.544837\pi$
$84$	0	0
$85$	3.95906e8	0.822635
$86$	0	0
$87$	6.28157e8	1.17552
$88$	0	0
$89$	4.63605e8	0.783236	0.391618	−	0.920128i	$-0.371915\pi$
$89$	4.63605e8	0.783236	0.391618	+	0.920128i	$0.371915\pi$
$90$	0	0
$91$	1.09094e8	0.166768
$92$	0	0
$93$	−7.99054e8	−1.10765
$94$	0	0
$95$	−7.47481e7	−0.0941551
$96$	0	0
$97$	−3.61385e8	−0.414474	−0.207237	−	0.978291i	$-0.566447\pi$
$97$	−3.61385e8	−0.414474	−0.207237	+	0.978291i	$0.566447\pi$
$98$	0	0
$99$	5.23620e8	0.547845
$100$	0	0
$101$	9.77607e8	0.934799	0.467399	−	0.884046i	$-0.345191\pi$
$101$	9.77607e8	0.934799	0.467399	+	0.884046i	$0.345191\pi$
$102$	0	0
$103$	1.62927e9	1.42635	0.713175	−	0.700986i	$-0.247257\pi$
$103$	1.62927e9	1.42635	0.713175	+	0.700986i	$0.247257\pi$
$104$	0	0
$105$	4.03917e8	0.324295
$106$	0	0
$107$	2.81592e8	0.207679	0.103840	−	0.994594i	$-0.466887\pi$
$107$	2.81592e8	0.207679	0.103840	+	0.994594i	$0.466887\pi$
$108$	0	0
$109$	1.87886e9	1.27490	0.637449	−	0.770493i	$-0.279990\pi$
$109$	1.87886e9	1.27490	0.637449	+	0.770493i	$0.279990\pi$
$110$	0	0
$111$	−1.43682e9	−0.898354
$112$	0	0
$113$	−2.70805e8	−0.156244	−0.0781220	−	0.996944i	$-0.524892\pi$
$113$	−2.70805e8	−0.156244	−0.0781220	+	0.996944i	$0.524892\pi$
$114$	0	0
$115$	2.52365e9	1.34551
$116$	0	0
$117$	−3.36003e8	−0.165771
$118$	0	0
$119$	−1.27255e9	−0.581720
$120$	0	0
$121$	−3.76916e8	−0.159849
$122$	0	0
$123$	−3.24122e8	−0.127684
$124$	0	0
$125$	2.96383e9	1.08582
$126$	0	0
$127$	−2.88479e9	−0.984004	−0.492002	−	0.870594i	$-0.663735\pi$
$127$	−2.88479e9	−0.984004	−0.492002	+	0.870594i	$0.663735\pi$
$128$	0	0
$129$	2.86267e9	0.910160
$130$	0	0
$131$	−2.52604e9	−0.749410	−0.374705	−	0.927144i	$-0.622256\pi$
$131$	−2.52604e9	−0.749410	−0.374705	+	0.927144i	$0.622256\pi$
$132$	0	0
$133$	2.40261e8	0.0665811
$134$	0	0
$135$	−3.32546e9	−0.861687
$136$	0	0
$137$	−7.12497e9	−1.72799	−0.863993	−	0.503503i	$-0.832044\pi$
$137$	−7.12497e9	−1.72799	−0.863993	+	0.503503i	$0.832044\pi$
$138$	0	0
$139$	−3.00490e9	−0.682753	−0.341376	−	0.939927i	$-0.610893\pi$
$139$	−3.00490e9	−0.682753	−0.341376	+	0.939927i	$0.610893\pi$
$140$	0	0
$141$	−1.23974e9	−0.264146
$142$	0	0
$143$	−1.27122e9	−0.254218
$144$	0	0
$145$	8.38856e9	1.57591
$146$	0	0
$147$	2.29262e9	0.404953
$148$	0	0
$149$	7.81131e8	0.129833	0.0649166	−	0.997891i	$-0.479322\pi$
$149$	7.81131e8	0.129833	0.0649166	+	0.997891i	$0.479322\pi$
$150$	0	0
$151$	−1.23384e10	−1.93136	−0.965679	−	0.259738i	$-0.916364\pi$
$151$	−1.23384e10	−1.93136	−0.965679	+	0.259738i	$0.916364\pi$
$152$	0	0
$153$	3.91940e9	0.578240
$154$	0	0
$155$	−1.06708e10	−1.48492
$156$	0	0
$157$	−4.82363e9	−0.633615	−0.316807	−	0.948490i	$-0.602611\pi$
$157$	−4.82363e9	−0.633615	−0.316807	+	0.948490i	$0.602611\pi$
$158$	0	0
$159$	−6.40459e9	−0.794702
$160$	0	0
$161$	−8.11170e9	−0.951470
$162$	0	0
$163$	−4.92519e9	−0.546486	−0.273243	−	0.961945i	$-0.588096\pi$
$163$	−4.92519e9	−0.546486	−0.273243	+	0.961945i	$0.588096\pi$
$164$	0	0
$165$	−4.70666e9	−0.494350
$166$	0	0
$167$	1.02301e8	0.0101778	0.00508891	−	0.999987i	$-0.498380\pi$
$167$	1.02301e8	0.0101778	0.00508891	+	0.999987i	$0.498380\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−7.39993e8	−0.0661828
$172$	0	0
$173$	1.54869e10	1.31449	0.657245	−	0.753677i	$-0.271722\pi$
$173$	1.54869e10	1.31449	0.657245	+	0.753677i	$0.271722\pi$
$174$	0	0
$175$	−2.06628e9	−0.166540
$176$	0	0
$177$	6.69726e9	0.512882
$178$	0	0
$179$	−1.04266e9	−0.0759112	−0.0379556	−	0.999279i	$-0.512085\pi$
$179$	−1.04266e9	−0.0759112	−0.0379556	+	0.999279i	$0.512085\pi$
$180$	0	0
$181$	1.20656e10	0.835594	0.417797	−	0.908540i	$-0.362802\pi$
$181$	1.20656e10	0.835594	0.417797	+	0.908540i	$0.362802\pi$
$182$	0	0
$183$	−1.86344e10	−1.22825
$184$	0	0
$185$	−1.91876e10	−1.20434
$186$	0	0
$187$	1.48284e10	0.886763
$188$	0	0
$189$	1.06889e10	0.609335
$190$	0	0
$191$	1.59078e9	0.0864887	0.0432443	−	0.999065i	$-0.486231\pi$
$191$	1.59078e9	0.0864887	0.0432443	+	0.999065i	$0.486231\pi$
$192$	0	0
$193$	1.50731e10	0.781979	0.390990	−	0.920395i	$-0.372133\pi$
$193$	1.50731e10	0.781979	0.390990	+	0.920395i	$0.372133\pi$
$194$	0	0
$195$	3.02023e9	0.149584
$196$	0	0
$197$	1.84500e10	0.872766	0.436383	−	0.899761i	$-0.356259\pi$
$197$	1.84500e10	0.872766	0.436383	+	0.899761i	$0.356259\pi$
$198$	0	0
$199$	−3.53832e10	−1.59940	−0.799701	−	0.600398i	$-0.795009\pi$
$199$	−3.53832e10	−1.59940	−0.799701	+	0.600398i	$0.795009\pi$
$200$	0	0
$201$	−2.37340e10	−1.02562
$202$	0	0
$203$	−2.69631e10	−1.11439
$204$	0	0
$205$	−4.32840e9	−0.171173
$206$	0	0
$207$	2.49837e10	0.945779
$208$	0	0
$209$	−2.79964e9	−0.101495
$210$	0	0
$211$	5.60226e9	0.194577	0.0972887	−	0.995256i	$-0.468983\pi$
$211$	5.60226e9	0.194577	0.0972887	+	0.995256i	$0.468983\pi$
$212$	0	0
$213$	−1.28065e10	−0.426307
$214$	0	0
$215$	3.82288e10	1.22016
$216$	0	0
$217$	3.42988e10	1.05005
$218$	0	0
$219$	−3.10606e10	−0.912456
$220$	0	0
$221$	−9.51531e9	−0.268323
$222$	0	0
$223$	6.34802e10	1.71896	0.859481	−	0.511168i	$-0.170787\pi$
$223$	6.34802e10	1.71896	0.859481	+	0.511168i	$0.170787\pi$
$224$	0	0
$225$	6.36406e9	0.165544
$226$	0	0
$227$	−5.82160e8	−0.0145521	−0.00727606	−	0.999974i	$-0.502316\pi$
$227$	−5.82160e8	−0.0145521	−0.00727606	+	0.999974i	$0.502316\pi$
$228$	0	0
$229$	−2.38821e10	−0.573869	−0.286935	−	0.957950i	$-0.592636\pi$
$229$	−2.38821e10	−0.573869	−0.286935	+	0.957950i	$0.592636\pi$
$230$	0	0
$231$	1.51285e10	0.349576
$232$	0	0
$233$	6.06514e10	1.34815	0.674076	−	0.738662i	$-0.264542\pi$
$233$	6.06514e10	1.34815	0.674076	+	0.738662i	$0.264542\pi$
$234$	0	0
$235$	−1.65557e10	−0.354114
$236$	0	0
$237$	4.96098e10	1.02141
$238$	0	0
$239$	−1.82283e10	−0.361374	−0.180687	−	0.983541i	$-0.557832\pi$
$239$	−1.82283e10	−0.361374	−0.180687	+	0.983541i	$0.557832\pi$
$240$	0	0
$241$	−5.23842e10	−1.00028	−0.500142	−	0.865943i	$-0.666719\pi$
$241$	−5.23842e10	−1.00028	−0.500142	+	0.865943i	$0.666719\pi$
$242$	0	0
$243$	−5.35271e10	−0.984794
$244$	0	0
$245$	3.06162e10	0.542881
$246$	0	0
$247$	1.79651e9	0.0307110
$248$	0	0
$249$	1.08031e10	0.178095
$250$	0	0
$251$	1.08399e11	1.72383	0.861917	−	0.507050i	$-0.169264\pi$
$251$	1.08399e11	1.72383	0.861917	+	0.507050i	$0.169264\pi$
$252$	0	0
$253$	9.45217e10	1.45040
$254$	0	0
$255$	−3.52303e10	−0.521777
$256$	0	0
$257$	−4.77652e10	−0.682987	−0.341493	−	0.939884i	$-0.610933\pi$
$257$	−4.77652e10	−0.682987	−0.341493	+	0.939884i	$0.610933\pi$
$258$	0	0
$259$	6.16741e10	0.851636
$260$	0	0
$261$	8.30453e10	1.10773
$262$	0	0
$263$	5.81856e10	0.749919	0.374960	−	0.927041i	$-0.377657\pi$
$263$	5.81856e10	0.749919	0.374960	+	0.927041i	$0.377657\pi$
$264$	0	0
$265$	−8.55285e10	−1.06538
$266$	0	0
$267$	−4.12545e10	−0.496788
$268$	0	0
$269$	−4.87140e10	−0.567242	−0.283621	−	0.958936i	$-0.591536\pi$
$269$	−4.87140e10	−0.567242	−0.283621	+	0.958936i	$0.591536\pi$
$270$	0	0
$271$	8.84431e10	0.996098	0.498049	−	0.867149i	$-0.334050\pi$
$271$	8.84431e10	0.996098	0.498049	+	0.867149i	$0.334050\pi$
$272$	0	0
$273$	−9.70785e9	−0.105777
$274$	0	0
$275$	2.40774e10	0.253871
$276$	0	0
$277$	−1.60500e11	−1.63801	−0.819005	−	0.573786i	$-0.805474\pi$
$277$	−1.60500e11	−1.63801	−0.819005	+	0.573786i	$0.805474\pi$
$278$	0	0
$279$	−1.05639e11	−1.04377
$280$	0	0
$281$	1.72141e11	1.64705	0.823524	−	0.567281i	$-0.192005\pi$
$281$	1.72141e11	1.64705	0.823524	+	0.567281i	$0.192005\pi$
$282$	0	0
$283$	1.66637e11	1.54430	0.772150	−	0.635440i	$-0.219181\pi$
$283$	1.66637e11	1.54430	0.772150	+	0.635440i	$0.219181\pi$
$284$	0	0
$285$	6.65157e9	0.0597203
$286$	0	0
$287$	1.39127e10	0.121043
$288$	0	0
$289$	−7.59410e9	−0.0640378
$290$	0	0
$291$	3.21584e10	0.262891
$292$	0	0
$293$	1.04433e11	0.827813	0.413906	−	0.910319i	$-0.364164\pi$
$293$	1.04433e11	0.827813	0.413906	+	0.910319i	$0.364164\pi$
$294$	0	0
$295$	8.94368e10	0.687570
$296$	0	0
$297$	−1.24553e11	−0.928860
$298$	0	0
$299$	−6.06540e10	−0.438873
$300$	0	0
$301$	−1.22878e11	−0.862828
$302$	0	0
$303$	−8.69937e10	−0.592920
$304$	0	0
$305$	−2.48848e11	−1.64659
$306$	0	0
$307$	5.73512e9	0.0368485	0.0184242	−	0.999830i	$-0.494135\pi$
$307$	5.73512e9	0.0368485	0.0184242	+	0.999830i	$0.494135\pi$
$308$	0	0
$309$	−1.44983e11	−0.904699
$310$	0	0
$311$	7.23401e10	0.438488	0.219244	−	0.975670i	$-0.429641\pi$
$311$	7.23401e10	0.438488	0.219244	+	0.975670i	$0.429641\pi$
$312$	0	0
$313$	−9.44626e10	−0.556302	−0.278151	−	0.960537i	$-0.589722\pi$
$313$	−9.44626e10	−0.556302	−0.278151	+	0.960537i	$0.589722\pi$
$314$	0	0
$315$	5.33997e10	0.305592
$316$	0	0
$317$	3.09934e11	1.72386	0.861930	−	0.507027i	$-0.169255\pi$
$317$	3.09934e11	1.72386	0.861930	+	0.507027i	$0.169255\pi$
$318$	0	0
$319$	3.14188e11	1.69876
$320$	0	0
$321$	−2.50578e10	−0.131726
$322$	0	0
$323$	−2.09559e10	−0.107126
$324$	0	0
$325$	−1.54503e10	−0.0768178
$326$	0	0
$327$	−1.67193e11	−0.808637
$328$	0	0
$329$	5.32147e10	0.250409
$330$	0	0
$331$	−2.18235e11	−0.999304	−0.499652	−	0.866226i	$-0.666539\pi$
$331$	−2.18235e11	−0.999304	−0.499652	+	0.866226i	$0.666539\pi$
$332$	0	0
$333$	−1.89954e11	−0.846542
$334$	0	0
$335$	−3.16949e11	−1.37495
$336$	0	0
$337$	2.05459e11	0.867740	0.433870	−	0.900975i	$-0.357148\pi$
$337$	2.05459e11	0.867740	0.433870	+	0.900975i	$0.357148\pi$
$338$	0	0
$339$	2.40980e10	0.0991018
$340$	0	0
$341$	−3.99667e11	−1.60068
$342$	0	0
$343$	−2.52546e11	−0.985185
$344$	0	0
$345$	−2.24570e11	−0.853428
$346$	0	0
$347$	1.25071e11	0.463098	0.231549	−	0.972823i	$-0.425621\pi$
$347$	1.25071e11	0.463098	0.231549	+	0.972823i	$0.425621\pi$
$348$	0	0
$349$	2.54795e11	0.919339	0.459670	−	0.888090i	$-0.347968\pi$
$349$	2.54795e11	0.919339	0.459670	+	0.888090i	$0.347968\pi$
$350$	0	0
$351$	7.99249e10	0.281061
$352$	0	0
$353$	−1.54721e11	−0.530349	−0.265175	−	0.964200i	$-0.585430\pi$
$353$	−1.54721e11	−0.530349	−0.265175	+	0.964200i	$0.585430\pi$
$354$	0	0
$355$	−1.71021e11	−0.571508
$356$	0	0
$357$	1.13240e11	0.368971
$358$	0	0
$359$	−2.45696e11	−0.780680	−0.390340	−	0.920671i	$-0.627643\pi$
$359$	−2.45696e11	−0.780680	−0.390340	+	0.920671i	$0.627643\pi$
$360$	0	0
$361$	−3.18731e11	−0.987739
$362$	0	0
$363$	3.35404e10	0.101389
$364$	0	0
$365$	−4.14791e11	−1.22324
$366$	0	0
$367$	−2.32120e11	−0.667906	−0.333953	−	0.942590i	$-0.608383\pi$
$367$	−2.32120e11	−0.667906	−0.333953	+	0.942590i	$0.608383\pi$
$368$	0	0
$369$	−4.28504e10	−0.120319
$370$	0	0
$371$	2.74912e11	0.753375
$372$	0	0
$373$	−6.16163e11	−1.64818	−0.824092	−	0.566456i	$-0.808314\pi$
$373$	−6.16163e11	−1.64818	−0.824092	+	0.566456i	$0.808314\pi$
$374$	0	0
$375$	−2.63741e11	−0.688711
$376$	0	0
$377$	−2.01613e11	−0.514022
$378$	0	0
$379$	2.57145e11	0.640180	0.320090	−	0.947387i	$-0.396287\pi$
$379$	2.57145e11	0.640180	0.320090	+	0.947387i	$0.396287\pi$
$380$	0	0
$381$	2.56707e11	0.624130
$382$	0	0
$383$	2.69852e11	0.640812	0.320406	−	0.947280i	$-0.396181\pi$
$383$	2.69852e11	0.640812	0.320406	+	0.947280i	$0.396181\pi$
$384$	0	0
$385$	2.02029e11	0.468642
$386$	0	0
$387$	3.78458e11	0.857667
$388$	0	0
$389$	−5.78186e11	−1.28025	−0.640125	−	0.768271i	$-0.721117\pi$
$389$	−5.78186e11	−1.28025	−0.640125	+	0.768271i	$0.721117\pi$
$390$	0	0
$391$	7.07514e11	1.53088
$392$	0	0
$393$	2.24783e11	0.475333
$394$	0	0
$395$	6.62501e11	1.36930
$396$	0	0
$397$	2.51591e11	0.508320	0.254160	−	0.967162i	$-0.418201\pi$
$397$	2.51591e11	0.508320	0.254160	+	0.967162i	$0.418201\pi$
$398$	0	0
$399$	−2.13800e10	−0.0422308
$400$	0	0
$401$	4.38439e11	0.846758	0.423379	−	0.905953i	$-0.360844\pi$
$401$	4.38439e11	0.846758	0.423379	+	0.905953i	$0.360844\pi$
$402$	0	0
$403$	2.56464e11	0.484343
$404$	0	0
$405$	2.07485e10	0.0383212
$406$	0	0
$407$	−7.18659e11	−1.29822
$408$	0	0
$409$	5.90446e11	1.04334	0.521669	−	0.853148i	$-0.325310\pi$
$409$	5.90446e11	1.04334	0.521669	+	0.853148i	$0.325310\pi$
$410$	0	0
$411$	6.34026e11	1.09602
$412$	0	0
$413$	−2.87474e11	−0.486210
$414$	0	0
$415$	1.44268e11	0.238755
$416$	0	0
$417$	2.67395e11	0.433054
$418$	0	0
$419$	−7.05394e11	−1.11807	−0.559034	−	0.829144i	$-0.688828\pi$
$419$	−7.05394e11	−1.11807	−0.559034	+	0.829144i	$0.688828\pi$
$420$	0	0
$421$	3.20094e11	0.496602	0.248301	−	0.968683i	$-0.420128\pi$
$421$	3.20094e11	0.496602	0.248301	+	0.968683i	$0.420128\pi$
$422$	0	0
$423$	−1.63899e11	−0.248911
$424$	0	0
$425$	1.80224e11	0.267956
$426$	0	0
$427$	7.99867e11	1.16437
$428$	0	0
$429$	1.13121e11	0.161245
$430$	0	0
$431$	−5.88766e11	−0.821854	−0.410927	−	0.911668i	$-0.634795\pi$
$431$	−5.88766e11	−0.821854	−0.410927	+	0.911668i	$0.634795\pi$
$432$	0	0
$433$	−2.61673e11	−0.357737	−0.178868	−	0.983873i	$-0.557244\pi$
$433$	−2.61673e11	−0.357737	−0.178868	+	0.983873i	$0.557244\pi$
$434$	0	0
$435$	−7.46468e11	−0.999562
$436$	0	0
$437$	−1.33581e11	−0.175217
$438$	0	0
$439$	3.44746e11	0.443005	0.221503	−	0.975160i	$-0.428904\pi$
$439$	3.44746e11	0.443005	0.221503	+	0.975160i	$0.428904\pi$
$440$	0	0
$441$	3.03095e11	0.381597
$442$	0	0
$443$	2.57699e11	0.317904	0.158952	−	0.987286i	$-0.449189\pi$
$443$	2.57699e11	0.317904	0.158952	+	0.987286i	$0.449189\pi$
$444$	0	0
$445$	−5.50923e11	−0.665995
$446$	0	0
$447$	−6.95101e10	−0.0823500
$448$	0	0
$449$	1.08978e12	1.26541	0.632703	−	0.774395i	$-0.281946\pi$
$449$	1.08978e12	1.26541	0.632703	+	0.774395i	$0.281946\pi$
$450$	0	0
$451$	−1.62117e11	−0.184517
$452$	0	0
$453$	1.09795e12	1.22501
$454$	0	0
$455$	−1.29641e11	−0.141805
$456$	0	0
$457$	−6.12189e11	−0.656542	−0.328271	−	0.944584i	$-0.606466\pi$
$457$	−6.12189e11	−0.656542	−0.328271	+	0.944584i	$0.606466\pi$
$458$	0	0
$459$	−9.32305e11	−0.980395
$460$	0	0
$461$	−3.17379e10	−0.0327284	−0.0163642	−	0.999866i	$-0.505209\pi$
$461$	−3.17379e10	−0.0327284	−0.0163642	+	0.999866i	$0.505209\pi$
$462$	0	0
$463$	−1.07320e12	−1.08535	−0.542673	−	0.839944i	$-0.682588\pi$
$463$	−1.07320e12	−1.08535	−0.542673	+	0.839944i	$0.682588\pi$
$464$	0	0
$465$	9.49553e11	0.941848
$466$	0	0
$467$	5.58614e11	0.543483	0.271741	−	0.962370i	$-0.412400\pi$
$467$	5.58614e11	0.543483	0.271741	+	0.962370i	$0.412400\pi$
$468$	0	0
$469$	1.01876e12	0.972287
$470$	0	0
$471$	4.29237e11	0.401887
$472$	0	0
$473$	1.43184e12	1.31528
$474$	0	0
$475$	−3.40268e10	−0.0306690
$476$	0	0
$477$	−8.46717e11	−0.748868
$478$	0	0
$479$	−7.52934e11	−0.653502	−0.326751	−	0.945110i	$-0.605954\pi$
$479$	−7.52934e11	−0.653502	−0.326751	+	0.945110i	$0.605954\pi$
$480$	0	0
$481$	4.61159e11	0.392824
$482$	0	0
$483$	7.21831e11	0.603495
$484$	0	0
$485$	4.29451e11	0.352432
$486$	0	0
$487$	−1.60784e12	−1.29527	−0.647637	−	0.761949i	$-0.724243\pi$
$487$	−1.60784e12	−1.29527	−0.647637	+	0.761949i	$0.724243\pi$
$488$	0	0
$489$	4.38275e11	0.346623
$490$	0	0
$491$	2.94134e10	0.0228391	0.0114195	−	0.999935i	$-0.496365\pi$
$491$	2.94134e10	0.0228391	0.0114195	+	0.999935i	$0.496365\pi$
$492$	0	0
$493$	2.35176e12	1.79301
$494$	0	0
$495$	−6.22242e11	−0.465839
$496$	0	0
$497$	5.49709e11	0.404138
$498$	0	0
$499$	5.23368e11	0.377881	0.188940	−	0.981989i	$-0.439495\pi$
$499$	5.23368e11	0.377881	0.188940	+	0.981989i	$0.439495\pi$
$500$	0	0
$501$	−9.10338e9	−0.00645555
$502$	0	0
$503$	9.64150e11	0.671566	0.335783	−	0.941939i	$-0.390999\pi$
$503$	9.64150e11	0.671566	0.335783	+	0.941939i	$0.390999\pi$
$504$	0	0
$505$	−1.16174e12	−0.794870
$506$	0	0
$507$	−7.25890e10	−0.0487904
$508$	0	0
$509$	6.80583e11	0.449419	0.224709	−	0.974426i	$-0.427857\pi$
$509$	6.80583e11	0.449419	0.224709	+	0.974426i	$0.427857\pi$
$510$	0	0
$511$	1.33325e12	0.865005
$512$	0	0
$513$	1.76022e11	0.112212
$514$	0	0
$515$	−1.93614e12	−1.21284
$516$	0	0
$517$	−6.20085e11	−0.381719
$518$	0	0
$519$	−1.37812e12	−0.833749
$520$	0	0
$521$	−2.31132e12	−1.37433	−0.687164	−	0.726502i	$-0.741145\pi$
$521$	−2.31132e12	−1.37433	−0.687164	+	0.726502i	$0.741145\pi$
$522$	0	0
$523$	1.61100e12	0.941537	0.470768	−	0.882257i	$-0.343977\pi$
$523$	1.61100e12	0.941537	0.470768	+	0.882257i	$0.343977\pi$
$524$	0	0
$525$	1.83871e11	0.105632
$526$	0	0
$527$	−2.99159e12	−1.68948
$528$	0	0
$529$	2.70880e12	1.50392
$530$	0	0
$531$	8.85408e11	0.483302
$532$	0	0
$533$	1.04030e11	0.0558322
$534$	0	0
$535$	−3.34628e11	−0.176592
$536$	0	0
$537$	9.27830e10	0.0481486
$538$	0	0
$539$	1.14671e12	0.585201
$540$	0	0
$541$	3.08968e12	1.55070	0.775348	−	0.631535i	$-0.217575\pi$
$541$	3.08968e12	1.55070	0.775348	+	0.631535i	$0.217575\pi$
$542$	0	0
$543$	−1.07367e12	−0.529997
$544$	0	0
$545$	−2.23274e12	−1.08406
$546$	0	0
$547$	1.99321e12	0.951941	0.475971	−	0.879461i	$-0.342097\pi$
$547$	1.99321e12	0.951941	0.475971	+	0.879461i	$0.342097\pi$
$548$	0	0
$549$	−2.46356e12	−1.15741
$550$	0	0
$551$	−4.44019e11	−0.205220
$552$	0	0
$553$	−2.12946e12	−0.968292
$554$	0	0
$555$	1.70743e12	0.763881
$556$	0	0
$557$	−2.70768e12	−1.19192	−0.595962	−	0.803013i	$-0.703229\pi$
$557$	−2.70768e12	−1.19192	−0.595962	+	0.803013i	$0.703229\pi$
$558$	0	0
$559$	−9.18800e11	−0.397986
$560$	0	0
$561$	−1.31953e12	−0.562453
$562$	0	0
$563$	−4.11657e12	−1.72682	−0.863412	−	0.504500i	$-0.831677\pi$
$563$	−4.11657e12	−1.72682	−0.863412	+	0.504500i	$0.831677\pi$
$564$	0	0
$565$	3.21810e11	0.132856
$566$	0	0
$567$	−6.66914e10	−0.0270986
$568$	0	0
$569$	−2.54412e12	−1.01749	−0.508747	−	0.860916i	$-0.669891\pi$
$569$	−2.54412e12	−1.01749	−0.508747	+	0.860916i	$0.669891\pi$
$570$	0	0
$571$	−5.37420e10	−0.0211569	−0.0105784	−	0.999944i	$-0.503367\pi$
$571$	−5.37420e10	−0.0211569	−0.0105784	+	0.999944i	$0.503367\pi$
$572$	0	0
$573$	−1.41558e11	−0.0548577
$574$	0	0
$575$	1.14881e12	0.438273
$576$	0	0
$577$	8.17074e11	0.306881	0.153441	−	0.988158i	$-0.450965\pi$
$577$	8.17074e11	0.306881	0.153441	+	0.988158i	$0.450965\pi$
$578$	0	0
$579$	−1.34130e12	−0.495991
$580$	0	0
$581$	−4.63716e11	−0.168834
$582$	0	0
$583$	−3.20342e12	−1.14843
$584$	0	0
$585$	3.99288e11	0.140957
$586$	0	0
$587$	1.52642e12	0.530642	0.265321	−	0.964160i	$-0.414522\pi$
$587$	1.52642e12	0.530642	0.265321	+	0.964160i	$0.414522\pi$
$588$	0	0
$589$	5.64820e11	0.193371
$590$	0	0
$591$	−1.64180e12	−0.553574
$592$	0	0
$593$	−3.70454e12	−1.23023	−0.615117	−	0.788435i	$-0.710891\pi$
$593$	−3.70454e12	−1.23023	−0.615117	+	0.788435i	$0.710891\pi$
$594$	0	0
$595$	1.51223e12	0.494643
$596$	0	0
$597$	3.14862e12	1.01446
$598$	0	0
$599$	3.57840e12	1.13571	0.567856	−	0.823128i	$-0.307773\pi$
$599$	3.57840e12	1.13571	0.567856	+	0.823128i	$0.307773\pi$
$600$	0	0
$601$	−4.78400e10	−0.0149574	−0.00747869	−	0.999972i	$-0.502381\pi$
$601$	−4.78400e10	−0.0149574	−0.00747869	+	0.999972i	$0.502381\pi$
$602$	0	0
$603$	−3.13774e12	−0.966471
$604$	0	0
$605$	4.47907e11	0.135922
$606$	0	0
$607$	−5.17180e12	−1.54629	−0.773147	−	0.634226i	$-0.781319\pi$
$607$	−5.17180e12	−1.54629	−0.773147	+	0.634226i	$0.781319\pi$
$608$	0	0
$609$	2.39935e12	0.706832
$610$	0	0
$611$	3.97905e11	0.115503
$612$	0	0
$613$	−9.86282e11	−0.282117	−0.141058	−	0.990001i	$-0.545051\pi$
$613$	−9.86282e11	−0.282117	−0.141058	+	0.990001i	$0.545051\pi$
$614$	0	0
$615$	3.85169e11	0.108571
$616$	0	0
$617$	2.26657e11	0.0629631	0.0314816	−	0.999504i	$-0.489977\pi$
$617$	2.26657e11	0.0629631	0.0314816	+	0.999504i	$0.489977\pi$
$618$	0	0
$619$	−1.10516e12	−0.302565	−0.151283	−	0.988491i	$-0.548340\pi$
$619$	−1.10516e12	−0.302565	−0.151283	+	0.988491i	$0.548340\pi$
$620$	0	0
$621$	−5.94285e12	−1.60355
$622$	0	0
$623$	1.77082e12	0.470953
$624$	0	0
$625$	−2.46550e12	−0.646317
$626$	0	0
$627$	2.49130e11	0.0643758
$628$	0	0
$629$	−5.37931e12	−1.37025
$630$	0	0
$631$	2.66054e12	0.668094	0.334047	−	0.942556i	$-0.391586\pi$
$631$	2.66054e12	0.668094	0.334047	+	0.942556i	$0.391586\pi$
$632$	0	0
$633$	−4.98525e11	−0.123416
$634$	0	0
$635$	3.42812e12	0.836710
$636$	0	0
$637$	−7.35838e11	−0.177074
$638$	0	0
$639$	−1.69308e12	−0.401720
$640$	0	0
$641$	7.19055e12	1.68229	0.841145	−	0.540810i	$-0.181882\pi$
$641$	7.19055e12	1.68229	0.841145	+	0.540810i	$0.181882\pi$
$642$	0	0
$643$	6.05637e12	1.39722	0.698608	−	0.715505i	$-0.253803\pi$
$643$	6.05637e12	1.39722	0.698608	+	0.715505i	$0.253803\pi$
$644$	0	0
$645$	−3.40184e12	−0.773919
$646$	0	0
$647$	−5.07119e12	−1.13773	−0.568867	−	0.822430i	$-0.692618\pi$
$647$	−5.07119e12	−1.13773	−0.568867	+	0.822430i	$0.692618\pi$
$648$	0	0
$649$	3.34980e12	0.741170
$650$	0	0
$651$	−3.05212e12	−0.666021
$652$	0	0
$653$	6.51831e12	1.40290	0.701448	−	0.712721i	$-0.252537\pi$
$653$	6.51831e12	1.40290	0.701448	+	0.712721i	$0.252537\pi$
$654$	0	0
$655$	3.00181e12	0.637232
$656$	0	0
$657$	−4.10636e12	−0.859831
$658$	0	0
$659$	6.49661e12	1.34184	0.670922	−	0.741528i	$-0.265898\pi$
$659$	6.49661e12	1.34184	0.670922	+	0.741528i	$0.265898\pi$
$660$	0	0
$661$	1.39396e12	0.284016	0.142008	−	0.989866i	$-0.454644\pi$
$661$	1.39396e12	0.284016	0.142008	+	0.989866i	$0.454644\pi$
$662$	0	0
$663$	8.46733e11	0.170191
$664$	0	0
$665$	−2.85513e11	−0.0566146
$666$	0	0
$667$	1.49910e13	2.93268
$668$	0	0
$669$	−5.64887e12	−1.09030
$670$	0	0
$671$	−9.32047e12	−1.77495
$672$	0	0
$673$	1.53979e12	0.289330	0.144665	−	0.989481i	$-0.453790\pi$
$673$	1.53979e12	0.289330	0.144665	+	0.989481i	$0.453790\pi$
$674$	0	0
$675$	−1.51381e12	−0.280676
$676$	0	0
$677$	−8.58172e11	−0.157009	−0.0785047	−	0.996914i	$-0.525015\pi$
$677$	−8.58172e11	−0.157009	−0.0785047	+	0.996914i	$0.525015\pi$
$678$	0	0
$679$	−1.38037e12	−0.249220
$680$	0	0
$681$	5.18044e10	0.00923005
$682$	0	0
$683$	−2.08873e12	−0.367273	−0.183636	−	0.982994i	$-0.558787\pi$
$683$	−2.08873e12	−0.367273	−0.183636	+	0.982994i	$0.558787\pi$
$684$	0	0
$685$	8.46693e12	1.46933
$686$	0	0
$687$	2.12518e12	0.363991
$688$	0	0
$689$	2.05561e12	0.347500
$690$	0	0
$691$	4.11077e12	0.685918	0.342959	−	0.939350i	$-0.388571\pi$
$691$	4.11077e12	0.685918	0.342959	+	0.939350i	$0.388571\pi$
$692$	0	0
$693$	2.00005e12	0.329414
$694$	0	0
$695$	3.57086e12	0.580552
$696$	0	0
$697$	−1.21348e12	−0.194754
$698$	0	0
$699$	−5.39715e12	−0.855101
$700$	0	0
$701$	7.95355e12	1.24403	0.622013	−	0.783007i	$-0.286315\pi$
$701$	7.95355e12	1.24403	0.622013	+	0.783007i	$0.286315\pi$
$702$	0	0
$703$	1.01563e12	0.156832
$704$	0	0
$705$	1.47324e12	0.224606
$706$	0	0
$707$	3.73414e12	0.562086
$708$	0	0
$709$	−8.62653e12	−1.28212	−0.641059	−	0.767492i	$-0.721505\pi$
$709$	−8.62653e12	−1.28212	−0.641059	+	0.767492i	$0.721505\pi$
$710$	0	0
$711$	6.55865e12	0.962500
$712$	0	0
$713$	−1.90695e13	−2.76335
$714$	0	0
$715$	1.51064e12	0.216165
$716$	0	0
$717$	1.62208e12	0.229211
$718$	0	0
$719$	4.92453e11	0.0687203	0.0343601	−	0.999410i	$-0.489061\pi$
$719$	4.92453e11	0.0687203	0.0343601	+	0.999410i	$0.489061\pi$
$720$	0	0
$721$	6.22328e12	0.857651
$722$	0	0
$723$	4.66148e12	0.634456
$724$	0	0
$725$	3.81864e12	0.513319
$726$	0	0
$727$	−2.20983e12	−0.293396	−0.146698	−	0.989181i	$-0.546865\pi$
$727$	−2.20983e12	−0.293396	−0.146698	+	0.989181i	$0.546865\pi$
$728$	0	0
$729$	5.10685e12	0.669698
$730$	0	0
$731$	1.07176e13	1.38825
$732$	0	0
$733$	1.26204e13	1.61476	0.807378	−	0.590034i	$-0.200886\pi$
$733$	1.26204e13	1.61476	0.807378	+	0.590034i	$0.200886\pi$
$734$	0	0
$735$	−2.72443e12	−0.344336
$736$	0	0
$737$	−1.18711e13	−1.48214
$738$	0	0
$739$	1.41478e13	1.74498	0.872489	−	0.488633i	$-0.162504\pi$
$739$	1.41478e13	1.74498	0.872489	+	0.488633i	$0.162504\pi$
$740$	0	0
$741$	−1.59865e11	−0.0194793
$742$	0	0
$743$	−9.72293e12	−1.17044	−0.585218	−	0.810876i	$-0.698991\pi$
$743$	−9.72293e12	−1.17044	−0.585218	+	0.810876i	$0.698991\pi$
$744$	0	0
$745$	−9.28254e11	−0.110399
$746$	0	0
$747$	1.42822e12	0.167824
$748$	0	0
$749$	1.07559e12	0.124876
$750$	0	0
$751$	−1.77678e11	−0.0203823	−0.0101911	−	0.999948i	$-0.503244\pi$
$751$	−1.77678e11	−0.0203823	−0.0101911	+	0.999948i	$0.503244\pi$
$752$	0	0
$753$	−9.64608e12	−1.09339
$754$	0	0
$755$	1.46623e13	1.64226
$756$	0	0
$757$	−1.67323e11	−0.0185193	−0.00925963	−	0.999957i	$-0.502947\pi$
$757$	−1.67323e11	−0.0185193	−0.00925963	+	0.999957i	$0.502947\pi$
$758$	0	0
$759$	−8.41115e12	−0.919956
$760$	0	0
$761$	2.92101e12	0.315720	0.157860	−	0.987462i	$-0.449541\pi$
$761$	2.92101e12	0.315720	0.157860	+	0.987462i	$0.449541\pi$
$762$	0	0
$763$	7.17663e12	0.766584
$764$	0	0
$765$	−4.65761e12	−0.491684
$766$	0	0
$767$	−2.14955e12	−0.224268
$768$	0	0
$769$	3.68762e12	0.380258	0.190129	−	0.981759i	$-0.439109\pi$
$769$	3.68762e12	0.380258	0.190129	+	0.981759i	$0.439109\pi$
$770$	0	0
$771$	4.25045e12	0.433202
$772$	0	0
$773$	5.57994e12	0.562110	0.281055	−	0.959692i	$-0.409316\pi$
$773$	5.57994e12	0.562110	0.281055	+	0.959692i	$0.409316\pi$
$774$	0	0
$775$	−4.85754e12	−0.483681
$776$	0	0
$777$	−5.48816e12	−0.540172
$778$	0	0
$779$	2.29109e11	0.0222907
$780$	0	0
$781$	−6.40550e12	−0.616060
$782$	0	0
$783$	−1.97539e13	−1.87813
$784$	0	0
$785$	5.73214e12	0.538770
$786$	0	0
$787$	−7.85517e12	−0.729910	−0.364955	−	0.931025i	$-0.618916\pi$
$787$	−7.85517e12	−0.729910	−0.364955	+	0.931025i	$0.618916\pi$
$788$	0	0
$789$	−5.17773e12	−0.475656
$790$	0	0
$791$	−1.03439e12	−0.0939481
$792$	0	0
$793$	5.98089e12	0.537076
$794$	0	0
$795$	7.61087e12	0.675744
$796$	0	0
$797$	−1.92020e13	−1.68572	−0.842859	−	0.538134i	$-0.819129\pi$
$797$	−1.92020e13	−1.68572	−0.842859	+	0.538134i	$0.819129\pi$
$798$	0	0
$799$	−4.64147e12	−0.402898
$800$	0	0
$801$	−5.45404e12	−0.468136
$802$	0	0
$803$	−1.55358e13	−1.31860
$804$	0	0
$805$	9.63950e12	0.809046
$806$	0	0
$807$	4.33488e12	0.359788
$808$	0	0
$809$	3.61312e12	0.296561	0.148281	−	0.988945i	$-0.452626\pi$
$809$	3.61312e12	0.296561	0.148281	+	0.988945i	$0.452626\pi$
$810$	0	0
$811$	1.93239e13	1.56856	0.784279	−	0.620409i	$-0.213033\pi$
$811$	1.93239e13	1.56856	0.784279	+	0.620409i	$0.213033\pi$
$812$	0	0
$813$	−7.87023e12	−0.631801
$814$	0	0
$815$	5.85283e12	0.464683
$816$	0	0
$817$	−2.02351e12	−0.158893
$818$	0	0
$819$	−1.28342e12	−0.0996763
$820$	0	0
$821$	−1.20242e13	−0.923661	−0.461831	−	0.886968i	$-0.652807\pi$
$821$	−1.20242e13	−0.923661	−0.461831	+	0.886968i	$0.652807\pi$
$822$	0	0
$823$	2.25098e12	0.171030	0.0855150	−	0.996337i	$-0.472746\pi$
$823$	2.25098e12	0.171030	0.0855150	+	0.996337i	$0.472746\pi$
$824$	0	0
$825$	−2.14256e12	−0.161024
$826$	0	0
$827$	−1.64682e13	−1.22425	−0.612127	−	0.790760i	$-0.709686\pi$
$827$	−1.64682e13	−1.22425	−0.612127	+	0.790760i	$0.709686\pi$
$828$	0	0
$829$	−1.22802e13	−0.903048	−0.451524	−	0.892259i	$-0.649119\pi$
$829$	−1.22802e13	−0.903048	−0.451524	+	0.892259i	$0.649119\pi$
$830$	0	0
$831$	1.42823e13	1.03895
$832$	0	0
$833$	8.58337e12	0.617669
$834$	0	0
$835$	−1.21569e11	−0.00865432
$836$	0	0
$837$	2.51282e13	1.76969
$838$	0	0
$839$	−1.84244e13	−1.28370	−0.641850	−	0.766830i	$-0.721833\pi$
$839$	−1.84244e13	−1.28370	−0.641850	+	0.766830i	$0.721833\pi$
$840$	0	0
$841$	3.53226e13	2.43484
$842$	0	0
$843$	−1.53182e13	−1.04468
$844$	0	0
$845$	−9.69370e11	−0.0654086
$846$	0	0
$847$	−1.43970e12	−0.0961160
$848$	0	0
$849$	−1.48284e13	−0.979512
$850$	0	0
$851$	−3.42897e13	−2.24120
$852$	0	0
$853$	2.10694e12	0.136264	0.0681320	−	0.997676i	$-0.478296\pi$
$853$	2.10694e12	0.136264	0.0681320	+	0.997676i	$0.478296\pi$
$854$	0	0
$855$	8.79368e11	0.0562760
$856$	0	0
$857$	5.50650e12	0.348708	0.174354	−	0.984683i	$-0.444216\pi$
$857$	5.50650e12	0.348708	0.174354	+	0.984683i	$0.444216\pi$
$858$	0	0
$859$	1.14504e13	0.717550	0.358775	−	0.933424i	$-0.383194\pi$
$859$	1.14504e13	0.717550	0.358775	+	0.933424i	$0.383194\pi$
$860$	0	0
$861$	−1.23804e12	−0.0767750
$862$	0	0
$863$	−1.95971e13	−1.20266	−0.601329	−	0.799001i	$-0.705362\pi$
$863$	−1.95971e13	−1.20266	−0.601329	+	0.799001i	$0.705362\pi$
$864$	0	0
$865$	−1.84038e13	−1.11773
$866$	0	0
$867$	6.75772e11	0.0406176
$868$	0	0
$869$	2.48136e13	1.47605
$870$	0	0
$871$	7.61763e12	0.448475
$872$	0	0
$873$	4.25149e12	0.247729
$874$	0	0
$875$	1.13209e13	0.652895
$876$	0	0
$877$	1.21365e13	0.692783	0.346391	−	0.938090i	$-0.387407\pi$
$877$	1.21365e13	0.692783	0.346391	+	0.938090i	$0.387407\pi$
$878$	0	0
$879$	−9.29309e12	−0.525062
$880$	0	0
$881$	−5.07297e11	−0.0283708	−0.0141854	−	0.999899i	$-0.504515\pi$
$881$	−5.07297e11	−0.0283708	−0.0141854	+	0.999899i	$0.504515\pi$
$882$	0	0
$883$	7.97430e12	0.441438	0.220719	−	0.975337i	$-0.429160\pi$
$883$	7.97430e12	0.441438	0.220719	+	0.975337i	$0.429160\pi$
$884$	0	0
$885$	−7.95866e12	−0.436109
$886$	0	0
$887$	3.08847e12	0.167528	0.0837639	−	0.996486i	$-0.473306\pi$
$887$	3.08847e12	0.167528	0.0837639	+	0.996486i	$0.473306\pi$
$888$	0	0
$889$	−1.10189e13	−0.591673
$890$	0	0
$891$	7.77123e11	0.0413086
$892$	0	0
$893$	8.76321e11	0.0461139
$894$	0	0
$895$	1.23905e12	0.0645481
$896$	0	0
$897$	5.39738e12	0.278367
$898$	0	0
$899$	−6.33865e13	−3.23652
$900$	0	0
$901$	−2.39782e13	−1.21215
$902$	0	0
$903$	1.09345e13	0.547271
$904$	0	0
$905$	−1.43381e13	−0.710515
$906$	0	0
$907$	−1.65177e13	−0.810430	−0.405215	−	0.914221i	$-0.632803\pi$
$907$	−1.65177e13	−0.810430	−0.405215	+	0.914221i	$0.632803\pi$
$908$	0	0
$909$	−1.15010e13	−0.558724
$910$	0	0
$911$	−1.30186e13	−0.626226	−0.313113	−	0.949716i	$-0.601372\pi$
$911$	−1.30186e13	−0.626226	−0.313113	+	0.949716i	$0.601372\pi$
$912$	0	0
$913$	5.40346e12	0.257367
$914$	0	0
$915$	2.21441e13	1.04439
$916$	0	0
$917$	−9.64865e12	−0.450614
$918$	0	0
$919$	1.74711e13	0.807978	0.403989	−	0.914764i	$-0.367623\pi$
$919$	1.74711e13	0.807978	0.403989	+	0.914764i	$0.367623\pi$
$920$	0	0
$921$	−5.10347e11	−0.0233721
$922$	0	0
$923$	4.11037e12	0.186412
$924$	0	0
$925$	−8.73456e12	−0.392286
$926$	0	0
$927$	−1.91674e13	−0.852521
$928$	0	0
$929$	3.53491e12	0.155707	0.0778534	−	0.996965i	$-0.475193\pi$
$929$	3.53491e12	0.155707	0.0778534	+	0.996965i	$0.475193\pi$
$930$	0	0
$931$	−1.62056e12	−0.0706956
$932$	0	0
$933$	−6.43729e12	−0.278122
$934$	0	0
$935$	−1.76213e13	−0.754025
$936$	0	0
$937$	2.75317e13	1.16682	0.583411	−	0.812177i	$-0.301718\pi$
$937$	2.75317e13	1.16682	0.583411	+	0.812177i	$0.301718\pi$
$938$	0	0
$939$	8.40589e12	0.352849
$940$	0	0
$941$	3.18537e13	1.32436	0.662180	−	0.749345i	$-0.269631\pi$
$941$	3.18537e13	1.32436	0.662180	+	0.749345i	$0.269631\pi$
$942$	0	0
$943$	−7.73517e12	−0.318542
$944$	0	0
$945$	−1.27022e13	−0.518125
$946$	0	0
$947$	8.57276e12	0.346374	0.173187	−	0.984889i	$-0.444593\pi$
$947$	8.57276e12	0.346374	0.173187	+	0.984889i	$0.444593\pi$
$948$	0	0
$949$	9.96920e12	0.398990
$950$	0	0
$951$	−2.75799e13	−1.09340
$952$	0	0
$953$	3.85246e13	1.51294	0.756468	−	0.654031i	$-0.226923\pi$
$953$	3.85246e13	1.51294	0.756468	+	0.654031i	$0.226923\pi$
$954$	0	0
$955$	−1.89039e12	−0.0735423
$956$	0	0
$957$	−2.79585e13	−1.07748
$958$	0	0
$959$	−2.72150e13	−1.03902
$960$	0	0
$961$	5.41919e13	2.04965
$962$	0	0
$963$	−3.31276e12	−0.124129
$964$	0	0
$965$	−1.79121e13	−0.664926
$966$	0	0
$967$	4.40785e13	1.62109	0.810547	−	0.585674i	$-0.199170\pi$
$967$	4.40785e13	1.62109	0.810547	+	0.585674i	$0.199170\pi$
$968$	0	0
$969$	1.86479e12	0.0679475
$970$	0	0
$971$	−2.16995e13	−0.783364	−0.391682	−	0.920101i	$-0.628107\pi$
$971$	−2.16995e13	−0.783364	−0.391682	+	0.920101i	$0.628107\pi$
$972$	0	0
$973$	−1.14777e13	−0.410533
$974$	0	0
$975$	1.37487e12	0.0487237
$976$	0	0
$977$	2.73543e13	0.960507	0.480253	−	0.877130i	$-0.340545\pi$
$977$	2.73543e13	0.960507	0.480253	+	0.877130i	$0.340545\pi$
$978$	0	0
$979$	−2.06345e13	−0.717912
$980$	0	0
$981$	−2.21037e13	−0.761999
$982$	0	0
$983$	7.93965e12	0.271213	0.135607	−	0.990763i	$-0.456702\pi$
$983$	7.93965e12	0.271213	0.135607	+	0.990763i	$0.456702\pi$
$984$	0	0
$985$	−2.19250e13	−0.742123
$986$	0	0
$987$	−4.73539e12	−0.158828
$988$	0	0
$989$	6.83177e13	2.27065
$990$	0	0
$991$	−2.57860e13	−0.849282	−0.424641	−	0.905362i	$-0.639600\pi$
$991$	−2.57860e13	−0.849282	−0.424641	+	0.905362i	$0.639600\pi$
$992$	0	0
$993$	1.94199e13	0.633834
$994$	0	0
$995$	4.20474e13	1.35999
$996$	0	0
$997$	−2.22466e13	−0.713075	−0.356538	−	0.934281i	$-0.616043\pi$
$997$	−2.22466e13	−0.713075	−0.356538	+	0.934281i	$0.616043\pi$
$998$	0	0
$999$	4.51841e13	1.43530

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.a.d.1.3	✓	8
4.3	odd	2		208.10.a.m.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.d.1.3	✓	8	1.1	even	1	trivial
208.10.a.m.1.6		8	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:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	104.a (trivial)

\(f(q)\)	\(=\)	\(q-88.9864 q^{3} -1188.35 q^{5} +3819.67 q^{7} -11764.4 q^{9} -44508.8 q^{11} +28561.0 q^{13} +105747. q^{15} -333157. q^{17} +62901.0 q^{19} -339899. q^{21} -2.12366e6 q^{23} -540958. q^{25} +2.79839e6 q^{27} -7.05902e6 q^{29} +8.97951e6 q^{31} +3.96068e6 q^{33} -4.53909e6 q^{35} +1.61465e7 q^{37} -2.54154e6 q^{39} +3.64237e6 q^{41} -3.21697e7 q^{43} +1.39802e7 q^{45} +1.39318e7 q^{47} -2.57637e7 q^{49} +2.96465e7 q^{51} +7.19727e7 q^{53} +5.28918e7 q^{55} -5.59733e6 q^{57} -7.52616e7 q^{59} +2.09407e8 q^{61} -4.49362e7 q^{63} -3.39404e7 q^{65} +2.66714e8 q^{67} +1.88977e8 q^{69} +1.43915e8 q^{71} +3.49049e8 q^{73} +4.81380e7 q^{75} -1.70009e8 q^{77} -5.57499e8 q^{79} -1.74600e7 q^{81} -1.21402e8 q^{83} +3.95906e8 q^{85} +6.28157e8 q^{87} +4.63605e8 q^{89} +1.09094e8 q^{91} -7.99054e8 q^{93} -7.47481e7 q^{95} -3.61385e8 q^{97} +5.23620e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 141 q^{3} + 2051 q^{5} - 2417 q^{7} + 115741 q^{9} - 53118 q^{11} + 228488 q^{13} - 464555 q^{15} + 433095 q^{17} - 434954 q^{19} + 906875 q^{21} - 1124296 q^{23} + 5966065 q^{25} + 7820643 q^{27}+ \cdots - 641626736 q^{99}+O(q^{100}) \) 8 * q + 141 * q^3 + 2051 * q^5 - 2417 * q^7 + 115741 * q^9 - 53118 * q^11 + 228488 * q^13 - 464555 * q^15 + 433095 * q^17 - 434954 * q^19 + 906875 * q^21 - 1124296 * q^23 + 5966065 * q^25 + 7820643 * q^27 - 1913432 * q^29 + 5220148 * q^31 - 1626882 * q^33 + 950707 * q^35 - 419409 * q^37 + 4027101 * q^39 + 12148046 * q^41 + 32143187 * q^43 + 18893544 * q^45 + 57898411 * q^47 + 104302637 * q^49 + 198521477 * q^51 + 273106866 * q^53 + 141214662 * q^55 + 253032058 * q^57 + 50734502 * q^59 + 139520018 * q^61 - 20344158 * q^63 + 58578611 * q^65 + 528144786 * q^67 + 483506576 * q^69 + 334814973 * q^71 + 630864344 * q^73 + 2265796264 * q^75 + 939664146 * q^77 + 865037608 * q^79 + 3194281040 * q^81 + 1520229952 * q^83 + 2130158385 * q^85 + 1167867006 * q^87 + 894700240 * q^89 - 69031937 * q^91 + 3803209548 * q^93 - 2695926542 * q^95 + 980232164 * q^97 - 641626736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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