LMFDB - Embedded newform 104.10.f.a.25.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [104,10,Mod(25,104)] 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, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("104.25"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	104.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$53.5637269610$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			25.6
Character	$\chi$	$=$	104.25
Dual form			104.10.f.a.25.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-182.615 q^{3} -909.303i q^{5} -3898.48i q^{7} +13665.4 q^{9} +42009.9i q^{11} +(-15333.2 + 101830. i) q^{13} +166053. i q^{15} -405213. q^{17} -733733. i q^{19} +711923. i q^{21} -2.38431e6 q^{23} +1.12629e6 q^{25} +1.09891e6 q^{27} -6.21574e6 q^{29} +1.75978e6i q^{31} -7.67165e6i q^{33} -3.54490e6 q^{35} +1.58146e7i q^{37} +(2.80008e6 - 1.85958e7i) q^{39} +1.90152e7i q^{41} +1.39085e7 q^{43} -1.24260e7i q^{45} -1.76381e7i q^{47} +2.51555e7 q^{49} +7.39982e7 q^{51} +1.03876e8 q^{53} +3.81997e7 q^{55} +1.33991e8i q^{57} -1.34985e8i q^{59} -4.27452e7 q^{61} -5.32743e7i q^{63} +(9.25945e7 + 1.39425e7i) q^{65} -1.49940e8i q^{67} +4.35412e8 q^{69} +2.27379e8i q^{71} -1.48065e8i q^{73} -2.05679e8 q^{75} +1.63775e8 q^{77} +8.27879e7 q^{79} -4.69653e8 q^{81} -3.33426e8i q^{83} +3.68461e8i q^{85} +1.13509e9 q^{87} -2.12521e8i q^{89} +(3.96983e8 + 5.97762e7i) q^{91} -3.21362e8i q^{93} -6.67186e8 q^{95} +8.43207e8i q^{97} +5.74081e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) $ 32 * q - 162 * q^3 + 223074 * q^9 + 66270 * q^13 - 487902 * q^17 + 3171556 * q^23 - 13526722 * q^25 - 3694974 * q^27 + 8833508 * q^29 - 8281126 * q^35 - 12056860 * q^39 + 89959038 * q^43 - 172344874 * q^49 + 51206562 * q^51 + 139521112 * q^53 - 84550176 * q^55 + 372036836 * q^61 + 241471810 * q^65 - 486052516 * q^69 - 604473728 * q^75 - 1666197144 * q^77 + 1165982916 * q^79 + 859636544 * q^81 - 1738691296 * q^87 - 291718146 * q^91 + 1741143356 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/104\mathbb{Z}\right)^\times$.

$n$	$41$	$53$	$79$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	−182.615			−1.30164			−0.650821	−	0.759231i	$-0.725575\pi$
$3$	−182.615			−1.30164			−0.650821	+	0.759231i	$0.725575\pi$
$4$		0			0
$5$		−	909.303i		−	0.650644i	−0.945603	−	0.325322i	$-0.894527\pi$
$5$		−	909.303i		−	0.650644i	0.945603	−	0.325322i	$-0.105473\pi$
$6$		0			0
$7$		−	3898.48i		−	0.613697i	−0.951758	−	0.306849i	$-0.900725\pi$
$7$		−	3898.48i		−	0.613697i	0.951758	−	0.306849i	$-0.0992746\pi$
$8$		0			0
$9$	13665.4			0.694274
$10$		0			0
$11$			42009.9i			0.865136i	0.901601	+	0.432568i	$0.142392\pi$
$11$			42009.9i			0.865136i	−0.901601	+	0.432568i	$0.857608\pi$
$12$		0			0
$13$	−15333.2	+	101830.i	−0.148898	+	0.988853i
$13$	−15333.2	+	101830.i	−0.148898	+	0.988853i
$14$		0			0
$15$			166053.i			0.846906i
$16$		0			0
$17$	−405213.			−1.17669			−0.588347	−	0.808609i	$-0.700221\pi$
$17$	−405213.			−1.17669			−0.588347	+	0.808609i	$0.700221\pi$
$18$		0			0
$19$		−	733733.i		−	1.29166i	−0.763483	−	0.645828i	$-0.776512\pi$
$19$		−	733733.i		−	1.29166i	0.763483	−	0.645828i	$-0.223488\pi$
$20$		0			0
$21$			711923.i			0.798815i
$22$		0			0
$23$	−2.38431e6			−1.77659			−0.888296	−	0.459271i	$-0.848111\pi$
$23$	−2.38431e6			−1.77659			−0.888296	+	0.459271i	$0.848111\pi$
$24$		0			0
$25$	1.12629e6			0.576662
$26$		0			0
$27$	1.09891e6			0.397946
$28$		0			0
$29$	−6.21574e6			−1.63193			−0.815966	−	0.578100i	$-0.803794\pi$
$29$	−6.21574e6			−1.63193			−0.815966	+	0.578100i	$0.803794\pi$
$30$		0			0
$31$			1.75978e6i			0.342239i	0.985250	+	0.171120i	$0.0547385\pi$
$31$			1.75978e6i			0.342239i	−0.985250	+	0.171120i	$0.945262\pi$
$32$		0			0
$33$		−	7.67165e6i		−	1.12610i
$34$		0			0
$35$	−3.54490e6			−0.399299
$36$		0			0
$37$			1.58146e7i			1.38724i	0.720342	+	0.693619i	$0.243985\pi$
$37$			1.58146e7i			1.38724i	−0.720342	+	0.693619i	$0.756015\pi$
$38$		0			0
$39$	2.80008e6	−	1.85958e7i	0.193811	−	1.28713i
$40$		0			0
$41$			1.90152e7i			1.05093i	0.850815	+	0.525465i	$0.176109\pi$
$41$			1.90152e7i			1.05093i	−0.850815	+	0.525465i	$0.823891\pi$
$42$		0			0
$43$	1.39085e7			0.620402			0.310201	−	0.950671i	$-0.399604\pi$
$43$	1.39085e7			0.620402			0.310201	+	0.950671i	$0.399604\pi$
$44$		0			0
$45$		−	1.24260e7i		−	0.451725i
$46$		0			0
$47$		−	1.76381e7i		−	0.527244i	−0.964626	−	0.263622i	$-0.915083\pi$
$47$		−	1.76381e7i		−	0.527244i	0.964626	−	0.263622i	$-0.0849172\pi$
$48$		0			0
$49$	2.51555e7			0.623376
$50$		0			0
$51$	7.39982e7			1.53163
$52$		0			0
$53$	1.03876e8			1.80832			0.904160	−	0.427194i	$-0.140498\pi$
$53$	1.03876e8			1.80832			0.904160	+	0.427194i	$0.140498\pi$
$54$		0			0
$55$	3.81997e7			0.562896
$56$		0			0
$57$			1.33991e8i			1.68127i
$58$		0			0
$59$		−	1.34985e8i		−	1.45027i	−0.688604	−	0.725137i	$-0.741776\pi$
$59$		−	1.34985e8i		−	1.45027i	0.688604	−	0.725137i	$-0.258224\pi$
$60$		0			0
$61$	−4.27452e7			−0.395278			−0.197639	−	0.980275i	$-0.563327\pi$
$61$	−4.27452e7			−0.395278			−0.197639	+	0.980275i	$0.563327\pi$
$62$		0			0
$63$		−	5.32743e7i		−	0.426074i
$64$		0			0
$65$	9.25945e7	+	1.39425e7i	0.643391	+	0.0968794i
$66$		0			0
$67$		−	1.49940e8i		−	0.909034i	−0.890738	−	0.454517i	$-0.849812\pi$
$67$		−	1.49940e8i		−	0.909034i	0.890738	−	0.454517i	$-0.150188\pi$
$68$		0			0
$69$	4.35412e8			2.31249
$70$		0			0
$71$			2.27379e8i			1.06191i	0.847401	+	0.530954i	$0.178166\pi$
$71$			2.27379e8i			1.06191i	−0.847401	+	0.530954i	$0.821834\pi$
$72$		0			0
$73$		−	1.48065e8i		−	0.610237i	−0.952314	−	0.305118i	$-0.901304\pi$
$73$		−	1.48065e8i		−	0.610237i	0.952314	−	0.305118i	$-0.0986961\pi$
$74$		0			0
$75$	−2.05679e8			−0.750608
$76$		0			0
$77$	1.63775e8			0.530932
$78$		0			0
$79$	8.27879e7			0.239136			0.119568	−	0.992826i	$-0.461849\pi$
$79$	8.27879e7			0.239136			0.119568	+	0.992826i	$0.461849\pi$
$80$		0			0
$81$	−4.69653e8			−1.21226
$82$		0			0
$83$		−	3.33426e8i		−	0.771167i	−0.922673	−	0.385583i	$-0.874000\pi$
$83$		−	3.33426e8i		−	0.771167i	0.922673	−	0.385583i	$-0.126000\pi$
$84$		0			0
$85$			3.68461e8i			0.765609i
$86$		0			0
$87$	1.13509e9			2.12419
$88$		0			0
$89$		−	2.12521e8i		−	0.359043i	−0.983754	−	0.179522i	$-0.942545\pi$
$89$		−	2.12521e8i		−	0.359043i	0.983754	−	0.179522i	$-0.0574550\pi$
$90$		0			0
$91$	3.96983e8	+	5.97762e7i	0.606856	+	0.0913781i
$92$		0			0
$93$		−	3.21362e8i		−	0.445473i
$94$		0			0
$95$	−6.67186e8			−0.840409
$96$		0			0
$97$			8.43207e8i			0.967078i	0.875323	+	0.483539i	$0.160649\pi$
$97$			8.43207e8i			0.967078i	−0.875323	+	0.483539i	$0.839351\pi$
$98$		0			0
$99$			5.74081e8i			0.600641i
$100$		0			0
$101$	4.44271e8			0.424817			0.212408	−	0.977181i	$-0.431869\pi$
$101$	4.44271e8			0.424817			0.212408	+	0.977181i	$0.431869\pi$
$102$		0			0
$103$	1.26335e9			1.10600			0.553001	−	0.833180i	$-0.313483\pi$
$103$	1.26335e9			1.10600			0.553001	+	0.833180i	$0.313483\pi$
$104$		0			0
$105$	6.47353e8			0.519744
$106$		0			0
$107$	6.81300e8			0.502472			0.251236	−	0.967926i	$-0.419163\pi$
$107$	6.81300e8			0.502472			0.251236	+	0.967926i	$0.419163\pi$
$108$		0			0
$109$		−	1.62190e9i		−	1.10054i	−0.834987	−	0.550270i	$-0.814525\pi$
$109$		−	1.62190e9i		−	1.10054i	0.834987	−	0.550270i	$-0.185475\pi$
$110$		0			0
$111$		−	2.88799e9i		−	1.80569i
$112$		0			0
$113$	3.00696e9			1.73490			0.867449	−	0.497525i	$-0.165758\pi$
$113$	3.00696e9			1.73490			0.867449	+	0.497525i	$0.165758\pi$
$114$		0			0
$115$			2.16806e9i			1.15593i
$116$		0			0
$117$	−2.09534e8	+	1.39155e9i	−0.103376	+	0.686534i
$118$		0			0
$119$			1.57972e9i			0.722134i
$120$		0			0
$121$	5.93118e8			0.251540
$122$		0			0
$123$		−	3.47247e9i		−	1.36793i
$124$		0			0
$125$		−	2.80012e9i		−	1.02585i
$126$		0			0
$127$	3.02027e9			1.03022			0.515109	−	0.857125i	$-0.327752\pi$
$127$	3.02027e9			1.03022			0.515109	+	0.857125i	$0.327752\pi$
$128$		0			0
$129$	−2.53991e9			−0.807541
$130$		0			0
$131$	−3.41318e9			−1.01260			−0.506301	−	0.862357i	$-0.668987\pi$
$131$	−3.41318e9			−1.01260			−0.506301	+	0.862357i	$0.668987\pi$
$132$		0			0
$133$	−2.86045e9			−0.792686
$134$		0			0
$135$		−	9.99240e8i		−	0.258921i
$136$		0			0
$137$			5.82012e9i			1.41153i	0.708448	+	0.705763i	$0.249396\pi$
$137$			5.82012e9i			1.41153i	−0.708448	+	0.705763i	$0.750604\pi$
$138$		0			0
$139$	−4.69511e9			−1.06679			−0.533395	−	0.845866i	$-0.679084\pi$
$139$	−4.69511e9			−1.06679			−0.533395	+	0.845866i	$0.679084\pi$
$140$		0			0
$141$			3.22099e9i			0.686284i
$142$		0			0
$143$	−4.27787e9	−	6.44146e8i	−0.855492	−	0.128817i
$144$		0			0
$145$			5.65199e9i			1.06181i
$146$		0			0
$147$	−4.59377e9			−0.811412
$148$		0			0
$149$			6.81087e9i			1.13205i	0.824389	+	0.566024i	$0.191519\pi$
$149$			6.81087e9i			1.13205i	−0.824389	+	0.566024i	$0.808481\pi$
$150$		0			0
$151$		−	7.84276e9i		−	1.22764i	−0.789444	−	0.613822i	$-0.789631\pi$
$151$		−	7.84276e9i		−	1.22764i	0.789444	−	0.613822i	$-0.210369\pi$
$152$		0			0
$153$	−5.53740e9			−0.816948
$154$		0			0
$155$	1.60017e9			0.222676
$156$		0			0
$157$	3.38139e9			0.444168			0.222084	−	0.975028i	$-0.428714\pi$
$157$	3.38139e9			0.444168			0.222084	+	0.975028i	$0.428714\pi$
$158$		0			0
$159$	−1.89694e10			−2.35379
$160$		0			0
$161$			9.29520e9i			1.09029i
$162$		0			0
$163$		−	9.70092e9i		−	1.07639i	−0.842821	−	0.538194i	$-0.819107\pi$
$163$		−	9.70092e9i		−	1.07639i	0.842821	−	0.538194i	$-0.180893\pi$
$164$		0			0
$165$	−6.97585e9			−0.732689
$166$		0			0
$167$		−	6.26957e9i		−	0.623754i	−0.950122	−	0.311877i	$-0.899042\pi$
$167$		−	6.26957e9i		−	0.623754i	0.950122	−	0.311877i	$-0.100958\pi$
$168$		0			0
$169$	−1.01343e10	−	3.12277e9i	−0.955659	−	0.294476i
$170$		0			0
$171$		−	1.00268e10i		−	0.896763i
$172$		0			0
$173$	−7.80512e9			−0.662479			−0.331239	−	0.943547i	$-0.607467\pi$
$173$	−7.80512e9			−0.662479			−0.331239	+	0.943547i	$0.607467\pi$
$174$		0			0
$175$		−	4.39083e9i		−	0.353896i
$176$		0			0
$177$			2.46503e10i			1.88774i
$178$		0			0
$179$	4.17544e9			0.303993			0.151997	−	0.988381i	$-0.451430\pi$
$179$	4.17544e9			0.303993			0.151997	+	0.988381i	$0.451430\pi$
$180$		0			0
$181$	−4.14338e9			−0.286947			−0.143473	−	0.989654i	$-0.545827\pi$
$181$	−4.14338e9			−0.286947			−0.143473	+	0.989654i	$0.545827\pi$
$182$		0			0
$183$	7.80593e9			0.514511
$184$		0			0
$185$	1.43803e10			0.902598
$186$		0			0
$187$		−	1.70230e10i		−	1.01800i
$188$		0			0
$189$		−	4.28407e9i		−	0.244219i
$190$		0			0
$191$	1.32883e10			0.722468			0.361234	−	0.932475i	$-0.382356\pi$
$191$	1.32883e10			0.722468			0.361234	+	0.932475i	$0.382356\pi$
$192$		0			0
$193$			1.50789e10i			0.782281i	0.920331	+	0.391141i	$0.127919\pi$
$193$			1.50789e10i			0.782281i	−0.920331	+	0.391141i	$0.872081\pi$
$194$		0			0
$195$	−1.69092e10	−	2.54612e9i	−0.837465	−	0.126102i
$196$		0			0
$197$		−	5.37906e9i		−	0.254453i	−0.991874	−	0.127227i	$-0.959392\pi$
$197$		−	5.37906e9i		−	0.254453i	0.991874	−	0.127227i	$-0.0406075\pi$
$198$		0			0
$199$	3.42114e10			1.54644			0.773218	−	0.634140i	$-0.218646\pi$
$199$	3.42114e10			1.54644			0.773218	+	0.634140i	$0.218646\pi$
$200$		0			0
$201$			2.73813e10i			1.18324i
$202$		0			0
$203$			2.42319e10i			1.00151i
$204$		0			0
$205$	1.72906e10			0.683781
$206$		0			0
$207$	−3.25826e10			−1.23344
$208$		0			0
$209$	3.08240e10			1.11746
$210$		0			0
$211$	2.73218e9			0.0948938			0.0474469	−	0.998874i	$-0.484892\pi$
$211$	2.73218e9			0.0948938			0.0474469	+	0.998874i	$0.484892\pi$
$212$		0			0
$213$		−	4.15228e10i		−	1.38223i
$214$		0			0
$215$		−	1.26471e10i		−	0.403661i
$216$		0			0
$217$	6.86046e9			0.210031
$218$		0			0
$219$			2.70389e10i			0.794310i
$220$		0			0
$221$	6.21321e9	−	4.12629e10i	0.175207	−	1.16358i
$222$		0			0
$223$			1.13075e10i			0.306193i	0.988211	+	0.153097i	$0.0489245\pi$
$223$			1.13075e10i			0.306193i	−0.988211	+	0.153097i	$0.951075\pi$
$224$		0			0
$225$	1.53912e10			0.400361
$226$		0			0
$227$		−	5.39032e10i		−	1.34741i	−0.739003	−	0.673703i	$-0.764703\pi$
$227$		−	5.39032e10i		−	1.34741i	0.739003	−	0.673703i	$-0.235297\pi$
$228$		0			0
$229$			4.10990e10i			0.987579i	0.869581	+	0.493790i	$0.164389\pi$
$229$			4.10990e10i			0.987579i	−0.869581	+	0.493790i	$0.835611\pi$
$230$		0			0
$231$	−2.99078e10			−0.691083
$232$		0			0
$233$	−4.26271e10			−0.947510			−0.473755	−	0.880657i	$-0.657102\pi$
$233$	−4.26271e10			−0.947510			−0.473755	+	0.880657i	$0.657102\pi$
$234$		0			0
$235$	−1.60384e10			−0.343048
$236$		0			0
$237$	−1.51183e10			−0.311270
$238$		0			0
$239$			5.25503e10i			1.04180i	0.853617	+	0.520901i	$0.174404\pi$
$239$			5.25503e10i			1.04180i	−0.853617	+	0.520901i	$0.825596\pi$
$240$		0			0
$241$		−	9.43574e9i		−	0.180177i	−0.995934	−	0.0900885i	$-0.971285\pi$
$241$		−	9.43574e9i		−	0.180177i	0.995934	−	0.0900885i	$-0.0287150\pi$
$242$		0			0
$243$	6.41362e10			1.17998
$244$		0			0
$245$		−	2.28739e10i		−	0.405596i
$246$		0			0
$247$	7.47162e10	+	1.12505e10i	1.27726	+	0.192325i
$248$		0			0
$249$			6.08888e10i			1.00378i
$250$		0			0
$251$	1.83370e10			0.291606			0.145803	−	0.989314i	$-0.453423\pi$
$251$	1.83370e10			0.291606			0.145803	+	0.989314i	$0.453423\pi$
$252$		0			0
$253$		−	1.00165e11i		−	1.53699i
$254$		0			0
$255$		−	6.72867e10i		−	0.996549i
$256$		0			0
$257$	1.32113e11			1.88906			0.944531	−	0.328423i	$-0.106517\pi$
$257$	1.32113e11			1.88906			0.944531	+	0.328423i	$0.106517\pi$
$258$		0			0
$259$	6.16530e10			0.851344
$260$		0			0
$261$	−8.49405e10			−1.13301
$262$		0			0
$263$	−4.37038e10			−0.563272			−0.281636	−	0.959521i	$-0.590877\pi$
$263$	−4.37038e10			−0.563272			−0.281636	+	0.959521i	$0.590877\pi$
$264$		0			0
$265$		−	9.44551e10i		−	1.17657i
$266$		0			0
$267$			3.88096e10i			0.467346i
$268$		0			0
$269$	2.52591e10			0.294125			0.147062	−	0.989127i	$-0.453018\pi$
$269$	2.52591e10			0.294125			0.147062	+	0.989127i	$0.453018\pi$
$270$		0			0
$271$			1.47126e11i			1.65702i	0.559975	+	0.828509i	$0.310811\pi$
$271$			1.47126e11i			1.65702i	−0.559975	+	0.828509i	$0.689189\pi$
$272$		0			0
$273$	−7.24952e10	−	1.09161e10i	−0.789910	−	0.118942i
$274$		0			0
$275$			4.73154e10i			0.498891i
$276$		0			0
$277$	1.40026e11			1.42906			0.714528	−	0.699607i	$-0.246641\pi$
$277$	1.40026e11			1.42906			0.714528	+	0.699607i	$0.246641\pi$
$278$		0			0
$279$			2.40480e10i			0.237608i
$280$		0			0
$281$			1.33842e11i			1.28061i	0.768122	+	0.640303i	$0.221191\pi$
$281$			1.33842e11i			1.28061i	−0.768122	+	0.640303i	$0.778809\pi$
$282$		0			0
$283$	1.06836e11			0.990101			0.495050	−	0.868864i	$-0.335150\pi$
$283$	1.06836e11			0.990101			0.495050	+	0.868864i	$0.335150\pi$
$284$		0			0
$285$	1.21838e11			1.09391
$286$		0			0
$287$	7.41304e10			0.644953
$288$		0			0
$289$	4.56098e10			0.384608
$290$		0			0
$291$		−	1.53983e11i		−	1.25879i
$292$		0			0
$293$		−	5.82348e10i		−	0.461614i	−0.973000	−	0.230807i	$-0.925863\pi$
$293$		−	5.82348e10i		−	0.461614i	0.973000	−	0.230807i	$-0.0741365\pi$
$294$		0			0
$295$	−1.22742e11			−0.943613
$296$		0			0
$297$			4.61650e10i			0.344278i
$298$		0			0
$299$	3.65591e10	−	2.42795e11i	0.264530	−	1.75679i
$300$		0			0
$301$		−	5.42221e10i		−	0.380739i
$302$		0			0
$303$	−8.11307e10			−0.552959
$304$		0			0
$305$			3.88683e10i			0.257185i
$306$		0			0
$307$			1.46932e11i			0.944049i	0.881585	+	0.472025i	$0.156477\pi$
$307$			1.46932e11i			0.944049i	−0.881585	+	0.472025i	$0.843523\pi$
$308$		0			0
$309$	−2.30707e11			−1.43962
$310$		0			0
$311$	9.32542e10			0.565258			0.282629	−	0.959229i	$-0.408793\pi$
$311$	9.32542e10			0.565258			0.282629	+	0.959229i	$0.408793\pi$
$312$		0			0
$313$	−5.39361e10			−0.317636			−0.158818	−	0.987308i	$-0.550768\pi$
$313$	−5.39361e10			−0.317636			−0.158818	+	0.987308i	$0.550768\pi$
$314$		0			0
$315$	−4.84424e10			−0.277223
$316$		0			0
$317$		−	2.82868e11i		−	1.57332i	−0.617387	−	0.786660i	$-0.711809\pi$
$317$		−	2.82868e11i		−	1.57332i	0.617387	−	0.786660i	$-0.288191\pi$
$318$		0			0
$319$		−	2.61122e11i		−	1.41184i
$320$		0			0
$321$	−1.24416e11			−0.654038
$322$		0			0
$323$			2.97318e11i			1.51988i
$324$		0			0
$325$	−1.72697e10	+	1.14691e11i	−0.0858636	+	0.570234i
$326$		0			0
$327$			2.96185e11i			1.43251i
$328$		0			0
$329$	−6.87619e10			−0.323568
$330$		0			0
$331$			2.99064e11i			1.36943i	0.728813	+	0.684713i	$0.240072\pi$
$331$			2.99064e11i			1.36943i	−0.728813	+	0.684713i	$0.759928\pi$
$332$		0			0
$333$			2.16113e11i			0.963123i
$334$		0			0
$335$	−1.36341e11			−0.591458
$336$		0			0
$337$	−3.37725e11			−1.42636			−0.713179	−	0.700982i	$-0.752745\pi$
$337$	−3.37725e11			−1.42636			−0.713179	+	0.700982i	$0.752745\pi$
$338$		0			0
$339$	−5.49117e11			−2.25822
$340$		0			0
$341$	−7.39280e10			−0.296083
$342$		0			0
$343$		−	2.55386e11i		−	0.996261i
$344$		0			0
$345$		−	3.95921e11i		−	1.50461i
$346$		0			0
$347$	−3.50993e11			−1.29962			−0.649809	−	0.760097i	$-0.725151\pi$
$347$	−3.50993e11			−1.29962			−0.649809	+	0.760097i	$0.725151\pi$
$348$		0			0
$349$			2.24383e11i			0.809610i	0.914403	+	0.404805i	$0.132661\pi$
$349$			2.24383e11i			0.809610i	−0.914403	+	0.404805i	$0.867339\pi$
$350$		0			0
$351$	−1.68498e10	+	1.11902e11i	−0.0592533	+	0.393510i
$352$		0			0
$353$		−	3.14374e11i		−	1.07761i	−0.842431	−	0.538804i	$-0.818877\pi$
$353$		−	3.14374e11i		−	1.07761i	0.842431	−	0.538804i	$-0.181123\pi$
$354$		0			0
$355$	2.06756e11			0.690925
$356$		0			0
$357$		−	2.88480e11i		−	0.939960i
$358$		0			0
$359$			4.91921e11i			1.56304i	0.623880	+	0.781520i	$0.285555\pi$
$359$			4.91921e11i			1.56304i	−0.623880	+	0.781520i	$0.714445\pi$
$360$		0			0
$361$	−2.15677e11			−0.668376
$362$		0			0
$363$	−1.08313e11			−0.327415
$364$		0			0
$365$	−1.34636e11			−0.397047
$366$		0			0
$367$	−1.01245e11			−0.291324			−0.145662	−	0.989334i	$-0.546531\pi$
$367$	−1.01245e11			−0.291324			−0.145662	+	0.989334i	$0.546531\pi$
$368$		0			0
$369$			2.59850e11i			0.729633i
$370$		0			0
$371$		−	4.04960e11i		−	1.10976i
$372$		0			0
$373$	3.96421e11			1.06039			0.530197	−	0.847874i	$-0.322118\pi$
$373$	3.96421e11			1.06039			0.530197	+	0.847874i	$0.322118\pi$
$374$		0			0
$375$			5.11346e11i			1.33529i
$376$		0			0
$377$	9.53072e10	−	6.32950e11i	0.242991	−	1.61374i
$378$		0			0
$379$			2.72622e11i			0.678711i	0.940658	+	0.339356i	$0.110209\pi$
$379$			2.72622e11i			0.678711i	−0.940658	+	0.339356i	$0.889791\pi$
$380$		0			0
$381$	−5.51548e11			−1.34098
$382$		0			0
$383$			3.60816e11i			0.856823i	0.903584	+	0.428411i	$0.140927\pi$
$383$			3.60816e11i			0.856823i	−0.903584	+	0.428411i	$0.859073\pi$
$384$		0			0
$385$		−	1.48921e11i		−	0.345448i
$386$		0			0
$387$	1.90065e11			0.430729
$388$		0			0
$389$	5.71139e10			0.126464			0.0632322	−	0.997999i	$-0.479859\pi$
$389$	5.71139e10			0.126464			0.0632322	+	0.997999i	$0.479859\pi$
$390$		0			0
$391$	9.66155e11			2.09051
$392$		0			0
$393$	6.23300e11			1.31805
$394$		0			0
$395$		−	7.52793e10i		−	0.155592i
$396$		0			0
$397$			2.68622e11i			0.542730i	0.962476	+	0.271365i	$0.0874751\pi$
$397$			2.68622e11i			0.542730i	−0.962476	+	0.271365i	$0.912525\pi$
$398$		0			0
$399$	5.22361e11			1.03179
$400$		0			0
$401$			1.03947e11i			0.200754i	0.994949	+	0.100377i	$0.0320049\pi$
$401$			1.03947e11i			0.200754i	−0.994949	+	0.100377i	$0.967995\pi$
$402$		0			0
$403$	−1.79198e11	−	2.69830e10i	−0.338424	−	0.0509586i
$404$		0			0
$405$			4.27057e11i			0.788748i
$406$		0			0
$407$	−6.64370e11			−1.20015
$408$		0			0
$409$		−	4.50509e11i		−	0.796064i	−0.917371	−	0.398032i	$-0.869693\pi$
$409$		−	4.50509e11i		−	0.796064i	0.917371	−	0.398032i	$-0.130307\pi$
$410$		0			0
$411$		−	1.06284e12i		−	1.83730i
$412$		0			0
$413$	−5.26235e11			−0.890030
$414$		0			0
$415$	−3.03185e11			−0.501755
$416$		0			0
$417$	8.57399e11			1.38858
$418$		0			0
$419$	7.80469e11			1.23706			0.618532	−	0.785759i	$-0.287728\pi$
$419$	7.80469e11			1.23706			0.618532	+	0.785759i	$0.287728\pi$
$420$		0			0
$421$		−	1.18044e12i		−	1.83137i	−0.401897	−	0.915685i	$-0.631649\pi$
$421$		−	1.18044e12i		−	1.83137i	0.401897	−	0.915685i	$-0.368351\pi$
$422$		0			0
$423$		−	2.41032e11i		−	0.366052i
$424$		0			0
$425$	−4.56389e11			−0.678555
$426$		0			0
$427$			1.66641e11i			0.242581i
$428$		0			0
$429$	7.81206e11	+	1.17631e11i	1.11354	+	0.167673i
$430$		0			0
$431$			1.23904e12i			1.72957i	0.502138	+	0.864787i	$0.332547\pi$
$431$			1.23904e12i			1.72957i	−0.502138	+	0.864787i	$0.667453\pi$
$432$		0			0
$433$	4.05366e11			0.554181			0.277090	−	0.960844i	$-0.410630\pi$
$433$	4.05366e11			0.554181			0.277090	+	0.960844i	$0.410630\pi$
$434$		0			0
$435$		−	1.03214e12i		−	1.38209i
$436$		0			0
$437$			1.74945e12i			2.29475i
$438$		0			0
$439$	1.10179e12			1.41583			0.707913	−	0.706300i	$-0.249637\pi$
$439$	1.10179e12			1.41583			0.707913	+	0.706300i	$0.249637\pi$
$440$		0			0
$441$	3.43759e11			0.432793
$442$		0			0
$443$	3.79643e11			0.468338			0.234169	−	0.972196i	$-0.424763\pi$
$443$	3.79643e11			0.468338			0.234169	+	0.972196i	$0.424763\pi$
$444$		0			0
$445$	−1.93246e11			−0.233609
$446$		0			0
$447$		−	1.24377e12i		−	1.47352i
$448$		0			0
$449$		−	4.92061e11i		−	0.571361i	−0.958325	−	0.285681i	$-0.907780\pi$
$449$		−	4.92061e11i		−	0.571361i	0.958325	−	0.285681i	$-0.0922197\pi$
$450$		0			0
$451$	−7.98826e11			−0.909197
$452$		0			0
$453$			1.43221e12i			1.59796i
$454$		0			0
$455$	5.43547e10	−	3.60978e11i	0.0594546	−	0.394847i
$456$		0			0
$457$			2.32051e10i			0.0248863i	0.999923	+	0.0124432i	$0.00396089\pi$
$457$			2.32051e10i			0.0248863i	−0.999923	+	0.0124432i	$0.996039\pi$
$458$		0			0
$459$	−4.45292e11			−0.468261
$460$		0			0
$461$			6.45950e11i			0.666109i	0.942908	+	0.333054i	$0.108079\pi$
$461$			6.45950e11i			0.666109i	−0.942908	+	0.333054i	$0.891921\pi$
$462$		0			0
$463$			1.11111e12i			1.12368i	0.827244	+	0.561842i	$0.189907\pi$
$463$			1.11111e12i			1.12368i	−0.827244	+	0.561842i	$0.810093\pi$
$464$		0			0
$465$	−2.92216e11			−0.289845
$466$		0			0
$467$	5.61698e11			0.546483			0.273242	−	0.961945i	$-0.411904\pi$
$467$	5.61698e11			0.546483			0.273242	+	0.961945i	$0.411904\pi$
$468$		0			0
$469$	−5.84537e11			−0.557872
$470$		0			0
$471$	−6.17494e11			−0.578147
$472$		0			0
$473$			5.84295e11i			0.536732i
$474$		0			0
$475$		−	8.26399e11i		−	0.744849i
$476$		0			0
$477$	1.41951e12			1.25547
$478$		0			0
$479$		−	1.29785e12i		−	1.12646i	−0.826301	−	0.563229i	$-0.809559\pi$
$479$		−	1.29785e12i		−	1.12646i	0.826301	−	0.563229i	$-0.190441\pi$
$480$		0			0
$481$	−1.61041e12	−	2.42489e11i	−1.37177	−	0.206556i
$482$		0			0
$483$		−	1.69745e12i		−	1.41917i
$484$		0			0
$485$	7.66730e11			0.629223
$486$		0			0
$487$		−	1.87290e11i		−	0.150881i	−0.997150	−	0.0754403i	$-0.975964\pi$
$487$		−	1.87290e11i		−	0.150881i	0.997150	−	0.0754403i	$-0.0240362\pi$
$488$		0			0
$489$			1.77154e12i			1.40107i
$490$		0			0
$491$	−9.87933e11			−0.767115			−0.383558	−	0.923517i	$-0.625301\pi$
$491$	−9.87933e11			−0.767115			−0.383558	+	0.923517i	$0.625301\pi$
$492$		0			0
$493$	2.51870e12			1.92028
$494$		0			0
$495$	5.22014e11			0.390804
$496$		0			0
$497$	8.86431e11			0.651690
$498$		0			0
$499$		−	1.15974e12i		−	0.837356i	−0.908135	−	0.418678i	$-0.862494\pi$
$499$		−	1.15974e12i		−	0.837356i	0.908135	−	0.418678i	$-0.137506\pi$
$500$		0			0
$501$			1.14492e12i			0.811905i
$502$		0			0
$503$	−1.33699e12			−0.931266			−0.465633	−	0.884978i	$-0.654173\pi$
$503$	−1.33699e12			−0.931266			−0.465633	+	0.884978i	$0.654173\pi$
$504$		0			0
$505$		−	4.03977e11i		−	0.276404i
$506$		0			0
$507$	1.85068e12	+	5.70265e11i	1.24393	+	0.383302i
$508$		0			0
$509$			9.19496e11i			0.607183i	0.952802	+	0.303592i	$0.0981859\pi$
$509$			9.19496e11i			0.607183i	−0.952802	+	0.303592i	$0.901814\pi$
$510$		0			0
$511$	−5.77227e11			−0.374501
$512$		0			0
$513$		−	8.06306e11i		−	0.514010i
$514$		0			0
$515$		−	1.14877e12i		−	0.719614i
$516$		0			0
$517$	7.40975e11			0.456138
$518$		0			0
$519$	1.42533e12			0.862311
$520$		0			0
$521$	−2.46945e12			−1.46835			−0.734176	−	0.678959i	$-0.762432\pi$
$521$	−2.46945e12			−1.46835			−0.734176	+	0.678959i	$0.762432\pi$
$522$		0			0
$523$	−6.27344e11			−0.366647			−0.183324	−	0.983053i	$-0.558686\pi$
$523$	−6.27344e11			−0.366647			−0.183324	+	0.983053i	$0.558686\pi$
$524$		0			0
$525$			8.01834e11i			0.460646i
$526$		0			0
$527$		−	7.13085e11i		−	0.402711i
$528$		0			0
$529$	3.88379e12			2.15628
$530$		0			0
$531$		−	1.84462e12i		−	1.00689i
$532$		0			0
$533$	−1.93632e12	−	2.91564e11i	−1.03921	−	0.156481i
$534$		0			0
$535$		−	6.19508e11i		−	0.326930i
$536$		0			0
$537$	−7.62500e11			−0.395690
$538$		0			0
$539$			1.05678e12i			0.539304i
$540$		0			0
$541$		−	2.82649e11i		−	0.141860i	−0.997481	−	0.0709300i	$-0.977403\pi$
$541$		−	2.82649e11i		−	0.141860i	0.997481	−	0.0709300i	$-0.0225967\pi$
$542$		0			0
$543$	7.56645e11			0.373502
$544$		0			0
$545$	−1.47480e12			−0.716060
$546$		0			0
$547$	2.22390e12			1.06212			0.531059	−	0.847335i	$-0.321794\pi$
$547$	2.22390e12			1.06212			0.531059	+	0.847335i	$0.321794\pi$
$548$		0			0
$549$	−5.84130e11			−0.274431
$550$		0			0
$551$			4.56070e12i			2.10789i
$552$		0			0
$553$		−	3.22747e11i		−	0.146757i
$554$		0			0
$555$	−2.62606e12			−1.17486
$556$		0			0
$557$		−	2.80541e12i		−	1.23495i	−0.786592	−	0.617473i	$-0.788156\pi$
$557$		−	2.80541e12i		−	1.23495i	0.786592	−	0.617473i	$-0.211844\pi$
$558$		0			0
$559$	−2.13262e11	+	1.41631e12i	−0.0923763	+	0.613486i
$560$		0			0
$561$			3.10865e12i			1.32507i
$562$		0			0
$563$	3.09484e11			0.129823			0.0649113	−	0.997891i	$-0.479324\pi$
$563$	3.09484e11			0.129823			0.0649113	+	0.997891i	$0.479324\pi$
$564$		0			0
$565$		−	2.73423e12i		−	1.12880i
$566$		0			0
$567$			1.83094e12i			0.743959i
$568$		0			0
$569$	−3.28092e12			−1.31217			−0.656085	−	0.754687i	$-0.727789\pi$
$569$	−3.28092e12			−1.31217			−0.656085	+	0.754687i	$0.727789\pi$
$570$		0			0
$571$	−3.57422e12			−1.40708			−0.703541	−	0.710655i	$-0.748399\pi$
$571$	−3.57422e12			−1.40708			−0.703541	+	0.710655i	$0.748399\pi$
$572$		0			0
$573$	−2.42664e12			−0.940395
$574$		0			0
$575$	−2.68543e12			−1.02449
$576$		0			0
$577$			4.67081e12i			1.75429i	0.480225	+	0.877145i	$0.340555\pi$
$577$			4.67081e12i			1.75429i	−0.480225	+	0.877145i	$0.659445\pi$
$578$		0			0
$579$		−	2.75365e12i		−	1.01825i
$580$		0			0
$581$	−1.29986e12			−0.473263
$582$		0			0
$583$			4.36383e12i			1.56444i
$584$		0			0
$585$	1.26534e12	+	1.90530e11i	0.446690	+	0.0672608i
$586$		0			0
$587$		−	1.09141e12i		−	0.379417i	−0.981840	−	0.189708i	$-0.939246\pi$
$587$		−	1.09141e12i		−	0.379417i	0.981840	−	0.189708i	$-0.0607543\pi$
$588$		0			0
$589$	1.29121e12			0.442056
$590$		0			0
$591$			9.82298e11i			0.331207i
$592$		0			0
$593$		−	5.77157e12i		−	1.91667i	−0.285641	−	0.958337i	$-0.592207\pi$
$593$		−	5.77157e12i		−	1.91667i	0.285641	−	0.958337i	$-0.407793\pi$
$594$		0			0
$595$	1.43644e12			0.469852
$596$		0			0
$597$	−6.24753e12			−2.01291
$598$		0			0
$599$	−3.20466e12			−1.01709			−0.508547	−	0.861034i	$-0.669817\pi$
$599$	−3.20466e12			−1.01709			−0.508547	+	0.861034i	$0.669817\pi$
$600$		0			0
$601$	−6.63025e11			−0.207298			−0.103649	−	0.994614i	$-0.533052\pi$
$601$	−6.63025e11			−0.207298			−0.103649	+	0.994614i	$0.533052\pi$
$602$		0			0
$603$		−	2.04898e12i		−	0.631118i
$604$		0			0
$605$		−	5.39324e11i		−	0.163663i
$606$		0			0
$607$	1.71239e12			0.511980			0.255990	−	0.966679i	$-0.417599\pi$
$607$	1.71239e12			0.511980			0.255990	+	0.966679i	$0.417599\pi$
$608$		0			0
$609$		−	4.42513e12i		−	1.30361i
$610$		0			0
$611$	1.79609e12	+	2.70449e11i	0.521367	+	0.0785054i
$612$		0			0
$613$			1.44500e12i			0.413328i	0.978412	+	0.206664i	$0.0662608\pi$
$613$			1.44500e12i			0.413328i	−0.978412	+	0.206664i	$0.933739\pi$
$614$		0			0
$615$	−3.15753e12			−0.890039
$616$		0			0
$617$			4.63176e12i			1.28666i	0.765590	+	0.643329i	$0.222447\pi$
$617$			4.63176e12i			1.28666i	−0.765590	+	0.643329i	$0.777553\pi$
$618$		0			0
$619$		−	3.11245e11i		−	0.0852108i	−0.999092	−	0.0426054i	$-0.986434\pi$
$619$		−	3.11245e11i		−	0.0852108i	0.999092	−	0.0426054i	$-0.0135658\pi$
$620$		0			0
$621$	−2.62014e12			−0.706989
$622$		0			0
$623$	−8.28509e11			−0.220344
$624$		0			0
$625$	−3.46369e11			−0.0907985
$626$		0			0
$627$	−5.62894e12			−1.45453
$628$		0			0
$629$		−	6.40829e12i		−	1.63235i
$630$		0			0
$631$			4.27561e12i			1.07366i	0.843691	+	0.536829i	$0.180378\pi$
$631$			4.27561e12i			1.07366i	−0.843691	+	0.536829i	$0.819622\pi$
$632$		0			0
$633$	−4.98937e11			−0.123518
$634$		0			0
$635$		−	2.74634e12i		−	0.670305i
$636$		0			0
$637$	−3.85714e11	+	2.56158e12i	−0.0928191	+	0.616427i
$638$		0			0
$639$			3.10722e12i			0.737255i
$640$		0			0
$641$	2.34998e12			0.549798			0.274899	−	0.961473i	$-0.411356\pi$
$641$	2.34998e12			0.549798			0.274899	+	0.961473i	$0.411356\pi$
$642$		0			0
$643$		−	3.34793e12i		−	0.772372i	−0.922421	−	0.386186i	$-0.873792\pi$
$643$		−	3.34793e12i		−	0.772372i	0.922421	−	0.386186i	$-0.126208\pi$
$644$		0			0
$645$			2.30955e12i			0.525422i
$646$		0			0
$647$	6.99648e12			1.56968			0.784838	−	0.619701i	$-0.212746\pi$
$647$	6.99648e12			1.56968			0.784838	+	0.619701i	$0.212746\pi$
$648$		0			0
$649$	5.67069e12			1.25468
$650$		0			0
$651$	−1.25282e12			−0.273386
$652$		0			0
$653$	7.55308e12			1.62560			0.812802	−	0.582541i	$-0.197941\pi$
$653$	7.55308e12			1.62560			0.812802	+	0.582541i	$0.197941\pi$
$654$		0			0
$655$			3.10362e12i			0.658843i
$656$		0			0
$657$		−	2.02336e12i		−	0.423672i
$658$		0			0
$659$	−8.09522e12			−1.67203			−0.836015	−	0.548706i	$-0.815121\pi$
$659$	−8.09522e12			−1.67203			−0.836015	+	0.548706i	$0.815121\pi$
$660$		0			0
$661$		−	9.20416e12i		−	1.87533i	−0.347540	−	0.937665i	$-0.612983\pi$
$661$		−	9.20416e12i		−	1.87533i	0.347540	−	0.937665i	$-0.387017\pi$
$662$		0			0
$663$	−1.13463e12	+	7.53525e12i	−0.228057	+	1.51456i
$664$		0			0
$665$			2.60101e12i			0.515757i
$666$		0			0
$667$	1.48203e13			2.89928
$668$		0			0
$669$		−	2.06493e12i		−	0.398554i
$670$		0			0
$671$		−	1.79572e12i		−	0.341969i
$672$		0			0
$673$	1.55727e12			0.292616			0.146308	−	0.989239i	$-0.453261\pi$
$673$	1.55727e12			0.292616			0.146308	+	0.989239i	$0.453261\pi$
$674$		0			0
$675$	1.23769e12			0.229481
$676$		0			0
$677$	5.92542e12			1.08410			0.542051	−	0.840346i	$-0.317648\pi$
$677$	5.92542e12			1.08410			0.542051	+	0.840346i	$0.317648\pi$
$678$		0			0
$679$	3.28723e12			0.593493
$680$		0			0
$681$			9.84356e12i			1.75384i
$682$		0			0
$683$		−	6.94217e11i		−	0.122068i	−0.998136	−	0.0610340i	$-0.980560\pi$
$683$		−	6.94217e11i		−	0.122068i	0.998136	−	0.0610340i	$-0.0194398\pi$
$684$		0			0
$685$	5.29225e12			0.918402
$686$		0			0
$687$		−	7.50532e12i		−	1.28548i
$688$		0			0
$689$	−1.59276e12	+	1.05778e13i	−0.269255	+	1.78816i
$690$		0			0
$691$		−	2.58641e12i		−	0.431565i	−0.976441	−	0.215783i	$-0.930770\pi$
$691$		−	2.58641e12i		−	0.431565i	0.976441	−	0.215783i	$-0.0692302\pi$
$692$		0			0
$693$	2.23805e12			0.368612
$694$		0			0
$695$			4.26928e12i			0.694101i
$696$		0			0
$697$		−	7.70521e12i		−	1.23662i
$698$		0			0
$699$	7.78436e12			1.23332
$700$		0			0
$701$	−7.12448e12			−1.11435			−0.557176	−	0.830395i	$-0.688115\pi$
$701$	−7.12448e12			−1.11435			−0.557176	+	0.830395i	$0.688115\pi$
$702$		0			0
$703$	1.16037e13			1.79183
$704$		0			0
$705$	2.92886e12			0.446526
$706$		0			0
$707$		−	1.73198e12i		−	0.260709i
$708$		0			0
$709$		−	6.87275e12i		−	1.02146i	−0.859740	−	0.510731i	$-0.829375\pi$
$709$		−	6.87275e12i		−	1.02146i	0.859740	−	0.510731i	$-0.170625\pi$
$710$		0			0
$711$	1.13133e12			0.166026
$712$		0			0
$713$		−	4.19586e12i		−	0.608020i
$714$		0			0
$715$	−5.85724e11	+	3.88988e12i	−0.0838138	+	0.556621i
$716$		0			0
$717$		−	9.59650e12i		−	1.35605i
$718$		0			0
$719$	−4.97179e12			−0.693797			−0.346899	−	0.937903i	$-0.612765\pi$
$719$	−4.97179e12			−0.693797			−0.346899	+	0.937903i	$0.612765\pi$
$720$		0			0
$721$		−	4.92514e12i		−	0.678751i
$722$		0			0
$723$			1.72311e12i			0.234526i
$724$		0			0
$725$	−7.00075e12			−0.941073
$726$		0			0
$727$	7.02097e12			0.932164			0.466082	−	0.884742i	$-0.345665\pi$
$727$	7.02097e12			0.932164			0.466082	+	0.884742i	$0.345665\pi$
$728$		0			0
$729$	−2.46806e12			−0.323655
$730$		0			0
$731$	−5.63592e12			−0.730023
$732$		0			0
$733$			6.18653e12i			0.791551i	0.918347	+	0.395776i	$0.129524\pi$
$733$			6.18653e12i			0.791551i	−0.918347	+	0.395776i	$0.870476\pi$
$734$		0			0
$735$			4.17713e12i			0.527941i
$736$		0			0
$737$	6.29895e12			0.786438
$738$		0			0
$739$		−	1.52545e13i		−	1.88147i	−0.339145	−	0.940734i	$-0.610138\pi$
$739$		−	1.52545e13i		−	1.88147i	0.339145	−	0.940734i	$-0.389862\pi$
$740$		0			0
$741$	−1.36443e13	−	2.05451e12i	−1.66253	−	0.250338i
$742$		0			0
$743$			1.06221e13i			1.27868i	0.768924	+	0.639340i	$0.220792\pi$
$743$			1.06221e13i			1.27868i	−0.768924	+	0.639340i	$0.779208\pi$
$744$		0			0
$745$	6.19315e12			0.736560
$746$		0			0
$747$		−	4.55640e12i		−	0.535401i
$748$		0			0
$749$		−	2.65604e12i		−	0.308365i
$750$		0			0
$751$	5.25076e12			0.602341			0.301171	−	0.953570i	$-0.402623\pi$
$751$	5.25076e12			0.602341			0.301171	+	0.953570i	$0.402623\pi$
$752$		0			0
$753$	−3.34862e12			−0.379566
$754$		0			0
$755$	−7.13145e12			−0.798760
$756$		0			0
$757$	1.47163e13			1.62880			0.814399	−	0.580306i	$-0.197067\pi$
$757$	1.47163e13			1.62880			0.814399	+	0.580306i	$0.197067\pi$
$758$		0			0
$759$			1.82916e13i			2.00062i
$760$		0			0
$761$			2.54714e12i			0.275310i	0.990480	+	0.137655i	$0.0439566\pi$
$761$			2.54714e12i			0.275310i	−0.990480	+	0.137655i	$0.956043\pi$
$762$		0			0
$763$	−6.32297e12			−0.675399
$764$		0			0
$765$			5.03517e12i			0.531542i
$766$		0			0
$767$	1.37455e13	+	2.06975e12i	1.43411	+	0.215942i
$768$		0			0
$769$			8.88511e11i			0.0916209i	0.998950	+	0.0458104i	$0.0145870\pi$
$769$			8.88511e11i			0.0916209i	−0.998950	+	0.0458104i	$0.985413\pi$
$770$		0			0
$771$	−2.41258e13			−2.45888
$772$		0			0
$773$			3.41558e12i			0.344078i	0.985090	+	0.172039i	$0.0550355\pi$
$773$			3.41558e12i			0.344078i	−0.985090	+	0.172039i	$0.944964\pi$
$774$		0			0
$775$			1.98202e12i			0.197356i
$776$		0			0
$777$	−1.12588e13			−1.10815
$778$		0			0
$779$	1.39521e13			1.35744
$780$		0			0
$781$	−9.55215e12			−0.918695
$782$		0			0
$783$	−6.83053e12			−0.649421
$784$		0			0
$785$		−	3.07471e12i		−	0.288995i
$786$		0			0
$787$		−	1.32536e13i		−	1.23153i	−0.787929	−	0.615766i	$-0.788846\pi$
$787$		−	1.32536e13i		−	1.23153i	0.787929	−	0.615766i	$-0.211154\pi$
$788$		0			0
$789$	7.98099e12			0.733179
$790$		0			0
$791$		−	1.17226e13i		−	1.06470i
$792$		0			0
$793$	6.55420e11	−	4.35275e12i	0.0588560	−	0.390872i
$794$		0			0
$795$			1.72490e13i			1.53148i
$796$		0			0
$797$	7.27248e12			0.638440			0.319220	−	0.947681i	$-0.396579\pi$
$797$	7.27248e12			0.638440			0.319220	+	0.947681i	$0.396579\pi$
$798$		0			0
$799$			7.14720e12i			0.620405i
$800$		0			0
$801$		−	2.90418e12i		−	0.249274i
$802$		0			0
$803$	6.22018e12			0.527938
$804$		0			0
$805$	8.45215e12			0.709391
$806$		0			0
$807$	−4.61269e12			−0.382846
$808$		0			0
$809$	−9.08904e12			−0.746019			−0.373009	−	0.927828i	$-0.621674\pi$
$809$	−9.08904e12			−0.746019			−0.373009	+	0.927828i	$0.621674\pi$
$810$		0			0
$811$			1.91236e13i			1.55230i	0.630550	+	0.776149i	$0.282829\pi$
$811$			1.91236e13i			1.55230i	−0.630550	+	0.776149i	$0.717171\pi$
$812$		0			0
$813$		−	2.68675e13i		−	2.15685i
$814$		0			0
$815$	−8.82107e12			−0.700345
$816$		0			0
$817$		−	1.02051e13i		−	0.801346i
$818$		0			0
$819$	5.42493e12	+	8.16865e11i	0.421324	+	0.0634414i
$820$		0			0
$821$			1.65539e13i			1.27162i	0.771846	+	0.635810i	$0.219334\pi$
$821$			1.65539e13i			1.27162i	−0.771846	+	0.635810i	$0.780666\pi$
$822$		0			0
$823$	5.06045e12			0.384494			0.192247	−	0.981347i	$-0.438422\pi$
$823$	5.06045e12			0.384494			0.192247	+	0.981347i	$0.438422\pi$
$824$		0			0
$825$		−	8.64053e12i		−	0.649378i
$826$		0			0
$827$			1.43485e13i			1.06668i	0.845902	+	0.533339i	$0.179063\pi$
$827$			1.43485e13i			1.06668i	−0.845902	+	0.533339i	$0.820937\pi$
$828$		0			0
$829$	5.82329e12			0.428226			0.214113	−	0.976809i	$-0.431314\pi$
$829$	5.82329e12			0.428226			0.214113	+	0.976809i	$0.431314\pi$
$830$		0			0
$831$	−2.55709e13			−1.86012
$832$		0			0
$833$	−1.01933e13			−0.733522
$834$		0			0
$835$	−5.70094e12			−0.405842
$836$		0			0
$837$			1.93383e12i			0.136193i
$838$		0			0
$839$			7.61697e12i			0.530705i	0.964151	+	0.265353i	$0.0854884\pi$
$839$			7.61697e12i			0.530705i	−0.964151	+	0.265353i	$0.914512\pi$
$840$		0			0
$841$	2.41283e13			1.66320
$842$		0			0
$843$		−	2.44417e13i		−	1.66689i
$844$		0			0
$845$	−2.83954e12	+	9.21513e12i	−0.191599	+	0.621794i
$846$		0			0
$847$		−	2.31226e12i		−	0.154369i
$848$		0			0
$849$	−1.95099e13			−1.28876
$850$		0			0
$851$		−	3.77070e13i		−	2.46456i
$852$		0			0
$853$			2.97056e13i			1.92118i	0.277977	+	0.960588i	$0.410336\pi$
$853$			2.97056e13i			1.92118i	−0.277977	+	0.960588i	$0.589664\pi$
$854$		0			0
$855$	−9.11735e12			−0.583474
$856$		0			0
$857$	−3.97244e12			−0.251561			−0.125780	−	0.992058i	$-0.540144\pi$
$857$	−3.97244e12			−0.251561			−0.125780	+	0.992058i	$0.540144\pi$
$858$		0			0
$859$	3.05640e13			1.91532			0.957658	−	0.287907i	$-0.0929593\pi$
$859$	3.05640e13			1.91532			0.957658	+	0.287907i	$0.0929593\pi$
$860$		0			0
$861$	−1.35374e13			−0.839498
$862$		0			0
$863$		−	2.20925e13i		−	1.35580i	−0.735154	−	0.677900i	$-0.762890\pi$
$863$		−	2.20925e13i		−	1.35580i	0.735154	−	0.677900i	$-0.237110\pi$
$864$		0			0
$865$			7.09722e12i			0.431038i
$866$		0			0
$867$	−8.32906e12			−0.500622
$868$		0			0
$869$			3.47791e12i			0.206885i
$870$		0			0
$871$	1.52684e13	+	2.29906e12i	0.898901	+	0.135353i
$872$		0			0
$873$			1.15228e13i			0.671417i
$874$		0			0
$875$	−1.09162e13			−0.629559
$876$		0			0
$877$			6.40413e12i			0.365563i	0.983154	+	0.182781i	$0.0585101\pi$
$877$			6.40413e12i			0.365563i	−0.983154	+	0.182781i	$0.941490\pi$
$878$		0			0
$879$			1.06346e13i			0.600856i
$880$		0			0
$881$	−1.41451e13			−0.791067			−0.395534	−	0.918452i	$-0.629440\pi$
$881$	−1.41451e13			−0.791067			−0.395534	+	0.918452i	$0.629440\pi$
$882$		0			0
$883$	1.53312e13			0.848698			0.424349	−	0.905499i	$-0.360503\pi$
$883$	1.53312e13			0.848698			0.424349	+	0.905499i	$0.360503\pi$
$884$		0			0
$885$	2.24146e13			1.22825
$886$		0			0
$887$	−1.08759e13			−0.589941			−0.294971	−	0.955506i	$-0.595310\pi$
$887$	−1.08759e13			−0.589941			−0.294971	+	0.955506i	$0.595310\pi$
$888$		0			0
$889$		−	1.17745e13i		−	0.632242i
$890$		0			0
$891$		−	1.97301e13i		−	1.04877i
$892$		0			0
$893$	−1.29417e13			−0.681018
$894$		0			0
$895$		−	3.79674e12i		−	0.197791i
$896$		0			0
$897$	−6.67626e12	+	4.43381e13i	−0.344324	+	2.28671i
$898$		0			0
$899$		−	1.09383e13i		−	0.558511i
$900$		0			0
$901$	−4.20921e13			−2.12784
$902$		0			0
$903$			9.90179e12i			0.495586i
$904$		0			0
$905$			3.76759e12i			0.186700i
$906$		0			0
$907$	−3.66893e13			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$907$	−3.66893e13			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$908$		0			0
$909$	6.07113e12			0.294939
$910$		0			0
$911$	−1.61827e13			−0.778427			−0.389213	−	0.921148i	$-0.627253\pi$
$911$	−1.61827e13			−0.778427			−0.389213	+	0.921148i	$0.627253\pi$
$912$		0			0
$913$	1.40072e13			0.667164
$914$		0			0
$915$		−	7.09795e12i		−	0.334764i
$916$		0			0
$917$			1.33062e13i			0.621431i
$918$		0			0
$919$	−1.63706e13			−0.757086			−0.378543	−	0.925584i	$-0.623575\pi$
$919$	−1.63706e13			−0.757086			−0.378543	+	0.925584i	$0.623575\pi$
$920$		0			0
$921$		−	2.68321e13i		−	1.22881i
$922$		0			0
$923$	−2.31540e13	−	3.48644e12i	−1.05007	−	0.158116i
$924$		0			0
$925$			1.78119e13i			0.799968i
$926$		0			0
$927$	1.72642e13			0.767868
$928$		0			0
$929$			7.05525e12i			0.310772i	0.987854	+	0.155386i	$0.0496621\pi$
$929$			7.05525e12i			0.310772i	−0.987854	+	0.155386i	$0.950338\pi$
$930$		0			0
$931$		−	1.84574e13i		−	0.805187i
$932$		0			0
$933$	−1.70297e13			−0.735764
$934$		0			0
$935$	−1.54790e13			−0.662356
$936$		0			0
$937$	−1.25797e13			−0.533142			−0.266571	−	0.963815i	$-0.585891\pi$
$937$	−1.25797e13			−0.533142			−0.266571	+	0.963815i	$0.585891\pi$
$938$		0			0
$939$	9.84956e12			0.413449
$940$		0			0
$941$			3.28389e13i			1.36532i	0.730735	+	0.682661i	$0.239178\pi$
$941$			3.28389e13i			1.36532i	−0.730735	+	0.682661i	$0.760822\pi$
$942$		0			0
$943$		−	4.53382e13i		−	1.86707i
$944$		0			0
$945$	−3.89552e12			−0.158899
$946$		0			0
$947$		−	1.93983e12i		−	0.0783769i	−0.999232	−	0.0391885i	$-0.987523\pi$
$947$		−	1.93983e12i		−	0.0783769i	0.999232	−	0.0391885i	$-0.0124773\pi$
$948$		0			0
$949$	1.50775e13	+	2.27030e12i	0.603434	+	0.0908628i
$950$		0			0
$951$			5.16560e13i			2.04790i
$952$		0			0
$953$	3.50673e13			1.37716			0.688579	−	0.725161i	$-0.258235\pi$
$953$	3.50673e13			1.37716			0.688579	+	0.725161i	$0.258235\pi$
$954$		0			0
$955$		−	1.20831e13i		−	0.470069i
$956$		0			0
$957$			4.76850e13i			1.83771i
$958$		0			0
$959$	2.26896e13			0.866250
$960$		0			0
$961$	2.33428e13			0.882872
$962$		0			0
$963$	9.31023e12			0.348853
$964$		0			0
$965$	1.37113e13			0.508987
$966$		0			0
$967$		−	2.44116e13i		−	0.897794i	−0.893584	−	0.448897i	$-0.851817\pi$
$967$		−	2.44116e13i		−	0.897794i	0.893584	−	0.448897i	$-0.148183\pi$
$968$		0			0
$969$		−	5.42949e13i		−	1.97835i
$970$		0			0
$971$	−1.36946e13			−0.494384			−0.247192	−	0.968967i	$-0.579508\pi$
$971$	−1.36946e13			−0.494384			−0.247192	+	0.968967i	$0.579508\pi$
$972$		0			0
$973$			1.83038e13i			0.654686i
$974$		0			0
$975$	3.15371e12	−	2.09443e13i	0.111764	−	0.742241i
$976$		0			0
$977$			1.49319e13i			0.524313i	0.965025	+	0.262157i	$0.0844337\pi$
$977$			1.49319e13i			0.524313i	−0.965025	+	0.262157i	$0.915566\pi$
$978$		0			0
$979$	8.92797e12			0.310621
$980$		0			0
$981$		−	2.21640e13i		−	0.764077i
$982$		0			0
$983$			3.51470e13i			1.20060i	0.799776	+	0.600298i	$0.204951\pi$
$983$			3.51470e13i			1.20060i	−0.799776	+	0.600298i	$0.795049\pi$
$984$		0			0
$985$	−4.89119e12			−0.165559
$986$		0			0
$987$	1.25570e13			0.421170
$988$		0			0
$989$	−3.31623e13			−1.10220
$990$		0			0
$991$	1.96775e13			0.648095			0.324048	−	0.946041i	$-0.394956\pi$
$991$	1.96775e13			0.648095			0.324048	+	0.946041i	$0.394956\pi$
$992$		0			0
$993$		−	5.46137e13i		−	1.78250i
$994$		0			0
$995$		−	3.11085e13i		−	1.00618i
$996$		0			0
$997$	3.67522e13			1.17803			0.589013	−	0.808123i	$-0.299517\pi$
$997$	3.67522e13			1.17803			0.589013	+	0.808123i	$0.299517\pi$
$998$		0			0
$999$			1.73788e13i			0.552046i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.f.a.25.6	yes	32
4.3	odd	2		208.10.f.d.129.27		32
13.12	even	2	inner	104.10.f.a.25.5	✓	32
52.51	odd	2		208.10.f.d.129.28		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.5	✓	32	13.12	even	2	inner
104.10.f.a.25.6	yes	32	1.1	even	1	trivial
208.10.f.d.129.27		32	4.3	odd	2
208.10.f.d.129.28		32	52.51	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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-182.615 q^{3} -909.303i q^{5} -3898.48i q^{7} +13665.4 q^{9} +42009.9i q^{11} +(-15333.2 + 101830. i) q^{13} +166053. i q^{15} -405213. q^{17} -733733. i q^{19} +711923. i q^{21} -2.38431e6 q^{23} +1.12629e6 q^{25} +1.09891e6 q^{27} -6.21574e6 q^{29} +1.75978e6i q^{31} -7.67165e6i q^{33} -3.54490e6 q^{35} +1.58146e7i q^{37} +(2.80008e6 - 1.85958e7i) q^{39} +1.90152e7i q^{41} +1.39085e7 q^{43} -1.24260e7i q^{45} -1.76381e7i q^{47} +2.51555e7 q^{49} +7.39982e7 q^{51} +1.03876e8 q^{53} +3.81997e7 q^{55} +1.33991e8i q^{57} -1.34985e8i q^{59} -4.27452e7 q^{61} -5.32743e7i q^{63} +(9.25945e7 + 1.39425e7i) q^{65} -1.49940e8i q^{67} +4.35412e8 q^{69} +2.27379e8i q^{71} -1.48065e8i q^{73} -2.05679e8 q^{75} +1.63775e8 q^{77} +8.27879e7 q^{79} -4.69653e8 q^{81} -3.33426e8i q^{83} +3.68461e8i q^{85} +1.13509e9 q^{87} -2.12521e8i q^{89} +(3.96983e8 + 5.97762e7i) q^{91} -3.21362e8i q^{93} -6.67186e8 q^{95} +8.43207e8i q^{97} +5.74081e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) \) 32 * q - 162 * q^3 + 223074 * q^9 + 66270 * q^13 - 487902 * q^17 + 3171556 * q^23 - 13526722 * q^25 - 3694974 * q^27 + 8833508 * q^29 - 8281126 * q^35 - 12056860 * q^39 + 89959038 * q^43 - 172344874 * q^49 + 51206562 * q^51 + 139521112 * q^53 - 84550176 * q^55 + 372036836 * q^61 + 241471810 * q^65 - 486052516 * q^69 - 604473728 * q^75 - 1666197144 * q^77 + 1165982916 * q^79 + 859636544 * q^81 - 1738691296 * q^87 - 291718146 * q^91 + 1741143356 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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