LMFDB - Embedded newform 128.12.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,12,Mod(1,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("128.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	128.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$98.3479271116$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 4442x^{4} + 153566x^{3} - 1333532x^{2} - 4433532x + 49754286 $ x^6 - 2x^5 - 4442x^4 + 153566x^3 - 1333532x^2 - 4433532*x + 49754286
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{43}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-80.0003$ of defining polynomial
Character	$\chi$	$=$	128.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.4295 q^{3} -13262.1 q^{5} +35677.5 q^{7} -176339. q^{9} -851940. q^{11} -50601.4 q^{13} -377034. q^{15} +1.30790e6 q^{17} -1.84759e7 q^{19} +1.01429e6 q^{21} -3.49995e7 q^{23} +1.27054e8 q^{25} -1.00494e7 q^{27} -1.74392e8 q^{29} -1.31509e8 q^{31} -2.42202e7 q^{33} -4.73157e8 q^{35} -5.07196e8 q^{37} -1.43857e6 q^{39} -2.56635e8 q^{41} +1.53556e9 q^{43} +2.33862e9 q^{45} -1.17238e9 q^{47} -7.04446e8 q^{49} +3.71829e7 q^{51} +1.84474e9 q^{53} +1.12985e10 q^{55} -5.25260e8 q^{57} +6.00452e9 q^{59} -4.42882e9 q^{61} -6.29132e9 q^{63} +6.71080e8 q^{65} -1.09668e10 q^{67} -9.95018e8 q^{69} -1.03768e10 q^{71} +1.07011e10 q^{73} +3.61209e9 q^{75} -3.03951e10 q^{77} -2.37762e10 q^{79} +3.09522e10 q^{81} -3.91314e9 q^{83} -1.73454e10 q^{85} -4.95788e9 q^{87} +1.28889e10 q^{89} -1.80533e9 q^{91} -3.73873e9 q^{93} +2.45028e11 q^{95} +6.65565e9 q^{97} +1.50230e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 20 q^{3} - 1804 q^{5} + 49368 q^{7} + 313814 q^{9} - 688460 q^{11} - 2290348 q^{13} + 4828264 q^{15} + 4127636 q^{17} + 9936364 q^{19} + 20325616 q^{21} + 9921320 q^{23} + 51633002 q^{25} - 132503384 q^{27}+ \cdots + 240482467988 q^{99}+O(q^{100}) $ 6 * q - 20 * q^3 - 1804 * q^5 + 49368 * q^7 + 313814 * q^9 - 688460 * q^11 - 2290348 * q^13 + 4828264 * q^15 + 4127636 * q^17 + 9936364 * q^19 + 20325616 * q^21 + 9921320 * q^23 + 51633002 * q^25 - 132503384 * q^27 + 25222532 * q^29 - 246133120 * q^31 - 337991288 * q^33 + 227061200 * q^35 - 660146172 * q^37 + 648086056 * q^39 + 486915836 * q^41 + 1147261348 * q^43 + 626323028 * q^45 + 603975088 * q^47 + 1846879062 * q^49 - 1556268056 * q^51 - 428043996 * q^53 - 1355415976 * q^55 - 783352648 * q^57 + 809510676 * q^59 - 2822052332 * q^61 + 14526601496 * q^63 + 1376905688 * q^65 + 9742675036 * q^67 - 8441748208 * q^69 + 16181067512 * q^71 + 35963668692 * q^73 + 51500880116 * q^75 - 72754679408 * q^77 - 6103946192 * q^79 + 81240620990 * q^81 + 67929536124 * q^83 - 76741413928 * q^85 + 132686631816 * q^87 + 22721416116 * q^89 + 44438236496 * q^91 - 37482073344 * q^93 + 282717822568 * q^95 + 218896625492 * q^97 + 240482467988 * 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
$3$	28.4295	0.0675464	0.0337732	−	0.999430i	$-0.489248\pi$
$3$	28.4295	0.0675464	0.0337732	+	0.999430i	$0.489248\pi$
$4$	0	0
$5$	−13262.1	−1.89791	−0.948956	−	0.315407i	$-0.897859\pi$
$5$	−13262.1	−1.89791	−0.948956	+	0.315407i	$0.897859\pi$
$6$	0	0
$7$	35677.5	0.802333	0.401167	−	0.916005i	$-0.368605\pi$
$7$	35677.5	0.802333	0.401167	+	0.916005i	$0.368605\pi$
$8$	0	0
$9$	−176339.	−0.995437
$10$	0	0
$11$	−851940.	−1.59496	−0.797479	−	0.603346i	$-0.793834\pi$
$11$	−851940.	−1.59496	−0.797479	+	0.603346i	$0.793834\pi$
$12$	0	0
$13$	−50601.4	−0.0377985	−0.0188992	−	0.999821i	$-0.506016\pi$
$13$	−50601.4	−0.0377985	−0.0188992	+	0.999821i	$0.506016\pi$
$14$	0	0
$15$	−377034.	−0.128197
$16$	0	0
$17$	1.30790e6	0.223411	0.111706	−	0.993741i	$-0.464369\pi$
$17$	1.30790e6	0.223411	0.111706	+	0.993741i	$0.464369\pi$
$18$	0	0
$19$	−1.84759e7	−1.71183	−0.855914	−	0.517118i	$-0.827005\pi$
$19$	−1.84759e7	−1.71183	−0.855914	+	0.517118i	$0.827005\pi$
$20$	0	0
$21$	1.01429e6	0.0541947
$22$	0	0
$23$	−3.49995e7	−1.13386	−0.566929	−	0.823767i	$-0.691869\pi$
$23$	−3.49995e7	−1.13386	−0.566929	+	0.823767i	$0.691869\pi$
$24$	0	0
$25$	1.27054e8	2.60207
$26$	0	0
$27$	−1.00494e7	−0.134785
$28$	0	0
$29$	−1.74392e8	−1.57884	−0.789419	−	0.613854i	$-0.789618\pi$
$29$	−1.74392e8	−1.57884	−0.789419	+	0.613854i	$0.789618\pi$
$30$	0	0
$31$	−1.31509e8	−0.825023	−0.412511	−	0.910952i	$-0.635348\pi$
$31$	−1.31509e8	−0.825023	−0.412511	+	0.910952i	$0.635348\pi$
$32$	0	0
$33$	−2.42202e7	−0.107734
$34$	0	0
$35$	−4.73157e8	−1.52276
$36$	0	0
$37$	−5.07196e8	−1.20245	−0.601224	−	0.799080i	$-0.705320\pi$
$37$	−5.07196e8	−1.20245	−0.601224	+	0.799080i	$0.705320\pi$
$38$	0	0
$39$	−1.43857e6	−0.00255315
$40$	0	0
$41$	−2.56635e8	−0.345943	−0.172972	−	0.984927i	$-0.555337\pi$
$41$	−2.56635e8	−0.345943	−0.172972	+	0.984927i	$0.555337\pi$
$42$	0	0
$43$	1.53556e9	1.59291	0.796455	−	0.604697i	$-0.206706\pi$
$43$	1.53556e9	1.59291	0.796455	+	0.604697i	$0.206706\pi$
$44$	0	0
$45$	2.33862e9	1.88925
$46$	0	0
$47$	−1.17238e9	−0.745643	−0.372822	−	0.927903i	$-0.621610\pi$
$47$	−1.17238e9	−0.745643	−0.372822	+	0.927903i	$0.621610\pi$
$48$	0	0
$49$	−7.04446e8	−0.356262
$50$	0	0
$51$	3.71829e7	0.0150906
$52$	0	0
$53$	1.84474e9	0.605923	0.302961	−	0.953003i	$-0.402025\pi$
$53$	1.84474e9	0.605923	0.302961	+	0.953003i	$0.402025\pi$
$54$	0	0
$55$	1.12985e10	3.02709
$56$	0	0
$57$	−5.25260e8	−0.115628
$58$	0	0
$59$	6.00452e9	1.09343	0.546716	−	0.837318i	$-0.315878\pi$
$59$	6.00452e9	1.09343	0.546716	+	0.837318i	$0.315878\pi$
$60$	0	0
$61$	−4.42882e9	−0.671389	−0.335694	−	0.941971i	$-0.608971\pi$
$61$	−4.42882e9	−0.671389	−0.335694	+	0.941971i	$0.608971\pi$
$62$	0	0
$63$	−6.29132e9	−0.798672
$64$	0	0
$65$	6.71080e8	0.0717382
$66$	0	0
$67$	−1.09668e10	−0.992361	−0.496181	−	0.868219i	$-0.665265\pi$
$67$	−1.09668e10	−0.992361	−0.496181	+	0.868219i	$0.665265\pi$
$68$	0	0
$69$	−9.95018e8	−0.0765880
$70$	0	0
$71$	−1.03768e10	−0.682562	−0.341281	−	0.939961i	$-0.610861\pi$
$71$	−1.03768e10	−0.682562	−0.341281	+	0.939961i	$0.610861\pi$
$72$	0	0
$73$	1.07011e10	0.604160	0.302080	−	0.953283i	$-0.402319\pi$
$73$	1.07011e10	0.604160	0.302080	+	0.953283i	$0.402319\pi$
$74$	0	0
$75$	3.61209e9	0.175761
$76$	0	0
$77$	−3.03951e10	−1.27969
$78$	0	0
$79$	−2.37762e10	−0.869349	−0.434674	−	0.900588i	$-0.643137\pi$
$79$	−2.37762e10	−0.869349	−0.434674	+	0.900588i	$0.643137\pi$
$80$	0	0
$81$	3.09522e10	0.986333
$82$	0	0
$83$	−3.91314e9	−0.109043	−0.0545213	−	0.998513i	$-0.517363\pi$
$83$	−3.91314e9	−0.109043	−0.0545213	+	0.998513i	$0.517363\pi$
$84$	0	0
$85$	−1.73454e10	−0.424015
$86$	0	0
$87$	−4.95788e9	−0.106645
$88$	0	0
$89$	1.28889e10	0.244665	0.122333	−	0.992489i	$-0.460963\pi$
$89$	1.28889e10	0.244665	0.122333	+	0.992489i	$0.460963\pi$
$90$	0	0
$91$	−1.80533e9	−0.0303270
$92$	0	0
$93$	−3.73873e9	−0.0557273
$94$	0	0
$95$	2.45028e11	3.24890
$96$	0	0
$97$	6.65565e9	0.0786948	0.0393474	−	0.999226i	$-0.487472\pi$
$97$	6.65565e9	0.0786948	0.0393474	+	0.999226i	$0.487472\pi$
$98$	0	0
$99$	1.50230e11	1.58768
$100$	0	0
$101$	−6.98312e10	−0.661122	−0.330561	−	0.943785i	$-0.607238\pi$
$101$	−6.98312e10	−0.661122	−0.330561	+	0.943785i	$0.607238\pi$
$102$	0	0
$103$	2.88946e10	0.245591	0.122795	−	0.992432i	$-0.460814\pi$
$103$	2.88946e10	0.245591	0.122795	+	0.992432i	$0.460814\pi$
$104$	0	0
$105$	−1.34516e10	−0.102857
$106$	0	0
$107$	2.38747e10	0.164561	0.0822804	−	0.996609i	$-0.473780\pi$
$107$	2.38747e10	0.164561	0.0822804	+	0.996609i	$0.473780\pi$
$108$	0	0
$109$	2.00198e11	1.24628	0.623138	−	0.782112i	$-0.285857\pi$
$109$	2.00198e11	1.24628	0.623138	+	0.782112i	$0.285857\pi$
$110$	0	0
$111$	−1.44193e10	−0.0812211
$112$	0	0
$113$	−1.90785e9	−0.00974122	−0.00487061	−	0.999988i	$-0.501550\pi$
$113$	−1.90785e9	−0.00974122	−0.00487061	+	0.999988i	$0.501550\pi$
$114$	0	0
$115$	4.64166e11	2.15196
$116$	0	0
$117$	8.92300e9	0.0376260
$118$	0	0
$119$	4.66625e10	0.179250
$120$	0	0
$121$	4.40491e11	1.54389
$122$	0	0
$123$	−7.29600e9	−0.0233672
$124$	0	0
$125$	−1.03744e12	−3.04059
$126$	0	0
$127$	3.33442e11	0.895570	0.447785	−	0.894141i	$-0.352213\pi$
$127$	3.33442e11	0.895570	0.447785	+	0.894141i	$0.352213\pi$
$128$	0	0
$129$	4.36553e10	0.107595
$130$	0	0
$131$	−5.45163e11	−1.23462	−0.617312	−	0.786719i	$-0.711778\pi$
$131$	−5.45163e11	−1.23462	−0.617312	+	0.786719i	$0.711778\pi$
$132$	0	0
$133$	−6.59172e11	−1.37346
$134$	0	0
$135$	1.33276e11	0.255809
$136$	0	0
$137$	5.79293e11	1.02550	0.512749	−	0.858538i	$-0.328627\pi$
$137$	5.79293e11	1.02550	0.512749	+	0.858538i	$0.328627\pi$
$138$	0	0
$139$	9.38023e11	1.53332	0.766659	−	0.642055i	$-0.221918\pi$
$139$	9.38023e11	1.53332	0.766659	+	0.642055i	$0.221918\pi$
$140$	0	0
$141$	−3.33302e10	−0.0503655
$142$	0	0
$143$	4.31094e10	0.0602870
$144$	0	0
$145$	2.31280e12	2.99650
$146$	0	0
$147$	−2.00270e10	−0.0240642
$148$	0	0
$149$	−5.08299e11	−0.567016	−0.283508	−	0.958970i	$-0.591498\pi$
$149$	−5.08299e11	−0.567016	−0.283508	+	0.958970i	$0.591498\pi$
$150$	0	0
$151$	1.32466e12	1.37319	0.686594	−	0.727041i	$-0.259105\pi$
$151$	1.32466e12	1.37319	0.686594	+	0.727041i	$0.259105\pi$
$152$	0	0
$153$	−2.30633e11	−0.222392
$154$	0	0
$155$	1.74408e12	1.56582
$156$	0	0
$157$	5.64872e11	0.472609	0.236304	−	0.971679i	$-0.424064\pi$
$157$	5.64872e11	0.472609	0.236304	+	0.971679i	$0.424064\pi$
$158$	0	0
$159$	5.24450e10	0.0409279
$160$	0	0
$161$	−1.24869e12	−0.909732
$162$	0	0
$163$	−6.31830e11	−0.430099	−0.215050	−	0.976603i	$-0.568991\pi$
$163$	−6.31830e11	−0.430099	−0.215050	+	0.976603i	$0.568991\pi$
$164$	0	0
$165$	3.21210e11	0.204469
$166$	0	0
$167$	1.06226e12	0.632834	0.316417	−	0.948620i	$-0.397520\pi$
$167$	1.06226e12	0.632834	0.316417	+	0.948620i	$0.397520\pi$
$168$	0	0
$169$	−1.78960e12	−0.998571
$170$	0	0
$171$	3.25801e12	1.70402
$172$	0	0
$173$	−2.41005e12	−1.18242	−0.591211	−	0.806517i	$-0.701350\pi$
$173$	−2.41005e12	−1.18242	−0.591211	+	0.806517i	$0.701350\pi$
$174$	0	0
$175$	4.53298e12	2.08773
$176$	0	0
$177$	1.70705e11	0.0738574
$178$	0	0
$179$	2.21326e12	0.900204	0.450102	−	0.892977i	$-0.351388\pi$
$179$	2.21326e12	0.900204	0.450102	+	0.892977i	$0.351388\pi$
$180$	0	0
$181$	−7.50395e11	−0.287116	−0.143558	−	0.989642i	$-0.545854\pi$
$181$	−7.50395e11	−0.287116	−0.143558	+	0.989642i	$0.545854\pi$
$182$	0	0
$183$	−1.25909e11	−0.0453499
$184$	0	0
$185$	6.72647e12	2.28214
$186$	0	0
$187$	−1.11425e12	−0.356332
$188$	0	0
$189$	−3.58538e11	−0.108142
$190$	0	0
$191$	−2.98912e12	−0.850862	−0.425431	−	0.904991i	$-0.639877\pi$
$191$	−2.98912e12	−0.850862	−0.425431	+	0.904991i	$0.639877\pi$
$192$	0	0
$193$	−3.63183e12	−0.976248	−0.488124	−	0.872774i	$-0.662319\pi$
$193$	−3.63183e12	−0.976248	−0.488124	+	0.872774i	$0.662319\pi$
$194$	0	0
$195$	1.90785e10	0.00484566
$196$	0	0
$197$	−2.86938e12	−0.689008	−0.344504	−	0.938785i	$-0.611953\pi$
$197$	−2.86938e12	−0.689008	−0.344504	+	0.938785i	$0.611953\pi$
$198$	0	0
$199$	−4.15302e12	−0.943349	−0.471675	−	0.881773i	$-0.656350\pi$
$199$	−4.15302e12	−0.943349	−0.471675	+	0.881773i	$0.656350\pi$
$200$	0	0
$201$	−3.11782e11	−0.0670304
$202$	0	0
$203$	−6.22186e12	−1.26675
$204$	0	0
$205$	3.40351e12	0.656570
$206$	0	0
$207$	6.17177e12	1.12868
$208$	0	0
$209$	1.57403e13	2.73029
$210$	0	0
$211$	−4.58182e12	−0.754197	−0.377099	−	0.926173i	$-0.623078\pi$
$211$	−4.58182e12	−0.754197	−0.377099	+	0.926173i	$0.623078\pi$
$212$	0	0
$213$	−2.95007e11	−0.0461046
$214$	0	0
$215$	−2.03648e13	−3.02321
$216$	0	0
$217$	−4.69190e12	−0.661943
$218$	0	0
$219$	3.04226e11	0.0408088
$220$	0	0
$221$	−6.61816e10	−0.00844461
$222$	0	0
$223$	1.33092e13	1.61613	0.808065	−	0.589094i	$-0.200515\pi$
$223$	1.33092e13	1.61613	0.808065	+	0.589094i	$0.200515\pi$
$224$	0	0
$225$	−2.24046e13	−2.59020
$226$	0	0
$227$	−1.34550e13	−1.48163	−0.740817	−	0.671707i	$-0.765561\pi$
$227$	−1.34550e13	−1.48163	−0.740817	+	0.671707i	$0.765561\pi$
$228$	0	0
$229$	7.47737e12	0.784610	0.392305	−	0.919835i	$-0.371678\pi$
$229$	7.47737e12	0.784610	0.392305	+	0.919835i	$0.371678\pi$
$230$	0	0
$231$	−8.64116e11	−0.0864383
$232$	0	0
$233$	−1.27643e13	−1.21770	−0.608848	−	0.793287i	$-0.708368\pi$
$233$	−1.27643e13	−1.21770	−0.608848	+	0.793287i	$0.708368\pi$
$234$	0	0
$235$	1.55482e13	1.41517
$236$	0	0
$237$	−6.75946e11	−0.0587214
$238$	0	0
$239$	−5.85201e12	−0.485419	−0.242709	−	0.970099i	$-0.578036\pi$
$239$	−5.85201e12	−0.485419	−0.242709	+	0.970099i	$0.578036\pi$
$240$	0	0
$241$	7.45977e12	0.591060	0.295530	−	0.955333i	$-0.404504\pi$
$241$	7.45977e12	0.591060	0.295530	+	0.955333i	$0.404504\pi$
$242$	0	0
$243$	2.66018e12	0.201408
$244$	0	0
$245$	9.34241e12	0.676154
$246$	0	0
$247$	9.34906e11	0.0647045
$248$	0	0
$249$	−1.11249e11	−0.00736543
$250$	0	0
$251$	1.40102e13	0.887643	0.443821	−	0.896115i	$-0.353622\pi$
$251$	1.40102e13	0.887643	0.443821	+	0.896115i	$0.353622\pi$
$252$	0	0
$253$	2.98175e13	1.80846
$254$	0	0
$255$	−4.93122e11	−0.0286407
$256$	0	0
$257$	−2.27084e13	−1.26344	−0.631718	−	0.775198i	$-0.717650\pi$
$257$	−2.27084e13	−1.26344	−0.631718	+	0.775198i	$0.717650\pi$
$258$	0	0
$259$	−1.80955e13	−0.964764
$260$	0	0
$261$	3.07521e13	1.57163
$262$	0	0
$263$	−3.06356e13	−1.50131	−0.750654	−	0.660695i	$-0.770262\pi$
$263$	−3.06356e13	−1.50131	−0.750654	+	0.660695i	$0.770262\pi$
$264$	0	0
$265$	−2.44650e13	−1.14999
$266$	0	0
$267$	3.66426e11	0.0165263
$268$	0	0
$269$	−2.34794e13	−1.01637	−0.508183	−	0.861249i	$-0.669683\pi$
$269$	−2.34794e13	−1.01637	−0.508183	+	0.861249i	$0.669683\pi$
$270$	0	0
$271$	−1.59575e13	−0.663182	−0.331591	−	0.943423i	$-0.607585\pi$
$271$	−1.59575e13	−0.663182	−0.331591	+	0.943423i	$0.607585\pi$
$272$	0	0
$273$	−5.13246e10	−0.00204848
$274$	0	0
$275$	−1.08243e14	−4.15020
$276$	0	0
$277$	−7.95571e12	−0.293117	−0.146558	−	0.989202i	$-0.546820\pi$
$277$	−7.95571e12	−0.293117	−0.146558	+	0.989202i	$0.546820\pi$
$278$	0	0
$279$	2.31901e13	0.821259
$280$	0	0
$281$	3.25060e13	1.10682	0.553412	−	0.832908i	$-0.313326\pi$
$281$	3.25060e13	1.10682	0.553412	+	0.832908i	$0.313326\pi$
$282$	0	0
$283$	1.98411e12	0.0649742	0.0324871	−	0.999472i	$-0.489657\pi$
$283$	1.98411e12	0.0649742	0.0324871	+	0.999472i	$0.489657\pi$
$284$	0	0
$285$	6.96603e12	0.219451
$286$	0	0
$287$	−9.15609e12	−0.277562
$288$	0	0
$289$	−3.25613e13	−0.950087
$290$	0	0
$291$	1.89217e11	0.00531555
$292$	0	0
$293$	−3.80297e13	−1.02885	−0.514423	−	0.857536i	$-0.671994\pi$
$293$	−3.80297e13	−1.02885	−0.514423	+	0.857536i	$0.671994\pi$
$294$	0	0
$295$	−7.96323e13	−2.07524
$296$	0	0
$297$	8.56151e12	0.214976
$298$	0	0
$299$	1.77103e12	0.0428581
$300$	0	0
$301$	5.47850e13	1.27805
$302$	0	0
$303$	−1.98527e12	−0.0446564
$304$	0	0
$305$	5.87353e13	1.27424
$306$	0	0
$307$	1.83935e13	0.384950	0.192475	−	0.981302i	$-0.438349\pi$
$307$	1.83935e13	0.384950	0.192475	+	0.981302i	$0.438349\pi$
$308$	0	0
$309$	8.21459e11	0.0165888
$310$	0	0
$311$	−6.91084e13	−1.34694	−0.673470	−	0.739214i	$-0.735197\pi$
$311$	−6.91084e13	−1.34694	−0.673470	+	0.739214i	$0.735197\pi$
$312$	0	0
$313$	−4.83426e13	−0.909571	−0.454785	−	0.890601i	$-0.650284\pi$
$313$	−4.83426e13	−0.909571	−0.454785	+	0.890601i	$0.650284\pi$
$314$	0	0
$315$	8.34359e13	1.51581
$316$	0	0
$317$	−5.44259e13	−0.954948	−0.477474	−	0.878646i	$-0.658448\pi$
$317$	−5.44259e13	−0.954948	−0.477474	+	0.878646i	$0.658448\pi$
$318$	0	0
$319$	1.48572e14	2.51818
$320$	0	0
$321$	6.78745e11	0.0111155
$322$	0	0
$323$	−2.41646e13	−0.382442
$324$	0	0
$325$	−6.42913e12	−0.0983544
$326$	0	0
$327$	5.69153e12	0.0841815
$328$	0	0
$329$	−4.18276e13	−0.598254
$330$	0	0
$331$	3.20522e13	0.443409	0.221704	−	0.975114i	$-0.428838\pi$
$331$	3.20522e13	0.443409	0.221704	+	0.975114i	$0.428838\pi$
$332$	0	0
$333$	8.94383e13	1.19696
$334$	0	0
$335$	1.45443e14	1.88342
$336$	0	0
$337$	−1.13046e14	−1.41675	−0.708373	−	0.705838i	$-0.750571\pi$
$337$	−1.13046e14	−1.41675	−0.708373	+	0.705838i	$0.750571\pi$
$338$	0	0
$339$	−5.42393e10	−0.000657985 0
$340$	0	0
$341$	1.12038e14	1.31588
$342$	0	0
$343$	−9.56788e13	−1.08817
$344$	0	0
$345$	1.31960e13	0.145357
$346$	0	0
$347$	−3.64920e13	−0.389390	−0.194695	−	0.980864i	$-0.562372\pi$
$347$	−3.64920e13	−0.389390	−0.194695	+	0.980864i	$0.562372\pi$
$348$	0	0
$349$	−1.27749e14	−1.32074	−0.660372	−	0.750938i	$-0.729602\pi$
$349$	−1.27749e14	−1.32074	−0.660372	+	0.750938i	$0.729602\pi$
$350$	0	0
$351$	5.08515e11	0.00509466
$352$	0	0
$353$	1.13630e14	1.10340	0.551699	−	0.834043i	$-0.313980\pi$
$353$	1.13630e14	1.10340	0.551699	+	0.834043i	$0.313980\pi$
$354$	0	0
$355$	1.37618e14	1.29544
$356$	0	0
$357$	1.32659e12	0.0121077
$358$	0	0
$359$	1.28665e14	1.13879	0.569393	−	0.822065i	$-0.307178\pi$
$359$	1.28665e14	1.13879	0.569393	+	0.822065i	$0.307178\pi$
$360$	0	0
$361$	2.24867e14	1.93035
$362$	0	0
$363$	1.25229e13	0.104284
$364$	0	0
$365$	−1.41918e14	−1.14664
$366$	0	0
$367$	1.35315e14	1.06092	0.530459	−	0.847711i	$-0.322020\pi$
$367$	1.35315e14	1.06092	0.530459	+	0.847711i	$0.322020\pi$
$368$	0	0
$369$	4.52547e13	0.344365
$370$	0	0
$371$	6.58156e13	0.486152
$372$	0	0
$373$	−1.74152e14	−1.24891	−0.624453	−	0.781063i	$-0.714678\pi$
$373$	−1.74152e14	−1.24891	−0.624453	+	0.781063i	$0.714678\pi$
$374$	0	0
$375$	−2.94939e13	−0.205381
$376$	0	0
$377$	8.82449e12	0.0596777
$378$	0	0
$379$	9.30974e13	0.611536	0.305768	−	0.952106i	$-0.401087\pi$
$379$	9.30974e13	0.611536	0.305768	+	0.952106i	$0.401087\pi$
$380$	0	0
$381$	9.47958e12	0.0604925
$382$	0	0
$383$	−2.11399e14	−1.31072	−0.655359	−	0.755317i	$-0.727483\pi$
$383$	−2.11399e14	−1.31072	−0.655359	+	0.755317i	$0.727483\pi$
$384$	0	0
$385$	4.03101e14	2.42874
$386$	0	0
$387$	−2.70779e14	−1.58564
$388$	0	0
$389$	−1.99727e14	−1.13688	−0.568438	−	0.822726i	$-0.692452\pi$
$389$	−1.99727e14	−1.13688	−0.568438	+	0.822726i	$0.692452\pi$
$390$	0	0
$391$	−4.57758e13	−0.253317
$392$	0	0
$393$	−1.54987e13	−0.0833944
$394$	0	0
$395$	3.15322e14	1.64995
$396$	0	0
$397$	−1.28753e14	−0.655255	−0.327627	−	0.944807i	$-0.606249\pi$
$397$	−1.28753e14	−0.655255	−0.327627	+	0.944807i	$0.606249\pi$
$398$	0	0
$399$	−1.87399e13	−0.0927720
$400$	0	0
$401$	−1.95881e14	−0.943407	−0.471703	−	0.881757i	$-0.656361\pi$
$401$	−1.95881e14	−0.943407	−0.471703	+	0.881757i	$0.656361\pi$
$402$	0	0
$403$	6.65454e12	0.0311846
$404$	0	0
$405$	−4.10490e14	−1.87197
$406$	0	0
$407$	4.32101e14	1.91786
$408$	0	0
$409$	−2.68622e13	−0.116055	−0.0580273	−	0.998315i	$-0.518481\pi$
$409$	−2.68622e13	−0.116055	−0.0580273	+	0.998315i	$0.518481\pi$
$410$	0	0
$411$	1.64690e13	0.0692687
$412$	0	0
$413$	2.14226e14	0.877297
$414$	0	0
$415$	5.18963e13	0.206953
$416$	0	0
$417$	2.66675e13	0.103570
$418$	0	0
$419$	7.77684e13	0.294189	0.147094	−	0.989122i	$-0.453008\pi$
$419$	7.77684e13	0.294189	0.147094	+	0.989122i	$0.453008\pi$
$420$	0	0
$421$	2.69602e14	0.993508	0.496754	−	0.867891i	$-0.334525\pi$
$421$	2.69602e14	0.993508	0.496754	+	0.867891i	$0.334525\pi$
$422$	0	0
$423$	2.06737e14	0.742241
$424$	0	0
$425$	1.66174e14	0.581332
$426$	0	0
$427$	−1.58009e14	−0.538677
$428$	0	0
$429$	1.22558e12	0.00407217
$430$	0	0
$431$	4.84920e13	0.157053	0.0785263	−	0.996912i	$-0.474979\pi$
$431$	4.84920e13	0.157053	0.0785263	+	0.996912i	$0.474979\pi$
$432$	0	0
$433$	−5.92881e14	−1.87191	−0.935953	−	0.352126i	$-0.885459\pi$
$433$	−5.92881e14	−1.87191	−0.935953	+	0.352126i	$0.885459\pi$
$434$	0	0
$435$	6.57517e13	0.202403
$436$	0	0
$437$	6.46646e14	1.94097
$438$	0	0
$439$	1.00830e14	0.295145	0.147572	−	0.989051i	$-0.452854\pi$
$439$	1.00830e14	0.295145	0.147572	+	0.989051i	$0.452854\pi$
$440$	0	0
$441$	1.24221e14	0.354636
$442$	0	0
$443$	6.55676e14	1.82587	0.912933	−	0.408109i	$-0.133812\pi$
$443$	6.55676e14	1.82587	0.912933	+	0.408109i	$0.133812\pi$
$444$	0	0
$445$	−1.70934e14	−0.464353
$446$	0	0
$447$	−1.44507e13	−0.0382999
$448$	0	0
$449$	4.52693e14	1.17071	0.585354	−	0.810778i	$-0.300956\pi$
$449$	4.52693e14	1.17071	0.585354	+	0.810778i	$0.300956\pi$
$450$	0	0
$451$	2.18638e14	0.551765
$452$	0	0
$453$	3.76593e13	0.0927539
$454$	0	0
$455$	2.39424e13	0.0575580
$456$	0	0
$457$	3.05085e14	0.715948	0.357974	−	0.933732i	$-0.383468\pi$
$457$	3.05085e14	0.715948	0.357974	+	0.933732i	$0.383468\pi$
$458$	0	0
$459$	−1.31436e13	−0.0301124
$460$	0	0
$461$	1.73122e14	0.387255	0.193627	−	0.981075i	$-0.437975\pi$
$461$	1.73122e14	0.387255	0.193627	+	0.981075i	$0.437975\pi$
$462$	0	0
$463$	−4.87251e13	−0.106428	−0.0532142	−	0.998583i	$-0.516947\pi$
$463$	−4.87251e13	−0.106428	−0.0532142	+	0.998583i	$0.516947\pi$
$464$	0	0
$465$	4.95833e13	0.105766
$466$	0	0
$467$	4.70608e14	0.980431	0.490215	−	0.871601i	$-0.336918\pi$
$467$	4.70608e14	0.980431	0.490215	+	0.871601i	$0.336918\pi$
$468$	0	0
$469$	−3.91269e14	−0.796204
$470$	0	0
$471$	1.60590e13	0.0319230
$472$	0	0
$473$	−1.30821e15	−2.54063
$474$	0	0
$475$	−2.34744e15	−4.45430
$476$	0	0
$477$	−3.25299e14	−0.603158
$478$	0	0
$479$	−1.74144e14	−0.315547	−0.157774	−	0.987475i	$-0.550432\pi$
$479$	−1.74144e14	−0.315547	−0.157774	+	0.987475i	$0.550432\pi$
$480$	0	0
$481$	2.56649e13	0.0454507
$482$	0	0
$483$	−3.54997e13	−0.0614491
$484$	0	0
$485$	−8.82677e13	−0.149356
$486$	0	0
$487$	−7.26735e14	−1.20217	−0.601086	−	0.799184i	$-0.705265\pi$
$487$	−7.26735e14	−1.20217	−0.601086	+	0.799184i	$0.705265\pi$
$488$	0	0
$489$	−1.79626e13	−0.0290517
$490$	0	0
$491$	−9.40666e14	−1.48760	−0.743802	−	0.668400i	$-0.766979\pi$
$491$	−9.40666e14	−1.48760	−0.743802	+	0.668400i	$0.766979\pi$
$492$	0	0
$493$	−2.28087e14	−0.352730
$494$	0	0
$495$	−1.99236e15	−3.01328
$496$	0	0
$497$	−3.70217e14	−0.547642
$498$	0	0
$499$	1.04445e15	1.51124	0.755619	−	0.655012i	$-0.227336\pi$
$499$	1.04445e15	1.51124	0.755619	+	0.655012i	$0.227336\pi$
$500$	0	0
$501$	3.01995e13	0.0427456
$502$	0	0
$503$	−1.27010e14	−0.175879	−0.0879393	−	0.996126i	$-0.528028\pi$
$503$	−1.27010e14	−0.175879	−0.0879393	+	0.996126i	$0.528028\pi$
$504$	0	0
$505$	9.26106e14	1.25475
$506$	0	0
$507$	−5.08774e13	−0.0674499
$508$	0	0
$509$	1.46270e15	1.89762	0.948808	−	0.315852i	$-0.102290\pi$
$509$	1.46270e15	1.89762	0.948808	+	0.315852i	$0.102290\pi$
$510$	0	0
$511$	3.81787e14	0.484737
$512$	0	0
$513$	1.85672e14	0.230728
$514$	0	0
$515$	−3.83202e14	−0.466110
$516$	0	0
$517$	9.98800e14	1.18927
$518$	0	0
$519$	−6.85166e13	−0.0798684
$520$	0	0
$521$	−1.17275e15	−1.33844	−0.669218	−	0.743066i	$-0.733371\pi$
$521$	−1.17275e15	−1.33844	−0.669218	+	0.743066i	$0.733371\pi$
$522$	0	0
$523$	−9.04972e14	−1.01129	−0.505645	−	0.862742i	$-0.668745\pi$
$523$	−9.04972e14	−1.01129	−0.505645	+	0.862742i	$0.668745\pi$
$524$	0	0
$525$	1.28870e14	0.141019
$526$	0	0
$527$	−1.72000e14	−0.184319
$528$	0	0
$529$	2.72155e14	0.285634
$530$	0	0
$531$	−1.05883e15	−1.08844
$532$	0	0
$533$	1.29861e13	0.0130761
$534$	0	0
$535$	−3.16627e14	−0.312322
$536$	0	0
$537$	6.29219e13	0.0608056
$538$	0	0
$539$	6.00146e14	0.568223
$540$	0	0
$541$	3.44934e14	0.320000	0.160000	−	0.987117i	$-0.448851\pi$
$541$	3.44934e14	0.320000	0.160000	+	0.987117i	$0.448851\pi$
$542$	0	0
$543$	−2.13334e13	−0.0193937
$544$	0	0
$545$	−2.65504e15	−2.36532
$546$	0	0
$547$	1.52598e15	1.33235	0.666174	−	0.745796i	$-0.267931\pi$
$547$	1.52598e15	1.33235	0.666174	+	0.745796i	$0.267931\pi$
$548$	0	0
$549$	7.80973e14	0.668325
$550$	0	0
$551$	3.22204e15	2.70270
$552$	0	0
$553$	−8.48276e14	−0.697507
$554$	0	0
$555$	1.91230e14	0.154150
$556$	0	0
$557$	4.12629e14	0.326104	0.163052	−	0.986617i	$-0.447866\pi$
$557$	4.12629e14	0.326104	0.163052	+	0.986617i	$0.447866\pi$
$558$	0	0
$559$	−7.77018e13	−0.0602096
$560$	0	0
$561$	−3.16776e13	−0.0240689
$562$	0	0
$563$	−8.20945e14	−0.611671	−0.305835	−	0.952084i	$-0.598936\pi$
$563$	−8.20945e14	−0.611671	−0.305835	+	0.952084i	$0.598936\pi$
$564$	0	0
$565$	2.53021e13	0.0184880
$566$	0	0
$567$	1.10430e15	0.791368
$568$	0	0
$569$	3.96770e14	0.278883	0.139441	−	0.990230i	$-0.455469\pi$
$569$	3.96770e14	0.278883	0.139441	+	0.990230i	$0.455469\pi$
$570$	0	0
$571$	−4.80108e14	−0.331009	−0.165505	−	0.986209i	$-0.552925\pi$
$571$	−4.80108e14	−0.331009	−0.165505	+	0.986209i	$0.552925\pi$
$572$	0	0
$573$	−8.49790e13	−0.0574727
$574$	0	0
$575$	−4.44684e15	−2.95038
$576$	0	0
$577$	−1.13938e15	−0.741657	−0.370828	−	0.928701i	$-0.620926\pi$
$577$	−1.13938e15	−0.741657	−0.370828	+	0.928701i	$0.620926\pi$
$578$	0	0
$579$	−1.03251e14	−0.0659420
$580$	0	0
$581$	−1.39611e14	−0.0874884
$582$	0	0
$583$	−1.57161e15	−0.966422
$584$	0	0
$585$	−1.18337e14	−0.0714109
$586$	0	0
$587$	−6.07086e14	−0.359534	−0.179767	−	0.983709i	$-0.557534\pi$
$587$	−6.07086e14	−0.359534	−0.179767	+	0.983709i	$0.557534\pi$
$588$	0	0
$589$	2.42974e15	1.41230
$590$	0	0
$591$	−8.15751e13	−0.0465400
$592$	0	0
$593$	1.85101e15	1.03659	0.518295	−	0.855202i	$-0.326567\pi$
$593$	1.85101e15	1.03659	0.518295	+	0.855202i	$0.326567\pi$
$594$	0	0
$595$	−6.18841e14	−0.340201
$596$	0	0
$597$	−1.18068e14	−0.0637198
$598$	0	0
$599$	1.02140e15	0.541186	0.270593	−	0.962694i	$-0.412780\pi$
$599$	1.02140e15	0.541186	0.270593	+	0.962694i	$0.412780\pi$
$600$	0	0
$601$	−4.39352e14	−0.228561	−0.114281	−	0.993449i	$-0.536456\pi$
$601$	−4.39352e14	−0.228561	−0.114281	+	0.993449i	$0.536456\pi$
$602$	0	0
$603$	1.93388e15	0.987834
$604$	0	0
$605$	−5.84182e15	−2.93017
$606$	0	0
$607$	−9.02817e14	−0.444695	−0.222347	−	0.974968i	$-0.571372\pi$
$607$	−9.02817e14	−0.444695	−0.222347	+	0.974968i	$0.571372\pi$
$608$	0	0
$609$	−1.76884e14	−0.0855647
$610$	0	0
$611$	5.93243e13	0.0281842
$612$	0	0
$613$	−1.62435e15	−0.757959	−0.378980	−	0.925405i	$-0.623725\pi$
$613$	−1.62435e15	−0.757959	−0.378980	+	0.925405i	$0.623725\pi$
$614$	0	0
$615$	9.67601e13	0.0443489
$616$	0	0
$617$	2.90236e15	1.30672	0.653360	−	0.757048i	$-0.273359\pi$
$617$	2.90236e15	1.30672	0.653360	+	0.757048i	$0.273359\pi$
$618$	0	0
$619$	−5.41571e14	−0.239528	−0.119764	−	0.992802i	$-0.538214\pi$
$619$	−5.41571e14	−0.239528	−0.119764	+	0.992802i	$0.538214\pi$
$620$	0	0
$621$	3.51725e14	0.152827
$622$	0	0
$623$	4.59845e14	0.196303
$624$	0	0
$625$	7.55479e15	3.16871
$626$	0	0
$627$	4.47490e14	0.184422
$628$	0	0
$629$	−6.63361e14	−0.268640
$630$	0	0
$631$	1.05678e15	0.420554	0.210277	−	0.977642i	$-0.432563\pi$
$631$	1.05678e15	0.420554	0.210277	+	0.977642i	$0.432563\pi$
$632$	0	0
$633$	−1.30259e14	−0.0509433
$634$	0	0
$635$	−4.42213e15	−1.69971
$636$	0	0
$637$	3.56460e13	0.0134662
$638$	0	0
$639$	1.82983e15	0.679447
$640$	0	0
$641$	−5.55504e14	−0.202753	−0.101377	−	0.994848i	$-0.532325\pi$
$641$	−5.55504e14	−0.202753	−0.101377	+	0.994848i	$0.532325\pi$
$642$	0	0
$643$	4.84392e15	1.73795	0.868974	−	0.494858i	$-0.164780\pi$
$643$	4.84392e15	1.73795	0.868974	+	0.494858i	$0.164780\pi$
$644$	0	0
$645$	−5.78960e14	−0.204207
$646$	0	0
$647$	9.93347e13	0.0344451	0.0172225	−	0.999852i	$-0.494518\pi$
$647$	9.93347e13	0.0344451	0.0172225	+	0.999852i	$0.494518\pi$
$648$	0	0
$649$	−5.11549e15	−1.74398
$650$	0	0
$651$	−1.33388e14	−0.0447119
$652$	0	0
$653$	1.05230e15	0.346832	0.173416	−	0.984849i	$-0.444520\pi$
$653$	1.05230e15	0.346832	0.173416	+	0.984849i	$0.444520\pi$
$654$	0	0
$655$	7.23000e15	2.34321
$656$	0	0
$657$	−1.88701e15	−0.601403
$658$	0	0
$659$	1.32727e15	0.415997	0.207998	−	0.978129i	$-0.433305\pi$
$659$	1.32727e15	0.415997	0.207998	+	0.978129i	$0.433305\pi$
$660$	0	0
$661$	−4.60364e15	−1.41903	−0.709517	−	0.704688i	$-0.751087\pi$
$661$	−4.60364e15	−1.41903	−0.709517	+	0.704688i	$0.751087\pi$
$662$	0	0
$663$	−1.88151e12	−0.000570403 0
$664$	0	0
$665$	8.74198e15	2.60670
$666$	0	0
$667$	6.10363e15	1.79018
$668$	0	0
$669$	3.78375e14	0.109164
$670$	0	0
$671$	3.77309e15	1.07084
$672$	0	0
$673$	−4.35945e15	−1.21716	−0.608582	−	0.793491i	$-0.708261\pi$
$673$	−4.35945e15	−1.21716	−0.608582	+	0.793491i	$0.708261\pi$
$674$	0	0
$675$	−1.27682e15	−0.350719
$676$	0	0
$677$	6.30704e15	1.70446	0.852232	−	0.523164i	$-0.175249\pi$
$677$	6.30704e15	1.70446	0.852232	+	0.523164i	$0.175249\pi$
$678$	0	0
$679$	2.37457e14	0.0631395
$680$	0	0
$681$	−3.82518e14	−0.100079
$682$	0	0
$683$	6.82212e15	1.75633	0.878164	−	0.478360i	$-0.158768\pi$
$683$	6.82212e15	1.75633	0.878164	+	0.478360i	$0.158768\pi$
$684$	0	0
$685$	−7.68262e15	−1.94631
$686$	0	0
$687$	2.12578e14	0.0529976
$688$	0	0
$689$	−9.33464e13	−0.0229030
$690$	0	0
$691$	5.14474e15	1.24232	0.621161	−	0.783683i	$-0.286661\pi$
$691$	5.14474e15	1.24232	0.621161	+	0.783683i	$0.286661\pi$
$692$	0	0
$693$	5.35983e15	1.27385
$694$	0	0
$695$	−1.24401e16	−2.91010
$696$	0	0
$697$	−3.35653e14	−0.0772876
$698$	0	0
$699$	−3.62882e14	−0.0822509
$700$	0	0
$701$	4.93663e15	1.10149	0.550747	−	0.834673i	$-0.314343\pi$
$701$	4.93663e15	1.10149	0.550747	+	0.834673i	$0.314343\pi$
$702$	0	0
$703$	9.37089e15	2.05838
$704$	0	0
$705$	4.42028e14	0.0955894
$706$	0	0
$707$	−2.49140e15	−0.530440
$708$	0	0
$709$	−1.53781e15	−0.322366	−0.161183	−	0.986925i	$-0.551531\pi$
$709$	−1.53781e15	−0.322366	−0.161183	+	0.986925i	$0.551531\pi$
$710$	0	0
$711$	4.19267e15	0.865383
$712$	0	0
$713$	4.60275e15	0.935459
$714$	0	0
$715$	−5.71720e14	−0.114420
$716$	0	0
$717$	−1.66370e14	−0.0327883
$718$	0	0
$719$	−2.67529e15	−0.519233	−0.259616	−	0.965712i	$-0.583596\pi$
$719$	−2.67529e15	−0.519233	−0.259616	+	0.965712i	$0.583596\pi$
$720$	0	0
$721$	1.03089e15	0.197045
$722$	0	0
$723$	2.12077e14	0.0399240
$724$	0	0
$725$	−2.21573e16	−4.10825
$726$	0	0
$727$	8.79056e15	1.60538	0.802689	−	0.596398i	$-0.203402\pi$
$727$	8.79056e15	1.60538	0.802689	+	0.596398i	$0.203402\pi$
$728$	0	0
$729$	−5.40746e15	−0.972729
$730$	0	0
$731$	2.00836e15	0.355874
$732$	0	0
$733$	−4.31570e15	−0.753320	−0.376660	−	0.926352i	$-0.622928\pi$
$733$	−4.31570e15	−0.753320	−0.376660	+	0.926352i	$0.622928\pi$
$734$	0	0
$735$	2.65600e14	0.0456717
$736$	0	0
$737$	9.34309e15	1.58278
$738$	0	0
$739$	−8.86681e15	−1.47987	−0.739934	−	0.672680i	$-0.765143\pi$
$739$	−8.86681e15	−1.47987	−0.739934	+	0.672680i	$0.765143\pi$
$740$	0	0
$741$	2.65789e13	0.00437056
$742$	0	0
$743$	−2.00102e15	−0.324200	−0.162100	−	0.986774i	$-0.551827\pi$
$743$	−2.00102e15	−0.324200	−0.162100	+	0.986774i	$0.551827\pi$
$744$	0	0
$745$	6.74110e15	1.07615
$746$	0	0
$747$	6.90038e14	0.108545
$748$	0	0
$749$	8.51787e14	0.132033
$750$	0	0
$751$	−9.30295e15	−1.42102	−0.710511	−	0.703686i	$-0.751536\pi$
$751$	−9.30295e15	−1.42102	−0.710511	+	0.703686i	$0.751536\pi$
$752$	0	0
$753$	3.98302e14	0.0599571
$754$	0	0
$755$	−1.75677e16	−2.60619
$756$	0	0
$757$	8.24688e15	1.20576	0.602882	−	0.797830i	$-0.294019\pi$
$757$	8.24688e15	1.20576	0.602882	+	0.797830i	$0.294019\pi$
$758$	0	0
$759$	8.47696e14	0.122155
$760$	0	0
$761$	−5.01587e15	−0.712412	−0.356206	−	0.934408i	$-0.615930\pi$
$761$	−5.01587e15	−0.712412	−0.356206	+	0.934408i	$0.615930\pi$
$762$	0	0
$763$	7.14256e15	0.999929
$764$	0	0
$765$	3.05867e15	0.422080
$766$	0	0
$767$	−3.03837e14	−0.0413301
$768$	0	0
$769$	4.90706e15	0.658001	0.329001	−	0.944330i	$-0.393288\pi$
$769$	4.90706e15	0.658001	0.329001	+	0.944330i	$0.393288\pi$
$770$	0	0
$771$	−6.45587e14	−0.0853406
$772$	0	0
$773$	8.93310e15	1.16417	0.582083	−	0.813129i	$-0.302238\pi$
$773$	8.93310e15	1.16417	0.582083	+	0.813129i	$0.302238\pi$
$774$	0	0
$775$	−1.67088e16	−2.14677
$776$	0	0
$777$	−5.14445e14	−0.0651663
$778$	0	0
$779$	4.74156e15	0.592195
$780$	0	0
$781$	8.84040e15	1.08866
$782$	0	0
$783$	1.75254e15	0.212803
$784$	0	0
$785$	−7.49137e15	−0.896970
$786$	0	0
$787$	1.67113e15	0.197310	0.0986552	−	0.995122i	$-0.468546\pi$
$787$	1.67113e15	0.197310	0.0986552	+	0.995122i	$0.468546\pi$
$788$	0	0
$789$	−8.70955e14	−0.101408
$790$	0	0
$791$	−6.80674e13	−0.00781571
$792$	0	0
$793$	2.24105e14	0.0253775
$794$	0	0
$795$	−6.95529e14	−0.0776776
$796$	0	0
$797$	−1.39838e16	−1.54030	−0.770150	−	0.637863i	$-0.779819\pi$
$797$	−1.39838e16	−1.54030	−0.770150	+	0.637863i	$0.779819\pi$
$798$	0	0
$799$	−1.53336e15	−0.166585
$800$	0	0
$801$	−2.27282e15	−0.243549
$802$	0	0
$803$	−9.11668e15	−0.963610
$804$	0	0
$805$	1.65603e16	1.72659
$806$	0	0
$807$	−6.67508e14	−0.0686518
$808$	0	0
$809$	−1.89059e16	−1.91814	−0.959068	−	0.283176i	$-0.908612\pi$
$809$	−1.89059e16	−1.91814	−0.959068	+	0.283176i	$0.908612\pi$
$810$	0	0
$811$	1.19978e16	1.20084	0.600421	−	0.799684i	$-0.295000\pi$
$811$	1.19978e16	1.20084	0.600421	+	0.799684i	$0.295000\pi$
$812$	0	0
$813$	−4.53663e14	−0.0447956
$814$	0	0
$815$	8.37938e15	0.816291
$816$	0	0
$817$	−2.83709e16	−2.72679
$818$	0	0
$819$	3.18350e14	0.0301886
$820$	0	0
$821$	4.39123e15	0.410865	0.205432	−	0.978671i	$-0.434140\pi$
$821$	4.39123e15	0.410865	0.205432	+	0.978671i	$0.434140\pi$
$822$	0	0
$823$	6.82449e15	0.630044	0.315022	−	0.949084i	$-0.397988\pi$
$823$	6.82449e15	0.630044	0.315022	+	0.949084i	$0.397988\pi$
$824$	0	0
$825$	−3.07729e15	−0.280331
$826$	0	0
$827$	−7.08030e15	−0.636460	−0.318230	−	0.948013i	$-0.603088\pi$
$827$	−7.08030e15	−0.636460	−0.318230	+	0.948013i	$0.603088\pi$
$828$	0	0
$829$	1.01034e16	0.896228	0.448114	−	0.893976i	$-0.352096\pi$
$829$	1.01034e16	0.896228	0.448114	+	0.893976i	$0.352096\pi$
$830$	0	0
$831$	−2.26177e14	−0.0197990
$832$	0	0
$833$	−9.21344e14	−0.0795929
$834$	0	0
$835$	−1.40877e16	−1.20106
$836$	0	0
$837$	1.32159e15	0.111200
$838$	0	0
$839$	−8.34065e15	−0.692642	−0.346321	−	0.938116i	$-0.612569\pi$
$839$	−8.34065e15	−0.692642	−0.346321	+	0.938116i	$0.612569\pi$
$840$	0	0
$841$	1.82121e16	1.49273
$842$	0	0
$843$	9.24128e14	0.0747620
$844$	0	0
$845$	2.37338e16	1.89520
$846$	0	0
$847$	1.57156e16	1.23872
$848$	0	0
$849$	5.64073e13	0.00438877
$850$	0	0
$851$	1.77516e16	1.36341
$852$	0	0
$853$	−2.50294e16	−1.89771	−0.948857	−	0.315706i	$-0.897759\pi$
$853$	−2.50294e16	−1.89771	−0.948857	+	0.315706i	$0.897759\pi$
$854$	0	0
$855$	−4.32080e16	−3.23408
$856$	0	0
$857$	−2.29961e16	−1.69926	−0.849629	−	0.527381i	$-0.823174\pi$
$857$	−2.29961e16	−1.69926	−0.849629	+	0.527381i	$0.823174\pi$
$858$	0	0
$859$	4.89071e15	0.356787	0.178394	−	0.983959i	$-0.442910\pi$
$859$	4.89071e15	0.356787	0.178394	+	0.983959i	$0.442910\pi$
$860$	0	0
$861$	−2.60303e14	−0.0187483
$862$	0	0
$863$	6.59421e15	0.468925	0.234463	−	0.972125i	$-0.424667\pi$
$863$	6.59421e15	0.468925	0.234463	+	0.972125i	$0.424667\pi$
$864$	0	0
$865$	3.19623e16	2.24414
$866$	0	0
$867$	−9.25701e14	−0.0641750
$868$	0	0
$869$	2.02559e16	1.38658
$870$	0	0
$871$	5.54938e14	0.0375098
$872$	0	0
$873$	−1.17365e15	−0.0783358
$874$	0	0
$875$	−3.70133e16	−2.43957
$876$	0	0
$877$	−1.15524e16	−0.751924	−0.375962	−	0.926635i	$-0.622688\pi$
$877$	−1.15524e16	−0.751924	−0.375962	+	0.926635i	$0.622688\pi$
$878$	0	0
$879$	−1.08116e15	−0.0694949
$880$	0	0
$881$	−1.92226e16	−1.22024	−0.610120	−	0.792309i	$-0.708879\pi$
$881$	−1.92226e16	−1.22024	−0.610120	+	0.792309i	$0.708879\pi$
$882$	0	0
$883$	−2.96814e16	−1.86080	−0.930402	−	0.366542i	$-0.880542\pi$
$883$	−2.96814e16	−1.86080	−0.930402	+	0.366542i	$0.880542\pi$
$884$	0	0
$885$	−2.26391e15	−0.140175
$886$	0	0
$887$	−1.33218e16	−0.814674	−0.407337	−	0.913278i	$-0.633543\pi$
$887$	−1.33218e16	−0.814674	−0.407337	+	0.913278i	$0.633543\pi$
$888$	0	0
$889$	1.18964e16	0.718545
$890$	0	0
$891$	−2.63694e16	−1.57316
$892$	0	0
$893$	2.16608e16	1.27641
$894$	0	0
$895$	−2.93524e16	−1.70851
$896$	0	0
$897$	5.03493e13	0.00289491
$898$	0	0
$899$	2.29341e16	1.30258
$900$	0	0
$901$	2.41273e15	0.135370
$902$	0	0
$903$	1.55751e15	0.0863273
$904$	0	0
$905$	9.95179e15	0.544922
$906$	0	0
$907$	−1.12918e15	−0.0610833	−0.0305417	−	0.999533i	$-0.509723\pi$
$907$	−1.12918e15	−0.0610833	−0.0305417	+	0.999533i	$0.509723\pi$
$908$	0	0
$909$	1.23139e16	0.658106
$910$	0	0
$911$	−1.27425e16	−0.672827	−0.336414	−	0.941714i	$-0.609214\pi$
$911$	−1.27425e16	−0.672827	−0.336414	+	0.941714i	$0.609214\pi$
$912$	0	0
$913$	3.33376e15	0.173918
$914$	0	0
$915$	1.66982e15	0.0860701
$916$	0	0
$917$	−1.94500e16	−0.990579
$918$	0	0
$919$	2.69918e16	1.35830	0.679152	−	0.733998i	$-0.262348\pi$
$919$	2.69918e16	1.35830	0.679152	+	0.733998i	$0.262348\pi$
$920$	0	0
$921$	5.22919e14	0.0260020
$922$	0	0
$923$	5.25080e14	0.0257998
$924$	0	0
$925$	−6.44415e16	−3.12886
$926$	0	0
$927$	−5.09524e15	−0.244470
$928$	0	0
$929$	−1.49837e16	−0.710447	−0.355223	−	0.934781i	$-0.615595\pi$
$929$	−1.49837e16	−0.710447	−0.355223	+	0.934781i	$0.615595\pi$
$930$	0	0
$931$	1.30152e16	0.609859
$932$	0	0
$933$	−1.96472e15	−0.0909810
$934$	0	0
$935$	1.47773e16	0.676286
$936$	0	0
$937$	−6.16754e15	−0.278961	−0.139481	−	0.990225i	$-0.544543\pi$
$937$	−6.16754e15	−0.278961	−0.139481	+	0.990225i	$0.544543\pi$
$938$	0	0
$939$	−1.37436e15	−0.0614382
$940$	0	0
$941$	8.73576e15	0.385974	0.192987	−	0.981201i	$-0.438183\pi$
$941$	8.73576e15	0.385974	0.192987	+	0.981201i	$0.438183\pi$
$942$	0	0
$943$	8.98210e15	0.392250
$944$	0	0
$945$	4.75495e15	0.205244
$946$	0	0
$947$	1.17830e16	0.502726	0.251363	−	0.967893i	$-0.419121\pi$
$947$	1.17830e16	0.502726	0.251363	+	0.967893i	$0.419121\pi$
$948$	0	0
$949$	−5.41490e14	−0.0228363
$950$	0	0
$951$	−1.54730e15	−0.0645033
$952$	0	0
$953$	−1.96538e16	−0.809907	−0.404954	−	0.914337i	$-0.632712\pi$
$953$	−1.96538e16	−0.809907	−0.404954	+	0.914337i	$0.632712\pi$
$954$	0	0
$955$	3.96418e16	1.61486
$956$	0	0
$957$	4.22382e15	0.170094
$958$	0	0
$959$	2.06677e16	0.822791
$960$	0	0
$961$	−8.11388e15	−0.319337
$962$	0	0
$963$	−4.21003e15	−0.163810
$964$	0	0
$965$	4.81655e16	1.85283
$966$	0	0
$967$	4.28168e16	1.62843	0.814213	−	0.580566i	$-0.197169\pi$
$967$	4.28168e16	1.62843	0.814213	+	0.580566i	$0.197169\pi$
$968$	0	0
$969$	−6.86986e14	−0.0258326
$970$	0	0
$971$	4.18265e16	1.55506	0.777528	−	0.628849i	$-0.216473\pi$
$971$	4.18265e16	1.55506	0.777528	+	0.628849i	$0.216473\pi$
$972$	0	0
$973$	3.34663e16	1.23023
$974$	0	0
$975$	−1.82777e14	−0.00664349
$976$	0	0
$977$	2.02626e16	0.728240	0.364120	−	0.931352i	$-0.381370\pi$
$977$	2.02626e16	0.728240	0.364120	+	0.931352i	$0.381370\pi$
$978$	0	0
$979$	−1.09806e16	−0.390231
$980$	0	0
$981$	−3.53027e16	−1.24059
$982$	0	0
$983$	3.18980e16	1.10846	0.554229	−	0.832364i	$-0.313013\pi$
$983$	3.18980e16	1.10846	0.554229	+	0.832364i	$0.313013\pi$
$984$	0	0
$985$	3.80539e16	1.30768
$986$	0	0
$987$	−1.18914e15	−0.0404099
$988$	0	0
$989$	−5.37440e16	−1.80613
$990$	0	0
$991$	−3.06126e16	−1.01741	−0.508704	−	0.860942i	$-0.669875\pi$
$991$	−3.06126e16	−1.01741	−0.508704	+	0.860942i	$0.669875\pi$
$992$	0	0
$993$	9.11228e14	0.0299507
$994$	0	0
$995$	5.50777e16	1.79039
$996$	0	0
$997$	1.75855e16	0.565367	0.282683	−	0.959213i	$-0.408775\pi$
$997$	1.75855e16	0.565367	0.282683	+	0.959213i	$0.408775\pi$
$998$	0	0
$999$	5.09703e15	0.162072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.12.a.e.1.4	✓	6
4.3	odd	2		128.12.a.g.1.3	yes	6
8.3	odd	2		128.12.a.f.1.4	yes	6
8.5	even	2		128.12.a.h.1.3	yes	6
16.3	odd	4		256.12.b.q.129.6		12
16.5	even	4		256.12.b.p.129.6		12
16.11	odd	4		256.12.b.q.129.7		12
16.13	even	4		256.12.b.p.129.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.a.e.1.4	✓	6	1.1	even	1	trivial
128.12.a.f.1.4	yes	6	8.3	odd	2
128.12.a.g.1.3	yes	6	4.3	odd	2
128.12.a.h.1.3	yes	6	8.5	even	2
256.12.b.p.129.6		12	16.5	even	4
256.12.b.p.129.7		12	16.13	even	4
256.12.b.q.129.6		12	16.3	odd	4
256.12.b.q.129.7		12	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	128.a (trivial)

\(f(q)\)	\(=\)	\(q+28.4295 q^{3} -13262.1 q^{5} +35677.5 q^{7} -176339. q^{9} -851940. q^{11} -50601.4 q^{13} -377034. q^{15} +1.30790e6 q^{17} -1.84759e7 q^{19} +1.01429e6 q^{21} -3.49995e7 q^{23} +1.27054e8 q^{25} -1.00494e7 q^{27} -1.74392e8 q^{29} -1.31509e8 q^{31} -2.42202e7 q^{33} -4.73157e8 q^{35} -5.07196e8 q^{37} -1.43857e6 q^{39} -2.56635e8 q^{41} +1.53556e9 q^{43} +2.33862e9 q^{45} -1.17238e9 q^{47} -7.04446e8 q^{49} +3.71829e7 q^{51} +1.84474e9 q^{53} +1.12985e10 q^{55} -5.25260e8 q^{57} +6.00452e9 q^{59} -4.42882e9 q^{61} -6.29132e9 q^{63} +6.71080e8 q^{65} -1.09668e10 q^{67} -9.95018e8 q^{69} -1.03768e10 q^{71} +1.07011e10 q^{73} +3.61209e9 q^{75} -3.03951e10 q^{77} -2.37762e10 q^{79} +3.09522e10 q^{81} -3.91314e9 q^{83} -1.73454e10 q^{85} -4.95788e9 q^{87} +1.28889e10 q^{89} -1.80533e9 q^{91} -3.73873e9 q^{93} +2.45028e11 q^{95} +6.65565e9 q^{97} +1.50230e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 20 q^{3} - 1804 q^{5} + 49368 q^{7} + 313814 q^{9} - 688460 q^{11} - 2290348 q^{13} + 4828264 q^{15} + 4127636 q^{17} + 9936364 q^{19} + 20325616 q^{21} + 9921320 q^{23} + 51633002 q^{25} - 132503384 q^{27}+ \cdots + 240482467988 q^{99}+O(q^{100}) \) 6 * q - 20 * q^3 - 1804 * q^5 + 49368 * q^7 + 313814 * q^9 - 688460 * q^11 - 2290348 * q^13 + 4828264 * q^15 + 4127636 * q^17 + 9936364 * q^19 + 20325616 * q^21 + 9921320 * q^23 + 51633002 * q^25 - 132503384 * q^27 + 25222532 * q^29 - 246133120 * q^31 - 337991288 * q^33 + 227061200 * q^35 - 660146172 * q^37 + 648086056 * q^39 + 486915836 * q^41 + 1147261348 * q^43 + 626323028 * q^45 + 603975088 * q^47 + 1846879062 * q^49 - 1556268056 * q^51 - 428043996 * q^53 - 1355415976 * q^55 - 783352648 * q^57 + 809510676 * q^59 - 2822052332 * q^61 + 14526601496 * q^63 + 1376905688 * q^65 + 9742675036 * q^67 - 8441748208 * q^69 + 16181067512 * q^71 + 35963668692 * q^73 + 51500880116 * q^75 - 72754679408 * q^77 - 6103946192 * q^79 + 81240620990 * q^81 + 67929536124 * q^83 - 76741413928 * q^85 + 132686631816 * q^87 + 22721416116 * q^89 + 44438236496 * q^91 - 37482073344 * q^93 + 282717822568 * q^95 + 218896625492 * q^97 + 240482467988 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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