LMFDB - Embedded newform 32.16.a.c.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [32,16,Mod(1,32)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	32.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-2912] 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:	$45.6619216320$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10174x^{2} - 369720x - 3191805 $ x^4 - 10174x^2 - 369720x - 3191805
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{27}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-76.8271$ of defining polynomial
Character	$\chi$	$=$	32.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5704.21 q^{3} -70382.7 q^{5} -2.23695e6 q^{7} +1.81891e7 q^{9} +8.51679e7 q^{11} -3.82111e8 q^{13} -4.01478e8 q^{15} -2.06439e9 q^{17} +1.35210e9 q^{19} -1.27601e10 q^{21} -9.91058e9 q^{23} -2.55639e10 q^{25} +2.19055e10 q^{27} +2.07238e10 q^{29} -5.73247e10 q^{31} +4.85816e11 q^{33} +1.57443e11 q^{35} +5.43177e11 q^{37} -2.17964e12 q^{39} -1.83440e12 q^{41} +2.66702e12 q^{43} -1.28020e12 q^{45} -4.86752e12 q^{47} +2.56402e11 q^{49} -1.17757e13 q^{51} -1.10073e13 q^{53} -5.99435e12 q^{55} +7.71267e12 q^{57} -2.11481e13 q^{59} +4.68147e13 q^{61} -4.06883e13 q^{63} +2.68940e13 q^{65} -8.85142e13 q^{67} -5.65321e13 q^{69} +7.43818e13 q^{71} -2.38860e13 q^{73} -1.45822e14 q^{75} -1.90517e14 q^{77} -1.31202e14 q^{79} -1.36041e14 q^{81} +9.29060e13 q^{83} +1.45297e14 q^{85} +1.18213e14 q^{87} +7.29406e14 q^{89} +8.54765e14 q^{91} -3.26992e14 q^{93} -9.51645e13 q^{95} +8.62101e14 q^{97} +1.54913e15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2912 q^{3} - 98280 q^{5} - 2755776 q^{7} + 26457940 q^{9} + 110853856 q^{11} - 187741448 q^{13} - 442991680 q^{15} + 2121294984 q^{17} + 419203872 q^{19} + 8397459968 q^{21} - 5330808384 q^{23} + 8564955740 q^{25}+ \cdots + 26\!\cdots\!84 q^{99}+O(q^{100}) $ 4 * q - 2912 * q^3 - 98280 * q^5 - 2755776 * q^7 + 26457940 * q^9 + 110853856 * q^11 - 187741448 * q^13 - 442991680 * q^15 + 2121294984 * q^17 + 419203872 * q^19 + 8397459968 * q^21 - 5330808384 * q^23 + 8564955740 * q^25 - 41191036352 * q^27 - 86832359880 * q^29 + 96274977024 * q^31 + 41399413504 * q^33 - 929253215360 * q^35 - 112979590376 * q^37 - 2001683116352 * q^39 - 1534010535960 * q^41 - 3567397370400 * q^43 + 3155482037240 * q^45 - 9031658937472 * q^47 + 8694343447332 * q^49 - 37558794281664 * q^51 - 14370240161064 * q^53 - 46434781152960 * q^55 - 5813065407232 * q^57 - 68701112336800 * q^59 + 4806363233592 * q^61 - 192971404565440 * q^63 + 7651382051280 * q^65 - 167433288699744 * q^67 - 80410496862720 * q^69 - 180096642956992 * q^71 + 173676000667560 * q^73 - 227575010162080 * q^75 - 133277064067584 * q^77 - 232741753133952 * q^79 + 263803331371492 * q^81 + 20415834690336 * q^83 - 342055719314640 * q^85 + 870209244139712 * q^87 + 938354283775848 * q^89 + 1478922417113472 * q^91 - 2098341777250304 * q^93 + 2700969259238080 * q^95 + 1510271716715080 * q^97 + 2629725177868384 * 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
$3$	5704.21	1.50587	0.752933	−	0.658097i	$-0.228638\pi$
$3$	5704.21	1.50587	0.752933	+	0.658097i	$0.228638\pi$
$4$	0	0
$5$	−70382.7	−0.402894	−0.201447	−	0.979499i	$-0.564564\pi$
$5$	−70382.7	−0.402894	−0.201447	+	0.979499i	$0.564564\pi$
$6$	0	0
$7$	−2.23695e6	−1.02665	−0.513324	−	0.858195i	$-0.671586\pi$
$7$	−2.23695e6	−1.02665	−0.513324	+	0.858195i	$0.671586\pi$
$8$	0	0
$9$	1.81891e7	1.26763
$10$	0	0
$11$	8.51679e7	1.31774	0.658872	−	0.752255i	$-0.271034\pi$
$11$	8.51679e7	1.31774	0.658872	+	0.752255i	$0.271034\pi$
$12$	0	0
$13$	−3.82111e8	−1.68894	−0.844471	−	0.535601i	$-0.820085\pi$
$13$	−3.82111e8	−1.68894	−0.844471	+	0.535601i	$0.820085\pi$
$14$	0	0
$15$	−4.01478e8	−0.606704
$16$	0	0
$17$	−2.06439e9	−1.22018	−0.610091	−	0.792331i	$-0.708867\pi$
$17$	−2.06439e9	−1.22018	−0.610091	+	0.792331i	$0.708867\pi$
$18$	0	0
$19$	1.35210e9	0.347022	0.173511	−	0.984832i	$-0.444489\pi$
$19$	1.35210e9	0.347022	0.173511	+	0.984832i	$0.444489\pi$
$20$	0	0
$21$	−1.27601e10	−1.54600
$22$	0	0
$23$	−9.91058e9	−0.606933	−0.303466	−	0.952842i	$-0.598144\pi$
$23$	−9.91058e9	−0.606933	−0.303466	+	0.952842i	$0.598144\pi$
$24$	0	0
$25$	−2.55639e10	−0.837676
$26$	0	0
$27$	2.19055e10	0.403019
$28$	0	0
$29$	2.07238e10	0.223092	0.111546	−	0.993759i	$-0.464420\pi$
$29$	2.07238e10	0.223092	0.111546	+	0.993759i	$0.464420\pi$
$30$	0	0
$31$	−5.73247e10	−0.374222	−0.187111	−	0.982339i	$-0.559912\pi$
$31$	−5.73247e10	−0.374222	−0.187111	+	0.982339i	$0.559912\pi$
$32$	0	0
$33$	4.85816e11	1.98434
$34$	0	0
$35$	1.57443e11	0.413631
$36$	0	0
$37$	5.43177e11	0.940651	0.470325	−	0.882493i	$-0.344137\pi$
$37$	5.43177e11	0.940651	0.470325	+	0.882493i	$0.344137\pi$
$38$	0	0
$39$	−2.17964e12	−2.54332
$40$	0	0
$41$	−1.83440e12	−1.47101	−0.735504	−	0.677521i	$-0.763054\pi$
$41$	−1.83440e12	−1.47101	−0.735504	+	0.677521i	$0.763054\pi$
$42$	0	0
$43$	2.66702e12	1.49628	0.748139	−	0.663542i	$-0.230948\pi$
$43$	2.66702e12	1.49628	0.748139	+	0.663542i	$0.230948\pi$
$44$	0	0
$45$	−1.28020e12	−0.510722
$46$	0	0
$47$	−4.86752e12	−1.40144	−0.700718	−	0.713438i	$-0.747137\pi$
$47$	−4.86752e12	−1.40144	−0.700718	+	0.713438i	$0.747137\pi$
$48$	0	0
$49$	2.56402e11	0.0540070
$50$	0	0
$51$	−1.17757e13	−1.83743
$52$	0	0
$53$	−1.10073e13	−1.28710	−0.643552	−	0.765402i	$-0.722540\pi$
$53$	−1.10073e13	−1.28710	−0.643552	+	0.765402i	$0.722540\pi$
$54$	0	0
$55$	−5.99435e12	−0.530911
$56$	0	0
$57$	7.71267e12	0.522569
$58$	0	0
$59$	−2.11481e13	−1.10632	−0.553160	−	0.833075i	$-0.686578\pi$
$59$	−2.11481e13	−1.10632	−0.553160	+	0.833075i	$0.686578\pi$
$60$	0	0
$61$	4.68147e13	1.90725	0.953627	−	0.300991i	$-0.0973175\pi$
$61$	4.68147e13	1.90725	0.953627	+	0.300991i	$0.0973175\pi$
$62$	0	0
$63$	−4.06883e13	−1.30141
$64$	0	0
$65$	2.68940e13	0.680465
$66$	0	0
$67$	−8.85142e13	−1.78423	−0.892117	−	0.451804i	$-0.850781\pi$
$67$	−8.85142e13	−1.78423	−0.892117	+	0.451804i	$0.850781\pi$
$68$	0	0
$69$	−5.65321e13	−0.913959
$70$	0	0
$71$	7.43818e13	0.970575	0.485288	−	0.874355i	$-0.338715\pi$
$71$	7.43818e13	0.970575	0.485288	+	0.874355i	$0.338715\pi$
$72$	0	0
$73$	−2.38860e13	−0.253059	−0.126529	−	0.991963i	$-0.540384\pi$
$73$	−2.38860e13	−0.253059	−0.126529	+	0.991963i	$0.540384\pi$
$74$	0	0
$75$	−1.45822e14	−1.26143
$76$	0	0
$77$	−1.90517e14	−1.35286
$78$	0	0
$79$	−1.31202e14	−0.768664	−0.384332	−	0.923195i	$-0.625568\pi$
$79$	−1.31202e14	−0.768664	−0.384332	+	0.923195i	$0.625568\pi$
$80$	0	0
$81$	−1.36041e14	−0.660740
$82$	0	0
$83$	9.29060e13	0.375801	0.187901	−	0.982188i	$-0.439832\pi$
$83$	9.29060e13	0.375801	0.187901	+	0.982188i	$0.439832\pi$
$84$	0	0
$85$	1.45297e14	0.491604
$86$	0	0
$87$	1.18213e14	0.335947
$88$	0	0
$89$	7.29406e14	1.74801	0.874006	−	0.485915i	$-0.161513\pi$
$89$	7.29406e14	1.74801	0.874006	+	0.485915i	$0.161513\pi$
$90$	0	0
$91$	8.54765e14	1.73395
$92$	0	0
$93$	−3.26992e14	−0.563528
$94$	0	0
$95$	−9.51645e13	−0.139813
$96$	0	0
$97$	8.62101e14	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$97$	8.62101e14	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$98$	0	0
$99$	1.54913e15	1.67041
$100$	0	0
$101$	−8.30597e14	−0.770868	−0.385434	−	0.922735i	$-0.625948\pi$
$101$	−8.30597e14	−0.770868	−0.385434	+	0.922735i	$0.625948\pi$
$102$	0	0
$103$	−2.37256e13	−0.0190081	−0.00950405	−	0.999955i	$-0.503025\pi$
$103$	−2.37256e13	−0.0190081	−0.00950405	+	0.999955i	$0.503025\pi$
$104$	0	0
$105$	8.98088e14	0.622872
$106$	0	0
$107$	3.52032e14	0.211935	0.105968	−	0.994370i	$-0.466206\pi$
$107$	3.52032e14	0.211935	0.105968	+	0.994370i	$0.466206\pi$
$108$	0	0
$109$	−1.40730e15	−0.737373	−0.368686	−	0.929554i	$-0.620192\pi$
$109$	−1.40730e15	−0.737373	−0.368686	+	0.929554i	$0.620192\pi$
$110$	0	0
$111$	3.09840e15	1.41649
$112$	0	0
$113$	−1.17002e15	−0.467846	−0.233923	−	0.972255i	$-0.575156\pi$
$113$	−1.17002e15	−0.467846	−0.233923	+	0.972255i	$0.575156\pi$
$114$	0	0
$115$	6.97534e14	0.244530
$116$	0	0
$117$	−6.95028e15	−2.14096
$118$	0	0
$119$	4.61794e15	1.25270
$120$	0	0
$121$	3.07632e15	0.736447
$122$	0	0
$123$	−1.04638e16	−2.21514
$124$	0	0
$125$	3.94716e15	0.740389
$126$	0	0
$127$	−5.04212e15	−0.839624	−0.419812	−	0.907611i	$-0.637904\pi$
$127$	−5.04212e15	−0.839624	−0.419812	+	0.907611i	$0.637904\pi$
$128$	0	0
$129$	1.52132e16	2.25319
$130$	0	0
$131$	6.98891e15	0.922305	0.461152	−	0.887321i	$-0.347436\pi$
$131$	6.98891e15	0.922305	0.461152	+	0.887321i	$0.347436\pi$
$132$	0	0
$133$	−3.02459e15	−0.356270
$134$	0	0
$135$	−1.54177e15	−0.162374
$136$	0	0
$137$	1.10285e16	1.04019	0.520094	−	0.854109i	$-0.325897\pi$
$137$	1.10285e16	1.04019	0.520094	+	0.854109i	$0.325897\pi$
$138$	0	0
$139$	−2.00248e16	−1.69417	−0.847086	−	0.531455i	$-0.821645\pi$
$139$	−2.00248e16	−1.69417	−0.847086	+	0.531455i	$0.821645\pi$
$140$	0	0
$141$	−2.77653e16	−2.11038
$142$	0	0
$143$	−3.25436e16	−2.22559
$144$	0	0
$145$	−1.45860e15	−0.0898826
$146$	0	0
$147$	1.46257e15	0.0813273
$148$	0	0
$149$	1.17571e16	0.590750	0.295375	−	0.955381i	$-0.404555\pi$
$149$	1.17571e16	0.590750	0.295375	+	0.955381i	$0.404555\pi$
$150$	0	0
$151$	2.45743e16	1.11726	0.558630	−	0.829417i	$-0.311327\pi$
$151$	2.45743e16	1.11726	0.558630	+	0.829417i	$0.311327\pi$
$152$	0	0
$153$	−3.75495e16	−1.54674
$154$	0	0
$155$	4.03467e15	0.150772
$156$	0	0
$157$	−8.88183e15	−0.301478	−0.150739	−	0.988574i	$-0.548165\pi$
$157$	−8.88183e15	−0.301478	−0.150739	+	0.988574i	$0.548165\pi$
$158$	0	0
$159$	−6.27883e16	−1.93821
$160$	0	0
$161$	2.21695e16	0.623106
$162$	0	0
$163$	3.08616e16	0.790700	0.395350	−	0.918531i	$-0.370623\pi$
$163$	3.08616e16	0.790700	0.395350	+	0.918531i	$0.370623\pi$
$164$	0	0
$165$	−3.41930e16	−0.799481
$166$	0	0
$167$	−7.11004e15	−0.151879	−0.0759396	−	0.997112i	$-0.524196\pi$
$167$	−7.11004e15	−0.151879	−0.0759396	+	0.997112i	$0.524196\pi$
$168$	0	0
$169$	9.48231e16	1.85253
$170$	0	0
$171$	2.45936e16	0.439897
$172$	0	0
$173$	6.43957e16	1.05563	0.527814	−	0.849360i	$-0.323012\pi$
$173$	6.43957e16	1.05563	0.527814	+	0.849360i	$0.323012\pi$
$174$	0	0
$175$	5.71852e16	0.859999
$176$	0	0
$177$	−1.20633e17	−1.66597
$178$	0	0
$179$	5.50414e16	0.698701	0.349351	−	0.936992i	$-0.386402\pi$
$179$	5.50414e16	0.698701	0.349351	+	0.936992i	$0.386402\pi$
$180$	0	0
$181$	1.60345e17	1.87269	0.936345	−	0.351080i	$-0.114186\pi$
$181$	1.60345e17	1.87269	0.936345	+	0.351080i	$0.114186\pi$
$182$	0	0
$183$	2.67041e17	2.87207
$184$	0	0
$185$	−3.82303e16	−0.378983
$186$	0	0
$187$	−1.75820e17	−1.60789
$188$	0	0
$189$	−4.90016e16	−0.413759
$190$	0	0
$191$	3.74053e16	0.291865	0.145933	−	0.989295i	$-0.453382\pi$
$191$	3.74053e16	0.291865	0.145933	+	0.989295i	$0.453382\pi$
$192$	0	0
$193$	4.99587e15	0.0360522	0.0180261	−	0.999838i	$-0.494262\pi$
$193$	4.99587e15	0.0360522	0.0180261	+	0.999838i	$0.494262\pi$
$194$	0	0
$195$	1.53409e17	1.02469
$196$	0	0
$197$	−1.35723e17	−0.839762	−0.419881	−	0.907579i	$-0.637928\pi$
$197$	−1.35723e17	−0.839762	−0.419881	+	0.907579i	$0.637928\pi$
$198$	0	0
$199$	5.60008e16	0.321215	0.160608	−	0.987018i	$-0.448655\pi$
$199$	5.60008e16	0.321215	0.160608	+	0.987018i	$0.448655\pi$
$200$	0	0
$201$	−5.04904e17	−2.68682
$202$	0	0
$203$	−4.63582e16	−0.229038
$204$	0	0
$205$	1.29110e17	0.592660
$206$	0	0
$207$	−1.80265e17	−0.769368
$208$	0	0
$209$	1.15156e17	0.457286
$210$	0	0
$211$	2.22873e17	0.824022	0.412011	−	0.911179i	$-0.364827\pi$
$211$	2.22873e17	0.824022	0.412011	+	0.911179i	$0.364827\pi$
$212$	0	0
$213$	4.24290e17	1.46156
$214$	0	0
$215$	−1.87712e17	−0.602842
$216$	0	0
$217$	1.28233e17	0.384194
$218$	0	0
$219$	−1.36251e17	−0.381073
$220$	0	0
$221$	7.88826e17	2.06082
$222$	0	0
$223$	−3.53475e17	−0.863122	−0.431561	−	0.902084i	$-0.642037\pi$
$223$	−3.53475e17	−0.863122	−0.431561	+	0.902084i	$0.642037\pi$
$224$	0	0
$225$	−4.64985e17	−1.06187
$226$	0	0
$227$	−2.61043e17	−0.557850	−0.278925	−	0.960313i	$-0.589978\pi$
$227$	−2.61043e17	−0.557850	−0.278925	+	0.960313i	$0.589978\pi$
$228$	0	0
$229$	−3.06658e17	−0.613603	−0.306802	−	0.951773i	$-0.599259\pi$
$229$	−3.06658e17	−0.613603	−0.306802	+	0.951773i	$0.599259\pi$
$230$	0	0
$231$	−1.08675e18	−2.03722
$232$	0	0
$233$	4.33375e17	0.761543	0.380771	−	0.924669i	$-0.375658\pi$
$233$	4.33375e17	0.761543	0.380771	+	0.924669i	$0.375658\pi$
$234$	0	0
$235$	3.42589e17	0.564630
$236$	0	0
$237$	−7.48403e17	−1.15751
$238$	0	0
$239$	5.62819e17	0.817306	0.408653	−	0.912690i	$-0.365999\pi$
$239$	5.62819e17	0.817306	0.408653	+	0.912690i	$0.365999\pi$
$240$	0	0
$241$	−3.81291e17	−0.520150	−0.260075	−	0.965588i	$-0.583747\pi$
$241$	−3.81291e17	−0.520150	−0.260075	+	0.965588i	$0.583747\pi$
$242$	0	0
$243$	−1.09032e18	−1.39801
$244$	0	0
$245$	−1.80462e16	−0.0217591
$246$	0	0
$247$	−5.16653e17	−0.586100
$248$	0	0
$249$	5.29956e17	0.565906
$250$	0	0
$251$	−4.05961e17	−0.408255	−0.204128	−	0.978944i	$-0.565436\pi$
$251$	−4.05961e17	−0.408255	−0.204128	+	0.978944i	$0.565436\pi$
$252$	0	0
$253$	−8.44064e17	−0.799781
$254$	0	0
$255$	8.28807e17	0.740290
$256$	0	0
$257$	−1.45212e18	−1.22322	−0.611611	−	0.791159i	$-0.709478\pi$
$257$	−1.45212e18	−1.22322	−0.611611	+	0.791159i	$0.709478\pi$
$258$	0	0
$259$	−1.21506e18	−0.965717
$260$	0	0
$261$	3.76949e17	0.282799
$262$	0	0
$263$	1.88916e18	1.33845	0.669223	−	0.743061i	$-0.266627\pi$
$263$	1.88916e18	1.33845	0.669223	+	0.743061i	$0.266627\pi$
$264$	0	0
$265$	7.74727e17	0.518567
$266$	0	0
$267$	4.16069e18	2.63227
$268$	0	0
$269$	−5.57791e17	−0.333679	−0.166840	−	0.985984i	$-0.553356\pi$
$269$	−5.57791e17	−0.333679	−0.166840	+	0.985984i	$0.553356\pi$
$270$	0	0
$271$	1.02398e18	0.579457	0.289728	−	0.957109i	$-0.406435\pi$
$271$	1.02398e18	0.579457	0.289728	+	0.957109i	$0.406435\pi$
$272$	0	0
$273$	4.87576e18	2.61110
$274$	0	0
$275$	−2.17722e18	−1.10384
$276$	0	0
$277$	4.87253e17	0.233968	0.116984	−	0.993134i	$-0.462677\pi$
$277$	4.87253e17	0.233968	0.116984	+	0.993134i	$0.462677\pi$
$278$	0	0
$279$	−1.04269e18	−0.474376
$280$	0	0
$281$	−6.14307e16	−0.0264904	−0.0132452	−	0.999912i	$-0.504216\pi$
$281$	−6.14307e16	−0.0264904	−0.0132452	+	0.999912i	$0.504216\pi$
$282$	0	0
$283$	−9.04926e17	−0.370011	−0.185006	−	0.982737i	$-0.559230\pi$
$283$	−9.04926e17	−0.370011	−0.185006	+	0.982737i	$0.559230\pi$
$284$	0	0
$285$	−5.42839e17	−0.210540
$286$	0	0
$287$	4.10347e18	1.51021
$288$	0	0
$289$	1.39928e18	0.488844
$290$	0	0
$291$	4.91761e18	1.63138
$292$	0	0
$293$	−4.37736e18	−1.37945	−0.689723	−	0.724073i	$-0.742268\pi$
$293$	−4.37736e18	−1.37945	−0.689723	+	0.724073i	$0.742268\pi$
$294$	0	0
$295$	1.48846e18	0.445730
$296$	0	0
$297$	1.86565e18	0.531075
$298$	0	0
$299$	3.78695e18	1.02507
$300$	0	0
$301$	−5.96599e18	−1.53615
$302$	0	0
$303$	−4.73790e18	−1.16082
$304$	0	0
$305$	−3.29494e18	−0.768421
$306$	0	0
$307$	−1.73602e17	−0.0385494	−0.0192747	−	0.999814i	$-0.506136\pi$
$307$	−1.73602e17	−0.0385494	−0.0192747	+	0.999814i	$0.506136\pi$
$308$	0	0
$309$	−1.35336e17	−0.0286236
$310$	0	0
$311$	−8.84130e18	−1.78161	−0.890806	−	0.454383i	$-0.849860\pi$
$311$	−8.84130e18	−1.78161	−0.890806	+	0.454383i	$0.849860\pi$
$312$	0	0
$313$	−2.11332e18	−0.405866	−0.202933	−	0.979193i	$-0.565047\pi$
$313$	−2.11332e18	−0.405866	−0.202933	+	0.979193i	$0.565047\pi$
$314$	0	0
$315$	2.86375e18	0.524332
$316$	0	0
$317$	−5.02926e17	−0.0878131	−0.0439066	−	0.999036i	$-0.513980\pi$
$317$	−5.02926e17	−0.0878131	−0.0439066	+	0.999036i	$0.513980\pi$
$318$	0	0
$319$	1.76500e18	0.293979
$320$	0	0
$321$	2.00806e18	0.319146
$322$	0	0
$323$	−2.79126e18	−0.423430
$324$	0	0
$325$	9.76824e18	1.41479
$326$	0	0
$327$	−8.02753e18	−1.11038
$328$	0	0
$329$	1.08884e19	1.43878
$330$	0	0
$331$	−1.44046e19	−1.81883	−0.909414	−	0.415891i	$-0.863470\pi$
$331$	−1.44046e19	−1.81883	−0.909414	+	0.415891i	$0.863470\pi$
$332$	0	0
$333$	9.87992e18	1.19240
$334$	0	0
$335$	6.22987e18	0.718858
$336$	0	0
$337$	−1.03979e19	−1.14742	−0.573708	−	0.819060i	$-0.694495\pi$
$337$	−1.03979e19	−1.14742	−0.573708	+	0.819060i	$0.694495\pi$
$338$	0	0
$339$	−6.67402e18	−0.704514
$340$	0	0
$341$	−4.88222e18	−0.493128
$342$	0	0
$343$	1.00465e19	0.971202
$344$	0	0
$345$	3.97888e18	0.368229
$346$	0	0
$347$	−1.80887e19	−1.60301	−0.801504	−	0.597989i	$-0.795966\pi$
$347$	−1.80887e19	−1.60301	−0.801504	+	0.597989i	$0.795966\pi$
$348$	0	0
$349$	−1.29191e19	−1.09658	−0.548292	−	0.836287i	$-0.684722\pi$
$349$	−1.29191e19	−1.09658	−0.548292	+	0.836287i	$0.684722\pi$
$350$	0	0
$351$	−8.37035e18	−0.680675
$352$	0	0
$353$	−9.78419e18	−0.762455	−0.381228	−	0.924481i	$-0.624499\pi$
$353$	−9.78419e18	−0.762455	−0.381228	+	0.924481i	$0.624499\pi$
$354$	0	0
$355$	−5.23519e18	−0.391039
$356$	0	0
$357$	2.63417e19	1.88640
$358$	0	0
$359$	1.42154e19	0.976228	0.488114	−	0.872780i	$-0.337685\pi$
$359$	1.42154e19	0.976228	0.488114	+	0.872780i	$0.337685\pi$
$360$	0	0
$361$	−1.33529e19	−0.879576
$362$	0	0
$363$	1.75480e19	1.10899
$364$	0	0
$365$	1.68116e18	0.101956
$366$	0	0
$367$	−2.92681e19	−1.70372	−0.851861	−	0.523768i	$-0.824526\pi$
$367$	−2.92681e19	−1.70372	−0.851861	+	0.523768i	$0.824526\pi$
$368$	0	0
$369$	−3.33661e19	−1.86470
$370$	0	0
$371$	2.46229e19	1.32140
$372$	0	0
$373$	1.09345e19	0.563616	0.281808	−	0.959471i	$-0.409066\pi$
$373$	1.09345e19	0.563616	0.281808	+	0.959471i	$0.409066\pi$
$374$	0	0
$375$	2.25155e19	1.11493
$376$	0	0
$377$	−7.91881e18	−0.376790
$378$	0	0
$379$	−2.09513e19	−0.958114	−0.479057	−	0.877784i	$-0.659021\pi$
$379$	−2.09513e19	−0.958114	−0.479057	+	0.877784i	$0.659021\pi$
$380$	0	0
$381$	−2.87613e19	−1.26436
$382$	0	0
$383$	−3.05510e18	−0.129132	−0.0645660	−	0.997913i	$-0.520566\pi$
$383$	−3.05510e18	−0.129132	−0.0645660	+	0.997913i	$0.520566\pi$
$384$	0	0
$385$	1.34091e19	0.545059
$386$	0	0
$387$	4.85108e19	1.89673
$388$	0	0
$389$	2.40886e19	0.906126	0.453063	−	0.891478i	$-0.350331\pi$
$389$	2.40886e19	0.906126	0.453063	+	0.891478i	$0.350331\pi$
$390$	0	0
$391$	2.04593e19	0.740568
$392$	0	0
$393$	3.98662e19	1.38887
$394$	0	0
$395$	9.23434e18	0.309690
$396$	0	0
$397$	4.66337e19	1.50581	0.752906	−	0.658128i	$-0.228651\pi$
$397$	4.66337e19	1.50581	0.752906	+	0.658128i	$0.228651\pi$
$398$	0	0
$399$	−1.72529e19	−0.536494
$400$	0	0
$401$	−1.04599e19	−0.313289	−0.156645	−	0.987655i	$-0.550068\pi$
$401$	−1.04599e19	−0.313289	−0.156645	+	0.987655i	$0.550068\pi$
$402$	0	0
$403$	2.19044e19	0.632039
$404$	0	0
$405$	9.57490e18	0.266208
$406$	0	0
$407$	4.62612e19	1.23954
$408$	0	0
$409$	8.13448e16	0.00210090	0.00105045	−	0.999999i	$-0.499666\pi$
$409$	8.13448e16	0.00210090	0.00105045	+	0.999999i	$0.499666\pi$
$410$	0	0
$411$	6.29089e19	1.56638
$412$	0	0
$413$	4.73073e19	1.13580
$414$	0	0
$415$	−6.53897e18	−0.151408
$416$	0	0
$417$	−1.14226e20	−2.55120
$418$	0	0
$419$	−2.86495e19	−0.617322	−0.308661	−	0.951172i	$-0.599881\pi$
$419$	−2.86495e19	−0.617322	−0.308661	+	0.951172i	$0.599881\pi$
$420$	0	0
$421$	−2.77675e19	−0.577326	−0.288663	−	0.957431i	$-0.593211\pi$
$421$	−2.77675e19	−0.577326	−0.288663	+	0.957431i	$0.593211\pi$
$422$	0	0
$423$	−8.85359e19	−1.77651
$424$	0	0
$425$	5.27737e19	1.02212
$426$	0	0
$427$	−1.04722e20	−1.95808
$428$	0	0
$429$	−1.85636e20	−3.35144
$430$	0	0
$431$	6.05133e19	1.05505	0.527523	−	0.849541i	$-0.323121\pi$
$431$	6.05133e19	1.05505	0.527523	+	0.849541i	$0.323121\pi$
$432$	0	0
$433$	8.90996e19	1.50043	0.750217	−	0.661192i	$-0.229949\pi$
$433$	8.90996e19	1.50043	0.750217	+	0.661192i	$0.229949\pi$
$434$	0	0
$435$	−8.32016e18	−0.135351
$436$	0	0
$437$	−1.34001e19	−0.210619
$438$	0	0
$439$	1.71798e19	0.260936	0.130468	−	0.991453i	$-0.458352\pi$
$439$	1.71798e19	0.260936	0.130468	+	0.991453i	$0.458352\pi$
$440$	0	0
$441$	4.66372e18	0.0684610
$442$	0	0
$443$	1.18416e19	0.168028	0.0840141	−	0.996465i	$-0.473226\pi$
$443$	1.18416e19	0.168028	0.0840141	+	0.996465i	$0.473226\pi$
$444$	0	0
$445$	−5.13376e19	−0.704264
$446$	0	0
$447$	6.70651e19	0.889591
$448$	0	0
$449$	9.46407e18	0.121403	0.0607016	−	0.998156i	$-0.480666\pi$
$449$	9.46407e18	0.121403	0.0607016	+	0.998156i	$0.480666\pi$
$450$	0	0
$451$	−1.56232e20	−1.93841
$452$	0	0
$453$	1.40177e20	1.68245
$454$	0	0
$455$	−6.01607e19	−0.698598
$456$	0	0
$457$	7.59975e19	0.853941	0.426971	−	0.904265i	$-0.359581\pi$
$457$	7.59975e19	0.853941	0.426971	+	0.904265i	$0.359581\pi$
$458$	0	0
$459$	−4.52215e19	−0.491756
$460$	0	0
$461$	−1.27777e20	−1.34492	−0.672458	−	0.740135i	$-0.734761\pi$
$461$	−1.27777e20	−1.34492	−0.672458	+	0.740135i	$0.734761\pi$
$462$	0	0
$463$	−1.63571e20	−1.66666	−0.833332	−	0.552773i	$-0.813570\pi$
$463$	−1.63571e20	−1.66666	−0.833332	+	0.552773i	$0.813570\pi$
$464$	0	0
$465$	2.30146e19	0.227042
$466$	0	0
$467$	−2.20439e19	−0.210577	−0.105289	−	0.994442i	$-0.533577\pi$
$467$	−2.20439e19	−0.210577	−0.105289	+	0.994442i	$0.533577\pi$
$468$	0	0
$469$	1.98002e20	1.83178
$470$	0	0
$471$	−5.06639e19	−0.453985
$472$	0	0
$473$	2.27144e20	1.97171
$474$	0	0
$475$	−3.45649e19	−0.290692
$476$	0	0
$477$	−2.00214e20	−1.63158
$478$	0	0
$479$	2.27890e20	1.79974	0.899869	−	0.436161i	$-0.143662\pi$
$479$	2.27890e20	1.79974	0.899869	+	0.436161i	$0.143662\pi$
$480$	0	0
$481$	−2.07554e20	−1.58870
$482$	0	0
$483$	1.26460e20	0.938315
$484$	0	0
$485$	−6.06770e19	−0.436476
$486$	0	0
$487$	−1.96991e20	−1.37398	−0.686988	−	0.726669i	$-0.741068\pi$
$487$	−1.96991e20	−1.37398	−0.686988	+	0.726669i	$0.741068\pi$
$488$	0	0
$489$	1.76041e20	1.19069
$490$	0	0
$491$	1.05068e20	0.689223	0.344612	−	0.938745i	$-0.388011\pi$
$491$	1.05068e20	0.689223	0.344612	+	0.938745i	$0.388011\pi$
$492$	0	0
$493$	−4.27820e19	−0.272213
$494$	0	0
$495$	−1.09032e20	−0.673000
$496$	0	0
$497$	−1.66389e20	−0.996440
$498$	0	0
$499$	−1.77775e19	−0.103304	−0.0516521	−	0.998665i	$-0.516449\pi$
$499$	−1.77775e19	−0.103304	−0.0516521	+	0.998665i	$0.516449\pi$
$500$	0	0
$501$	−4.05572e19	−0.228710
$502$	0	0
$503$	−6.45190e19	−0.353124	−0.176562	−	0.984289i	$-0.556498\pi$
$503$	−6.45190e19	−0.353124	−0.176562	+	0.984289i	$0.556498\pi$
$504$	0	0
$505$	5.84597e19	0.310578
$506$	0	0
$507$	5.40891e20	2.78965
$508$	0	0
$509$	−2.76478e19	−0.138445	−0.0692224	−	0.997601i	$-0.522052\pi$
$509$	−2.76478e19	−0.138445	−0.0692224	+	0.997601i	$0.522052\pi$
$510$	0	0
$511$	5.34318e19	0.259802
$512$	0	0
$513$	2.96185e19	0.139856
$514$	0	0
$515$	1.66987e18	0.00765825
$516$	0	0
$517$	−4.14556e20	−1.84673
$518$	0	0
$519$	3.67327e20	1.58963
$520$	0	0
$521$	−2.68296e20	−1.12806	−0.564029	−	0.825755i	$-0.690749\pi$
$521$	−2.68296e20	−1.12806	−0.564029	+	0.825755i	$0.690749\pi$
$522$	0	0
$523$	1.11046e20	0.453672	0.226836	−	0.973933i	$-0.427162\pi$
$523$	1.11046e20	0.453672	0.226836	+	0.973933i	$0.427162\pi$
$524$	0	0
$525$	3.26196e20	1.29504
$526$	0	0
$527$	1.18340e20	0.456619
$528$	0	0
$529$	−1.68416e20	−0.631633
$530$	0	0
$531$	−3.84665e20	−1.40241
$532$	0	0
$533$	7.00944e20	2.48445
$534$	0	0
$535$	−2.47769e19	−0.0853875
$536$	0	0
$537$	3.13968e20	1.05215
$538$	0	0
$539$	2.18372e19	0.0711674
$540$	0	0
$541$	−1.13223e20	−0.358886	−0.179443	−	0.983768i	$-0.557430\pi$
$541$	−1.13223e20	−0.358886	−0.179443	+	0.983768i	$0.557430\pi$
$542$	0	0
$543$	9.14642e20	2.82002
$544$	0	0
$545$	9.90494e19	0.297083
$546$	0	0
$547$	−3.98701e20	−1.16344	−0.581718	−	0.813390i	$-0.697619\pi$
$547$	−3.98701e20	−1.16344	−0.581718	+	0.813390i	$0.697619\pi$
$548$	0	0
$549$	8.51519e20	2.41770
$550$	0	0
$551$	2.80207e19	0.0774180
$552$	0	0
$553$	2.93492e20	0.789148
$554$	0	0
$555$	−2.18074e20	−0.570697
$556$	0	0
$557$	−6.08629e20	−1.55038	−0.775191	−	0.631727i	$-0.782347\pi$
$557$	−6.08629e20	−1.55038	−0.775191	+	0.631727i	$0.782347\pi$
$558$	0	0
$559$	−1.01910e21	−2.52713
$560$	0	0
$561$	−1.00291e21	−2.42126
$562$	0	0
$563$	7.42176e20	1.74459	0.872296	−	0.488977i	$-0.162630\pi$
$563$	7.42176e20	1.74459	0.872296	+	0.488977i	$0.162630\pi$
$564$	0	0
$565$	8.23488e19	0.188493
$566$	0	0
$567$	3.04316e20	0.678348
$568$	0	0
$569$	−5.95158e20	−1.29208	−0.646041	−	0.763303i	$-0.723577\pi$
$569$	−5.95158e20	−1.29208	−0.646041	+	0.763303i	$0.723577\pi$
$570$	0	0
$571$	−5.51613e20	−1.16644	−0.583222	−	0.812313i	$-0.698208\pi$
$571$	−5.51613e20	−1.16644	−0.583222	+	0.812313i	$0.698208\pi$
$572$	0	0
$573$	2.13368e20	0.439510
$574$	0	0
$575$	2.53353e20	0.508413
$576$	0	0
$577$	−4.71991e20	−0.922815	−0.461408	−	0.887188i	$-0.652655\pi$
$577$	−4.71991e20	−0.922815	−0.461408	+	0.887188i	$0.652655\pi$
$578$	0	0
$579$	2.84975e19	0.0542897
$580$	0	0
$581$	−2.07826e20	−0.385816
$582$	0	0
$583$	−9.37473e20	−1.69607
$584$	0	0
$585$	4.89179e20	0.862579
$586$	0	0
$587$	−1.39003e20	−0.238912	−0.119456	−	0.992840i	$-0.538115\pi$
$587$	−1.39003e20	−0.238912	−0.119456	+	0.992840i	$0.538115\pi$
$588$	0	0
$589$	−7.75088e19	−0.129863
$590$	0	0
$591$	−7.74192e20	−1.26457
$592$	0	0
$593$	−1.40651e20	−0.223992	−0.111996	−	0.993709i	$-0.535724\pi$
$593$	−1.40651e20	−0.223992	−0.111996	+	0.993709i	$0.535724\pi$
$594$	0	0
$595$	−3.25023e20	−0.504705
$596$	0	0
$597$	3.19441e20	0.483707
$598$	0	0
$599$	9.48542e18	0.0140073	0.00700366	−	0.999975i	$-0.497771\pi$
$599$	9.48542e18	0.0140073	0.00700366	+	0.999975i	$0.497771\pi$
$600$	0	0
$601$	−1.01702e20	−0.146477	−0.0732386	−	0.997314i	$-0.523333\pi$
$601$	−1.01702e20	−0.146477	−0.0732386	+	0.997314i	$0.523333\pi$
$602$	0	0
$603$	−1.61000e21	−2.26175
$604$	0	0
$605$	−2.16520e20	−0.296710
$606$	0	0
$607$	−5.86605e20	−0.784207	−0.392104	−	0.919921i	$-0.628253\pi$
$607$	−5.86605e20	−0.784207	−0.392104	+	0.919921i	$0.628253\pi$
$608$	0	0
$609$	−2.64437e20	−0.344900
$610$	0	0
$611$	1.85993e21	2.36694
$612$	0	0
$613$	1.49336e21	1.85443	0.927214	−	0.374532i	$-0.122197\pi$
$613$	1.49336e21	1.85443	0.927214	+	0.374532i	$0.122197\pi$
$614$	0	0
$615$	7.36471e20	0.892467
$616$	0	0
$617$	7.20003e20	0.851522	0.425761	−	0.904836i	$-0.360006\pi$
$617$	7.20003e20	0.851522	0.425761	+	0.904836i	$0.360006\pi$
$618$	0	0
$619$	2.51978e20	0.290859	0.145429	−	0.989369i	$-0.453544\pi$
$619$	2.51978e20	0.290859	0.145429	+	0.989369i	$0.453544\pi$
$620$	0	0
$621$	−2.17096e20	−0.244605
$622$	0	0
$623$	−1.63165e21	−1.79459
$624$	0	0
$625$	5.02335e20	0.539378
$626$	0	0
$627$	6.56872e20	0.688611
$628$	0	0
$629$	−1.12133e21	−1.14777
$630$	0	0
$631$	7.35089e20	0.734717	0.367358	−	0.930079i	$-0.380262\pi$
$631$	7.35089e20	0.734717	0.367358	+	0.930079i	$0.380262\pi$
$632$	0	0
$633$	1.27131e21	1.24087
$634$	0	0
$635$	3.54878e20	0.338280
$636$	0	0
$637$	−9.79739e19	−0.0912147
$638$	0	0
$639$	1.35294e21	1.23033
$640$	0	0
$641$	1.20788e21	1.07298	0.536488	−	0.843908i	$-0.319751\pi$
$641$	1.20788e21	1.07298	0.536488	+	0.843908i	$0.319751\pi$
$642$	0	0
$643$	−1.01046e21	−0.876870	−0.438435	−	0.898763i	$-0.644467\pi$
$643$	−1.01046e21	−0.876870	−0.438435	+	0.898763i	$0.644467\pi$
$644$	0	0
$645$	−1.07075e21	−0.907799
$646$	0	0
$647$	−8.40486e20	−0.696223	−0.348111	−	0.937453i	$-0.613177\pi$
$647$	−8.40486e20	−0.696223	−0.348111	+	0.937453i	$0.613177\pi$
$648$	0	0
$649$	−1.80114e21	−1.45785
$650$	0	0
$651$	7.31467e20	0.578545
$652$	0	0
$653$	1.22949e20	0.0950337	0.0475169	−	0.998870i	$-0.484869\pi$
$653$	1.22949e20	0.0950337	0.0475169	+	0.998870i	$0.484869\pi$
$654$	0	0
$655$	−4.91898e20	−0.371591
$656$	0	0
$657$	−4.34465e20	−0.320786
$658$	0	0
$659$	3.33134e20	0.240425	0.120212	−	0.992748i	$-0.461642\pi$
$659$	3.33134e20	0.240425	0.120212	+	0.992748i	$0.461642\pi$
$660$	0	0
$661$	−5.96923e20	−0.421121	−0.210561	−	0.977581i	$-0.567529\pi$
$661$	−5.96923e20	−0.421121	−0.210561	+	0.977581i	$0.567529\pi$
$662$	0	0
$663$	4.49963e21	3.10331
$664$	0	0
$665$	2.12879e20	0.143539
$666$	0	0
$667$	−2.05385e20	−0.135402
$668$	0	0
$669$	−2.01630e21	−1.29975
$670$	0	0
$671$	3.98711e21	2.51327
$672$	0	0
$673$	−2.79848e20	−0.172508	−0.0862541	−	0.996273i	$-0.527490\pi$
$673$	−2.79848e20	−0.172508	−0.0862541	+	0.996273i	$0.527490\pi$
$674$	0	0
$675$	−5.59989e20	−0.337599
$676$	0	0
$677$	8.44142e20	0.497738	0.248869	−	0.968537i	$-0.419941\pi$
$677$	8.44142e20	0.497738	0.248869	+	0.968537i	$0.419941\pi$
$678$	0	0
$679$	−1.92848e21	−1.11222
$680$	0	0
$681$	−1.48904e21	−0.840048
$682$	0	0
$683$	−2.23988e20	−0.123615	−0.0618073	−	0.998088i	$-0.519686\pi$
$683$	−2.23988e20	−0.123615	−0.0618073	+	0.998088i	$0.519686\pi$
$684$	0	0
$685$	−7.76215e20	−0.419086
$686$	0	0
$687$	−1.74924e21	−0.924004
$688$	0	0
$689$	4.20603e21	2.17384
$690$	0	0
$691$	−1.64756e21	−0.833211	−0.416606	−	0.909087i	$-0.636780\pi$
$691$	−1.64756e21	−0.833211	−0.416606	+	0.909087i	$0.636780\pi$
$692$	0	0
$693$	−3.46533e21	−1.71493
$694$	0	0
$695$	1.40940e21	0.682572
$696$	0	0
$697$	3.78691e21	1.79490
$698$	0	0
$699$	2.47206e21	1.14678
$700$	0	0
$701$	1.22646e21	0.556890	0.278445	−	0.960452i	$-0.410181\pi$
$701$	1.22646e21	0.556890	0.278445	+	0.960452i	$0.410181\pi$
$702$	0	0
$703$	7.34430e20	0.326427
$704$	0	0
$705$	1.95420e21	0.850258
$706$	0	0
$707$	1.85801e21	0.791410
$708$	0	0
$709$	1.14964e21	0.479418	0.239709	−	0.970845i	$-0.422948\pi$
$709$	1.14964e21	0.479418	0.239709	+	0.970845i	$0.422948\pi$
$710$	0	0
$711$	−2.38645e21	−0.974384
$712$	0	0
$713$	5.68121e20	0.227127
$714$	0	0
$715$	2.29051e21	0.896678
$716$	0	0
$717$	3.21044e21	1.23075
$718$	0	0
$719$	−3.73156e21	−1.40095	−0.700477	−	0.713675i	$-0.747030\pi$
$719$	−3.73156e21	−1.40095	−0.700477	+	0.713675i	$0.747030\pi$
$720$	0	0
$721$	5.30731e19	0.0195146
$722$	0	0
$723$	−2.17496e21	−0.783276
$724$	0	0
$725$	−5.29781e20	−0.186879
$726$	0	0
$727$	−3.93565e21	−1.35990	−0.679952	−	0.733256i	$-0.737999\pi$
$727$	−3.93565e21	−1.35990	−0.679952	+	0.733256i	$0.737999\pi$
$728$	0	0
$729$	−4.26741e21	−1.44447
$730$	0	0
$731$	−5.50576e21	−1.82573
$732$	0	0
$733$	1.28588e21	0.417753	0.208877	−	0.977942i	$-0.433019\pi$
$733$	1.28588e21	0.417753	0.208877	+	0.977942i	$0.433019\pi$
$734$	0	0
$735$	−1.02940e20	−0.0327663
$736$	0	0
$737$	−7.53857e21	−2.35116
$738$	0	0
$739$	1.84753e21	0.564621	0.282311	−	0.959323i	$-0.408899\pi$
$739$	1.84753e21	0.564621	0.282311	+	0.959323i	$0.408899\pi$
$740$	0	0
$741$	−2.94710e21	−0.882588
$742$	0	0
$743$	−3.46263e21	−1.01623	−0.508113	−	0.861291i	$-0.669657\pi$
$743$	−3.46263e21	−1.01623	−0.508113	+	0.861291i	$0.669657\pi$
$744$	0	0
$745$	−8.27498e20	−0.238010
$746$	0	0
$747$	1.68988e21	0.476378
$748$	0	0
$749$	−7.87479e20	−0.217583
$750$	0	0
$751$	−1.55571e20	−0.0421336	−0.0210668	−	0.999778i	$-0.506706\pi$
$751$	−1.55571e20	−0.0421336	−0.0210668	+	0.999778i	$0.506706\pi$
$752$	0	0
$753$	−2.31569e21	−0.614778
$754$	0	0
$755$	−1.72961e21	−0.450138
$756$	0	0
$757$	1.89778e21	0.484201	0.242101	−	0.970251i	$-0.422164\pi$
$757$	1.89778e21	0.484201	0.242101	+	0.970251i	$0.422164\pi$
$758$	0	0
$759$	−4.81472e21	−1.20436
$760$	0	0
$761$	4.57291e21	1.12152	0.560760	−	0.827978i	$-0.310509\pi$
$761$	4.57291e21	1.12152	0.560760	+	0.827978i	$0.310509\pi$
$762$	0	0
$763$	3.14806e21	0.757023
$764$	0	0
$765$	2.64283e21	0.623173
$766$	0	0
$767$	8.08092e21	1.86851
$768$	0	0
$769$	5.26461e21	1.19376	0.596882	−	0.802329i	$-0.296406\pi$
$769$	5.26461e21	1.19376	0.596882	+	0.802329i	$0.296406\pi$
$770$	0	0
$771$	−8.28322e21	−1.84201
$772$	0	0
$773$	−6.89284e21	−1.50332	−0.751661	−	0.659550i	$-0.770747\pi$
$773$	−6.89284e21	−1.50332	−0.751661	+	0.659550i	$0.770747\pi$
$774$	0	0
$775$	1.46544e21	0.313477
$776$	0	0
$777$	−6.93097e21	−1.45424
$778$	0	0
$779$	−2.48029e21	−0.510472
$780$	0	0
$781$	6.33494e21	1.27897
$782$	0	0
$783$	4.53966e20	0.0899105
$784$	0	0
$785$	6.25127e20	0.121464
$786$	0	0
$787$	−2.48699e21	−0.474093	−0.237047	−	0.971498i	$-0.576179\pi$
$787$	−2.48699e21	−0.474093	−0.237047	+	0.971498i	$0.576179\pi$
$788$	0	0
$789$	1.07762e22	2.01552
$790$	0	0
$791$	2.61727e21	0.480314
$792$	0	0
$793$	−1.78884e22	−3.22124
$794$	0	0
$795$	4.41921e21	0.780892
$796$	0	0
$797$	5.50037e21	0.953793	0.476897	−	0.878959i	$-0.341762\pi$
$797$	5.50037e21	0.953793	0.476897	+	0.878959i	$0.341762\pi$
$798$	0	0
$799$	1.00484e22	1.71001
$800$	0	0
$801$	1.32673e22	2.21584
$802$	0	0
$803$	−2.03432e21	−0.333467
$804$	0	0
$805$	−1.56035e21	−0.251046
$806$	0	0
$807$	−3.18176e21	−0.502476
$808$	0	0
$809$	4.82590e21	0.748109	0.374054	−	0.927407i	$-0.377967\pi$
$809$	4.82590e21	0.748109	0.374054	+	0.927407i	$0.377967\pi$
$810$	0	0
$811$	−4.74535e21	−0.722124	−0.361062	−	0.932542i	$-0.617586\pi$
$811$	−4.74535e21	−0.722124	−0.361062	+	0.932542i	$0.617586\pi$
$812$	0	0
$813$	5.84099e21	0.872584
$814$	0	0
$815$	−2.17212e21	−0.318568
$816$	0	0
$817$	3.60608e21	0.519242
$818$	0	0
$819$	1.55474e22	2.19801
$820$	0	0
$821$	−6.64783e21	−0.922798	−0.461399	−	0.887193i	$-0.652652\pi$
$821$	−6.64783e21	−0.922798	−0.461399	+	0.887193i	$0.652652\pi$
$822$	0	0
$823$	7.74614e21	1.05581	0.527906	−	0.849303i	$-0.322977\pi$
$823$	7.74614e21	1.05581	0.527906	+	0.849303i	$0.322977\pi$
$824$	0	0
$825$	−1.24193e22	−1.66224
$826$	0	0
$827$	1.25902e22	1.65478	0.827389	−	0.561629i	$-0.189825\pi$
$827$	1.25902e22	1.65478	0.827389	+	0.561629i	$0.189825\pi$
$828$	0	0
$829$	2.43881e20	0.0314789	0.0157395	−	0.999876i	$-0.494990\pi$
$829$	2.43881e20	0.0314789	0.0157395	+	0.999876i	$0.494990\pi$
$830$	0	0
$831$	2.77939e21	0.352324
$832$	0	0
$833$	−5.29313e20	−0.0658984
$834$	0	0
$835$	5.00424e20	0.0611913
$836$	0	0
$837$	−1.25573e21	−0.150818
$838$	0	0
$839$	4.98445e21	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$839$	4.98445e21	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$840$	0	0
$841$	−8.19971e21	−0.950230
$842$	0	0
$843$	−3.50414e20	−0.0398910
$844$	0	0
$845$	−6.67391e21	−0.746371
$846$	0	0
$847$	−6.88159e21	−0.756072
$848$	0	0
$849$	−5.16189e21	−0.557187
$850$	0	0
$851$	−5.38320e21	−0.570912
$852$	0	0
$853$	−1.68031e22	−1.75094	−0.875472	−	0.483269i	$-0.839449\pi$
$853$	−1.68031e22	−1.75094	−0.875472	+	0.483269i	$0.839449\pi$
$854$	0	0
$855$	−1.73096e21	−0.177232
$856$	0	0
$857$	5.85956e21	0.589534	0.294767	−	0.955569i	$-0.404758\pi$
$857$	5.85956e21	0.589534	0.294767	+	0.955569i	$0.404758\pi$
$858$	0	0
$859$	−1.88515e20	−0.0186379	−0.00931895	−	0.999957i	$-0.502966\pi$
$859$	−1.88515e20	−0.0186379	−0.00931895	+	0.999957i	$0.502966\pi$
$860$	0	0
$861$	2.34070e22	2.27417
$862$	0	0
$863$	−1.84482e22	−1.76146	−0.880729	−	0.473620i	$-0.842947\pi$
$863$	−1.84482e22	−1.76146	−0.880729	+	0.473620i	$0.842947\pi$
$864$	0	0
$865$	−4.53234e21	−0.425306
$866$	0	0
$867$	7.98179e21	0.736134
$868$	0	0
$869$	−1.11742e22	−1.01290
$870$	0	0
$871$	3.38223e22	3.01347
$872$	0	0
$873$	1.56809e22	1.37329
$874$	0	0
$875$	−8.82962e21	−0.760119
$876$	0	0
$877$	−8.31583e21	−0.703734	−0.351867	−	0.936050i	$-0.614453\pi$
$877$	−8.31583e21	−0.703734	−0.351867	+	0.936050i	$0.614453\pi$
$878$	0	0
$879$	−2.49694e22	−2.07726
$880$	0	0
$881$	1.02417e22	0.837630	0.418815	−	0.908071i	$-0.362445\pi$
$881$	1.02417e22	0.837630	0.418815	+	0.908071i	$0.362445\pi$
$882$	0	0
$883$	1.85637e22	1.49265	0.746327	−	0.665579i	$-0.231815\pi$
$883$	1.85637e22	1.49265	0.746327	+	0.665579i	$0.231815\pi$
$884$	0	0
$885$	8.49049e21	0.671209
$886$	0	0
$887$	−1.74753e22	−1.35830	−0.679151	−	0.733999i	$-0.737652\pi$
$887$	−1.74753e22	−1.35830	−0.679151	+	0.733999i	$0.737652\pi$
$888$	0	0
$889$	1.12790e22	0.861999
$890$	0	0
$891$	−1.15863e22	−0.870686
$892$	0	0
$893$	−6.58137e21	−0.486329
$894$	0	0
$895$	−3.87396e21	−0.281503
$896$	0	0
$897$	2.16015e22	1.54362
$898$	0	0
$899$	−1.18799e21	−0.0834860
$900$	0	0
$901$	2.27235e22	1.57050
$902$	0	0
$903$	−3.40313e22	−2.31324
$904$	0	0
$905$	−1.12855e22	−0.754496
$906$	0	0
$907$	−7.17927e21	−0.472091	−0.236045	−	0.971742i	$-0.575851\pi$
$907$	−7.17927e21	−0.472091	−0.236045	+	0.971742i	$0.575851\pi$
$908$	0	0
$909$	−1.51078e22	−0.977177
$910$	0	0
$911$	1.70120e22	1.08235	0.541174	−	0.840911i	$-0.317980\pi$
$911$	1.70120e22	1.08235	0.541174	+	0.840911i	$0.317980\pi$
$912$	0	0
$913$	7.91261e21	0.495209
$914$	0	0
$915$	−1.87951e22	−1.15714
$916$	0	0
$917$	−1.56339e22	−0.946883
$918$	0	0
$919$	2.68367e22	1.59905	0.799526	−	0.600631i	$-0.205084\pi$
$919$	2.68367e22	1.59905	0.799526	+	0.600631i	$0.205084\pi$
$920$	0	0
$921$	−9.90264e20	−0.0580502
$922$	0	0
$923$	−2.84221e22	−1.63925
$924$	0	0
$925$	−1.38857e22	−0.787961
$926$	0	0
$927$	−4.31549e20	−0.0240953
$928$	0	0
$929$	6.54830e21	0.359758	0.179879	−	0.983689i	$-0.442429\pi$
$929$	6.54830e21	0.359758	0.179879	+	0.983689i	$0.442429\pi$
$930$	0	0
$931$	3.46681e20	0.0187416
$932$	0	0
$933$	−5.04327e22	−2.68287
$934$	0	0
$935$	1.23747e22	0.647808
$936$	0	0
$937$	−7.31180e21	−0.376684	−0.188342	−	0.982103i	$-0.560311\pi$
$937$	−7.31180e21	−0.376684	−0.188342	+	0.982103i	$0.560311\pi$
$938$	0	0
$939$	−1.20548e22	−0.611180
$940$	0	0
$941$	1.09415e22	0.545952	0.272976	−	0.962021i	$-0.411992\pi$
$941$	1.09415e22	0.545952	0.272976	+	0.962021i	$0.411992\pi$
$942$	0	0
$943$	1.81800e22	0.892802
$944$	0	0
$945$	3.44887e21	0.166701
$946$	0	0
$947$	−1.38913e22	−0.660871	−0.330436	−	0.943828i	$-0.607196\pi$
$947$	−1.38913e22	−0.660871	−0.330436	+	0.943828i	$0.607196\pi$
$948$	0	0
$949$	9.12710e21	0.427402
$950$	0	0
$951$	−2.86880e21	−0.132235
$952$	0	0
$953$	−2.78995e21	−0.126590	−0.0632951	−	0.997995i	$-0.520161\pi$
$953$	−2.78995e21	−0.126590	−0.0632951	+	0.997995i	$0.520161\pi$
$954$	0	0
$955$	−2.63268e21	−0.117591
$956$	0	0
$957$	1.00680e22	0.442692
$958$	0	0
$959$	−2.46702e22	−1.06791
$960$	0	0
$961$	−2.01791e22	−0.859958
$962$	0	0
$963$	6.40315e21	0.268656
$964$	0	0
$965$	−3.51623e20	−0.0145252
$966$	0	0
$967$	−2.82118e22	−1.14745	−0.573723	−	0.819050i	$-0.694501\pi$
$967$	−2.82118e22	−1.14745	−0.573723	+	0.819050i	$0.694501\pi$
$968$	0	0
$969$	−1.59220e22	−0.637629
$970$	0	0
$971$	2.34855e21	0.0926095	0.0463048	−	0.998927i	$-0.485255\pi$
$971$	2.34855e21	0.0926095	0.0463048	+	0.998927i	$0.485255\pi$
$972$	0	0
$973$	4.47946e22	1.73932
$974$	0	0
$975$	5.57201e22	2.13048
$976$	0	0
$977$	5.06144e22	1.90575	0.952873	−	0.303370i	$-0.0981118\pi$
$977$	5.06144e22	1.90575	0.952873	+	0.303370i	$0.0981118\pi$
$978$	0	0
$979$	6.21220e22	2.30343
$980$	0	0
$981$	−2.55975e22	−0.934718
$982$	0	0
$983$	1.05850e22	0.380663	0.190331	−	0.981720i	$-0.439044\pi$
$983$	1.05850e22	0.380663	0.190331	+	0.981720i	$0.439044\pi$
$984$	0	0
$985$	9.55254e21	0.338335
$986$	0	0
$987$	6.21098e22	2.16661
$988$	0	0
$989$	−2.64317e22	−0.908140
$990$	0	0
$991$	−2.21374e22	−0.749158	−0.374579	−	0.927195i	$-0.622213\pi$
$991$	−2.21374e22	−0.749158	−0.374579	+	0.927195i	$0.622213\pi$
$992$	0	0
$993$	−8.21670e22	−2.73891
$994$	0	0
$995$	−3.94149e21	−0.129416
$996$	0	0
$997$	5.95482e22	1.92600	0.962998	−	0.269510i	$-0.0868618\pi$
$997$	5.95482e22	1.92600	0.962998	+	0.269510i	$0.0868618\pi$
$998$	0	0
$999$	1.18986e22	0.379100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.16.a.c.1.4	✓	4
4.3	odd	2		32.16.a.e.1.1	yes	4
8.3	odd	2		64.16.a.o.1.4		4
8.5	even	2		64.16.a.q.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.16.a.c.1.4	✓	4	1.1	even	1	trivial
32.16.a.e.1.1	yes	4	4.3	odd	2
64.16.a.o.1.4		4	8.3	odd	2
64.16.a.q.1.1		4	8.5	even	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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)

\(f(q)\)	\(=\)	\(q+5704.21 q^{3} -70382.7 q^{5} -2.23695e6 q^{7} +1.81891e7 q^{9} +8.51679e7 q^{11} -3.82111e8 q^{13} -4.01478e8 q^{15} -2.06439e9 q^{17} +1.35210e9 q^{19} -1.27601e10 q^{21} -9.91058e9 q^{23} -2.55639e10 q^{25} +2.19055e10 q^{27} +2.07238e10 q^{29} -5.73247e10 q^{31} +4.85816e11 q^{33} +1.57443e11 q^{35} +5.43177e11 q^{37} -2.17964e12 q^{39} -1.83440e12 q^{41} +2.66702e12 q^{43} -1.28020e12 q^{45} -4.86752e12 q^{47} +2.56402e11 q^{49} -1.17757e13 q^{51} -1.10073e13 q^{53} -5.99435e12 q^{55} +7.71267e12 q^{57} -2.11481e13 q^{59} +4.68147e13 q^{61} -4.06883e13 q^{63} +2.68940e13 q^{65} -8.85142e13 q^{67} -5.65321e13 q^{69} +7.43818e13 q^{71} -2.38860e13 q^{73} -1.45822e14 q^{75} -1.90517e14 q^{77} -1.31202e14 q^{79} -1.36041e14 q^{81} +9.29060e13 q^{83} +1.45297e14 q^{85} +1.18213e14 q^{87} +7.29406e14 q^{89} +8.54765e14 q^{91} -3.26992e14 q^{93} -9.51645e13 q^{95} +8.62101e14 q^{97} +1.54913e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2912 q^{3} - 98280 q^{5} - 2755776 q^{7} + 26457940 q^{9} + 110853856 q^{11} - 187741448 q^{13} - 442991680 q^{15} + 2121294984 q^{17} + 419203872 q^{19} + 8397459968 q^{21} - 5330808384 q^{23} + 8564955740 q^{25}+ \cdots + 26\!\cdots\!84 q^{99}+O(q^{100}) \) 4 * q - 2912 * q^3 - 98280 * q^5 - 2755776 * q^7 + 26457940 * q^9 + 110853856 * q^11 - 187741448 * q^13 - 442991680 * q^15 + 2121294984 * q^17 + 419203872 * q^19 + 8397459968 * q^21 - 5330808384 * q^23 + 8564955740 * q^25 - 41191036352 * q^27 - 86832359880 * q^29 + 96274977024 * q^31 + 41399413504 * q^33 - 929253215360 * q^35 - 112979590376 * q^37 - 2001683116352 * q^39 - 1534010535960 * q^41 - 3567397370400 * q^43 + 3155482037240 * q^45 - 9031658937472 * q^47 + 8694343447332 * q^49 - 37558794281664 * q^51 - 14370240161064 * q^53 - 46434781152960 * q^55 - 5813065407232 * q^57 - 68701112336800 * q^59 + 4806363233592 * q^61 - 192971404565440 * q^63 + 7651382051280 * q^65 - 167433288699744 * q^67 - 80410496862720 * q^69 - 180096642956992 * q^71 + 173676000667560 * q^73 - 227575010162080 * q^75 - 133277064067584 * q^77 - 232741753133952 * q^79 + 263803331371492 * q^81 + 20415834690336 * q^83 - 342055719314640 * q^85 + 870209244139712 * q^87 + 938354283775848 * q^89 + 1478922417113472 * q^91 - 2098341777250304 * q^93 + 2700969259238080 * q^95 + 1510271716715080 * q^97 + 2629725177868384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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