LMFDB - Embedded newform 128.5.c.a.127.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,5,Mod(127,128)]

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([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("128.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	128.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$13.2313552747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.205520896.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 $ x^8 - 4x^7 + 12x^6 - 12x^5 - 8x^4 + 12x^3 + 12x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{39} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.4
Root			$-0.129640 - 0.312979i$ of defining polynomial
Character	$\chi$	$=$	128.127
Dual form			128.5.c.a.127.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.29514i q^{3} +22.7388 q^{5} -51.5147i q^{7} +41.3713 q^{9} +O(q^{10})$	$q-6.29514i q^{3} +22.7388 q^{5} -51.5147i q^{7} +41.3713 q^{9} +50.8148i q^{11} -160.493 q^{13} -143.144i q^{15} +416.835 q^{17} -515.673i q^{19} -324.292 q^{21} -15.8458i q^{23} -107.949 q^{25} -770.344i q^{27} -979.249 q^{29} -1906.37i q^{31} +319.886 q^{33} -1171.38i q^{35} -657.600 q^{37} +1010.33i q^{39} +2838.90 q^{41} +2532.93i q^{43} +940.731 q^{45} -940.191i q^{47} -252.760 q^{49} -2624.03i q^{51} +280.223 q^{53} +1155.46i q^{55} -3246.23 q^{57} +5036.91i q^{59} +2883.14 q^{61} -2131.23i q^{63} -3649.41 q^{65} -1425.50i q^{67} -99.7517 q^{69} +8926.20i q^{71} -4479.58 q^{73} +679.554i q^{75} +2617.71 q^{77} +6835.88i q^{79} -1498.35 q^{81} +4789.45i q^{83} +9478.31 q^{85} +6164.50i q^{87} +1706.23 q^{89} +8267.75i q^{91} -12000.8 q^{93} -11725.8i q^{95} +1896.21 q^{97} +2102.27i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 48 q^{5} - 216 q^{9}+O(q^{10}) $ 8 * q - 48 * q^5 - 216 * q^9	$ 8 q - 48 q^{5} - 216 q^{9} - 240 q^{13} + 240 q^{17} + 1216 q^{21} + 664 q^{25} - 432 q^{29} + 992 q^{33} - 2800 q^{37} - 2928 q^{41} + 4880 q^{45} - 5752 q^{49} - 1776 q^{53} + 8608 q^{57} - 12656 q^{61} + 672 q^{65} + 4416 q^{69} + 560 q^{73} + 31296 q^{77} + 14696 q^{81} + 14432 q^{85} - 22992 q^{89} - 56320 q^{93} - 3728 q^{97}+O(q^{100}) $ 8 * q - 48 * q^5 - 216 * q^9 - 240 * q^13 + 240 * q^17 + 1216 * q^21 + 664 * q^25 - 432 * q^29 + 992 * q^33 - 2800 * q^37 - 2928 * q^41 + 4880 * q^45 - 5752 * q^49 - 1776 * q^53 + 8608 * q^57 - 12656 * q^61 + 672 * q^65 + 4416 * q^69 + 560 * q^73 + 31296 * q^77 + 14696 * q^81 + 14432 * q^85 - 22992 * q^89 - 56320 * q^93 - 3728 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$1$	$-1$

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
$3$	− 6.29514i	− 0.699460i	−0.936851	−	0.349730i	$-0.886273\pi$
$3$	− 6.29514i	− 0.699460i	0.936851	−	0.349730i	$-0.113727\pi$
$4$	0	0
$5$	22.7388	0.909550	0.454775	−	0.890606i	$-0.349720\pi$
$5$	22.7388	0.909550	0.454775	+	0.890606i	$0.349720\pi$
$6$	0	0
$7$	− 51.5147i	− 1.05132i	−0.850695	−	0.525660i	$-0.823819\pi$
$7$	− 51.5147i	− 1.05132i	0.850695	−	0.525660i	$-0.176181\pi$
$8$	0	0
$9$	41.3713	0.510756
$10$	0	0
$11$	50.8148i	0.419957i	0.977706	+	0.209978i	$0.0673394\pi$
$11$	50.8148i	0.419957i	−0.977706	+	0.209978i	$0.932661\pi$
$12$	0	0
$13$	−160.493	−0.949663	−0.474832	−	0.880077i	$-0.657491\pi$
$13$	−160.493	−0.949663	−0.474832	+	0.880077i	$0.657491\pi$
$14$	0	0
$15$	− 143.144i	− 0.636194i
$16$	0	0
$17$	416.835	1.44234	0.721168	−	0.692760i	$-0.243606\pi$
$17$	416.835	1.44234	0.721168	+	0.692760i	$0.243606\pi$
$18$	0	0
$19$	− 515.673i	− 1.42846i	−0.699913	−	0.714228i	$-0.746778\pi$
$19$	− 515.673i	− 1.42846i	0.699913	−	0.714228i	$-0.253222\pi$
$20$	0	0
$21$	−324.292	−0.735356
$22$	0	0
$23$	− 15.8458i	− 0.0299543i	−0.999888	−	0.0149772i	$-0.995232\pi$
$23$	− 15.8458i	− 0.0299543i	0.999888	−	0.0149772i	$-0.00476755\pi$
$24$	0	0
$25$	−107.949	−0.172718
$26$	0	0
$27$	− 770.344i	− 1.05671i
$28$	0	0
$29$	−979.249	−1.16439	−0.582193	−	0.813051i	$-0.697805\pi$
$29$	−979.249	−1.16439	−0.582193	+	0.813051i	$0.697805\pi$
$30$	0	0
$31$	− 1906.37i	− 1.98373i	−0.127286	−	0.991866i	$-0.540627\pi$
$31$	− 1906.37i	− 1.98373i	0.127286	−	0.991866i	$-0.459373\pi$
$32$	0	0
$33$	319.886	0.293743
$34$	0	0
$35$	− 1171.38i	− 0.956228i
$36$	0	0
$37$	−657.600	−0.480351	−0.240175	−	0.970729i	$-0.577205\pi$
$37$	−657.600	−0.480351	−0.240175	+	0.970729i	$0.577205\pi$
$38$	0	0
$39$	1010.33i	0.664251i
$40$	0	0
$41$	2838.90	1.68882	0.844408	−	0.535701i	$-0.179953\pi$
$41$	2838.90	1.68882	0.844408	+	0.535701i	$0.179953\pi$
$42$	0	0
$43$	2532.93i	1.36989i	0.728595	+	0.684945i	$0.240174\pi$
$43$	2532.93i	1.36989i	−0.728595	+	0.684945i	$0.759826\pi$
$44$	0	0
$45$	940.731	0.464558
$46$	0	0
$47$	− 940.191i	− 0.425618i	−0.977094	−	0.212809i	$-0.931739\pi$
$47$	− 940.191i	− 0.425618i	0.977094	−	0.212809i	$-0.0682613\pi$
$48$	0	0
$49$	−252.760	−0.105273
$50$	0	0
$51$	− 2624.03i	− 1.00886i
$52$	0	0
$53$	280.223	0.0997590	0.0498795	−	0.998755i	$-0.484116\pi$
$53$	280.223	0.0997590	0.0498795	+	0.998755i	$0.484116\pi$
$54$	0	0
$55$	1155.46i	0.381972i
$56$	0	0
$57$	−3246.23	−0.999148
$58$	0	0
$59$	5036.91i	1.44697i	0.690339	+	0.723486i	$0.257461\pi$
$59$	5036.91i	1.44697i	−0.690339	+	0.723486i	$0.742539\pi$
$60$	0	0
$61$	2883.14	0.774830	0.387415	−	0.921905i	$-0.373368\pi$
$61$	2883.14	0.774830	0.387415	+	0.921905i	$0.373368\pi$
$62$	0	0
$63$	− 2131.23i	− 0.536968i
$64$	0	0
$65$	−3649.41	−0.863767
$66$	0	0
$67$	− 1425.50i	− 0.317555i	−0.987314	−	0.158777i	$-0.949245\pi$
$67$	− 1425.50i	− 0.317555i	0.987314	−	0.158777i	$-0.0507552\pi$
$68$	0	0
$69$	−99.7517	−0.0209518
$70$	0	0
$71$	8926.20i	1.77072i	0.464906	+	0.885360i	$0.346088\pi$
$71$	8926.20i	1.77072i	−0.464906	+	0.885360i	$0.653912\pi$
$72$	0	0
$73$	−4479.58	−0.840604	−0.420302	−	0.907384i	$-0.638076\pi$
$73$	−4479.58	−0.840604	−0.420302	+	0.907384i	$0.638076\pi$
$74$	0	0
$75$	679.554i	0.120810i
$76$	0	0
$77$	2617.71	0.441509
$78$	0	0
$79$	6835.88i	1.09532i	0.836701	+	0.547659i	$0.184481\pi$
$79$	6835.88i	1.09532i	−0.836701	+	0.547659i	$0.815519\pi$
$80$	0	0
$81$	−1498.35	−0.228372
$82$	0	0
$83$	4789.45i	0.695231i	0.937637	+	0.347616i	$0.113009\pi$
$83$	4789.45i	0.695231i	−0.937637	+	0.347616i	$0.886991\pi$
$84$	0	0
$85$	9478.31	1.31188
$86$	0	0
$87$	6164.50i	0.814441i
$88$	0	0
$89$	1706.23	0.215406	0.107703	−	0.994183i	$-0.465651\pi$
$89$	1706.23	0.215406	0.107703	+	0.994183i	$0.465651\pi$
$90$	0	0
$91$	8267.75i	0.998400i
$92$	0	0
$93$	−12000.8	−1.38754
$94$	0	0
$95$	− 11725.8i	− 1.29925i
$96$	0	0
$97$	1896.21	0.201532	0.100766	−	0.994910i	$-0.467871\pi$
$97$	1896.21	0.201532	0.100766	+	0.994910i	$0.467871\pi$
$98$	0	0
$99$	2102.27i	0.214496i
$100$	0	0
$101$	2734.82	0.268093	0.134047	−	0.990975i	$-0.457203\pi$
$101$	2734.82	0.268093	0.134047	+	0.990975i	$0.457203\pi$
$102$	0	0
$103$	− 6266.60i	− 0.590687i	−0.955391	−	0.295343i	$-0.904566\pi$
$103$	− 6266.60i	− 0.590687i	0.955391	−	0.295343i	$-0.0954341\pi$
$104$	0	0
$105$	−7373.99	−0.668843
$106$	0	0
$107$	− 8130.95i	− 0.710188i	−0.934831	−	0.355094i	$-0.884449\pi$
$107$	− 8130.95i	− 0.710188i	0.934831	−	0.355094i	$-0.115551\pi$
$108$	0	0
$109$	18754.6	1.57854	0.789269	−	0.614048i	$-0.210460\pi$
$109$	18754.6	1.57854	0.789269	+	0.614048i	$0.210460\pi$
$110$	0	0
$111$	4139.68i	0.335986i
$112$	0	0
$113$	4911.65	0.384654	0.192327	−	0.981331i	$-0.438397\pi$
$113$	4911.65	0.384654	0.192327	+	0.981331i	$0.438397\pi$
$114$	0	0
$115$	− 360.314i	− 0.0272449i
$116$	0	0
$117$	−6639.80	−0.485046
$118$	0	0
$119$	− 21473.1i	− 1.51636i
$120$	0	0
$121$	12058.9	0.823636
$122$	0	0
$123$	− 17871.3i	− 1.18126i
$124$	0	0
$125$	−16666.3	−1.06665
$126$	0	0
$127$	22112.7i	1.37099i	0.728078	+	0.685495i	$0.240414\pi$
$127$	22112.7i	1.37099i	−0.728078	+	0.685495i	$0.759586\pi$
$128$	0	0
$129$	15945.1	0.958182
$130$	0	0
$131$	− 7774.01i	− 0.453005i	−0.974011	−	0.226502i	$-0.927271\pi$
$131$	− 7774.01i	− 0.453005i	0.974011	−	0.226502i	$-0.0727291\pi$
$132$	0	0
$133$	−26564.7	−1.50176
$134$	0	0
$135$	− 17516.7i	− 0.961134i
$136$	0	0
$137$	10778.2	0.574254	0.287127	−	0.957893i	$-0.407300\pi$
$137$	10778.2	0.574254	0.287127	+	0.957893i	$0.407300\pi$
$138$	0	0
$139$	− 1538.28i	− 0.0796170i	−0.999207	−	0.0398085i	$-0.987325\pi$
$139$	− 1538.28i	− 0.0796170i	0.999207	−	0.0398085i	$-0.0126748\pi$
$140$	0	0
$141$	−5918.63	−0.297703
$142$	0	0
$143$	− 8155.42i	− 0.398818i
$144$	0	0
$145$	−22266.9	−1.05907
$146$	0	0
$147$	1591.16i	0.0736340i
$148$	0	0
$149$	−9301.21	−0.418955	−0.209477	−	0.977814i	$-0.567176\pi$
$149$	−9301.21	−0.418955	−0.209477	+	0.977814i	$0.567176\pi$
$150$	0	0
$151$	− 2447.40i	− 0.107337i	−0.998559	−	0.0536686i	$-0.982909\pi$
$151$	− 2447.40i	− 0.107337i	0.998559	−	0.0536686i	$-0.0170915\pi$
$152$	0	0
$153$	17245.0	0.736682
$154$	0	0
$155$	− 43348.4i	− 1.80430i
$156$	0	0
$157$	−24225.1	−0.982802	−0.491401	−	0.870933i	$-0.663515\pi$
$157$	−24225.1	−0.982802	−0.491401	+	0.870933i	$0.663515\pi$
$158$	0	0
$159$	− 1764.04i	− 0.0697774i
$160$	0	0
$161$	−816.292	−0.0314915
$162$	0	0
$163$	38521.1i	1.44985i	0.688827	+	0.724926i	$0.258126\pi$
$163$	38521.1i	1.44985i	−0.688827	+	0.724926i	$0.741874\pi$
$164$	0	0
$165$	7273.81	0.267174
$166$	0	0
$167$	− 15310.2i	− 0.548969i	−0.961592	−	0.274484i	$-0.911493\pi$
$167$	− 15310.2i	− 0.548969i	0.961592	−	0.274484i	$-0.0885072\pi$
$168$	0	0
$169$	−2802.96	−0.0981395
$170$	0	0
$171$	− 21334.0i	− 0.729593i
$172$	0	0
$173$	55399.9	1.85104	0.925522	−	0.378694i	$-0.123627\pi$
$173$	55399.9	1.85104	0.925522	+	0.378694i	$0.123627\pi$
$174$	0	0
$175$	5560.95i	0.181582i
$176$	0	0
$177$	31708.0	1.01210
$178$	0	0
$179$	− 4153.47i	− 0.129630i	−0.997897	−	0.0648150i	$-0.979354\pi$
$179$	− 4153.47i	− 0.129630i	0.997897	−	0.0648150i	$-0.0206457\pi$
$180$	0	0
$181$	50048.0	1.52767	0.763835	−	0.645411i	$-0.223314\pi$
$181$	50048.0	1.52767	0.763835	+	0.645411i	$0.223314\pi$
$182$	0	0
$183$	− 18149.8i	− 0.541963i
$184$	0	0
$185$	−14953.0	−0.436903
$186$	0	0
$187$	21181.4i	0.605719i
$188$	0	0
$189$	−39684.0	−1.11094
$190$	0	0
$191$	29434.0i	0.806831i	0.915017	+	0.403415i	$0.132177\pi$
$191$	29434.0i	0.806831i	−0.915017	+	0.403415i	$0.867823\pi$
$192$	0	0
$193$	−34940.3	−0.938021	−0.469010	−	0.883193i	$-0.655389\pi$
$193$	−34940.3	−0.938021	−0.469010	+	0.883193i	$0.655389\pi$
$194$	0	0
$195$	22973.6i	0.604170i
$196$	0	0
$197$	−55081.2	−1.41929	−0.709645	−	0.704559i	$-0.751145\pi$
$197$	−55081.2	−1.41929	−0.709645	+	0.704559i	$0.751145\pi$
$198$	0	0
$199$	18751.2i	0.473503i	0.971570	+	0.236751i	$0.0760827\pi$
$199$	18751.2i	0.473503i	−0.971570	+	0.236751i	$0.923917\pi$
$200$	0	0
$201$	−8973.74	−0.222117
$202$	0	0
$203$	50445.7i	1.22414i
$204$	0	0
$205$	64553.0	1.53606
$206$	0	0
$207$	− 655.562i	− 0.0152993i
$208$	0	0
$209$	26203.8	0.599890
$210$	0	0
$211$	56221.2i	1.26280i	0.775456	+	0.631401i	$0.217520\pi$
$211$	56221.2i	1.26280i	−0.775456	+	0.631401i	$0.782480\pi$
$212$	0	0
$213$	56191.7	1.23855
$214$	0	0
$215$	57595.6i	1.24598i
$216$	0	0
$217$	−98205.8	−2.08554
$218$	0	0
$219$	28199.6i	0.587968i
$220$	0	0
$221$	−66899.1	−1.36973
$222$	0	0
$223$	73830.1i	1.48465i	0.670041	+	0.742324i	$0.266277\pi$
$223$	73830.1i	1.48465i	−0.670041	+	0.742324i	$0.733723\pi$
$224$	0	0
$225$	−4465.98	−0.0882170
$226$	0	0
$227$	− 92455.9i	− 1.79425i	−0.441777	−	0.897125i	$-0.645652\pi$
$227$	− 92455.9i	− 1.79425i	0.441777	−	0.897125i	$-0.354348\pi$
$228$	0	0
$229$	52443.2	1.00004	0.500021	−	0.866014i	$-0.333326\pi$
$229$	52443.2	1.00004	0.500021	+	0.866014i	$0.333326\pi$
$230$	0	0
$231$	− 16478.8i	− 0.308818i
$232$	0	0
$233$	32172.5	0.592616	0.296308	−	0.955092i	$-0.404245\pi$
$233$	32172.5	0.592616	0.296308	+	0.955092i	$0.404245\pi$
$234$	0	0
$235$	− 21378.8i	− 0.387121i
$236$	0	0
$237$	43032.8	0.766131
$238$	0	0
$239$	− 56096.6i	− 0.982067i	−0.871141	−	0.491033i	$-0.836619\pi$
$239$	− 56096.6i	− 0.982067i	0.871141	−	0.491033i	$-0.163381\pi$
$240$	0	0
$241$	−95669.1	−1.64717	−0.823584	−	0.567195i	$-0.808029\pi$
$241$	−95669.1	−1.64717	−0.823584	+	0.567195i	$0.808029\pi$
$242$	0	0
$243$	− 52965.5i	− 0.896976i
$244$	0	0
$245$	−5747.44	−0.0957508
$246$	0	0
$247$	82761.9i	1.35655i
$248$	0	0
$249$	30150.2	0.486286
$250$	0	0
$251$	92926.2i	1.47500i	0.675350	+	0.737498i	$0.263993\pi$
$251$	92926.2i	1.47500i	−0.675350	+	0.737498i	$0.736007\pi$
$252$	0	0
$253$	805.202	0.0125795
$254$	0	0
$255$	− 59667.3i	− 0.917605i
$256$	0	0
$257$	−14146.4	−0.214180	−0.107090	−	0.994249i	$-0.534153\pi$
$257$	−14146.4	−0.214180	−0.107090	+	0.994249i	$0.534153\pi$
$258$	0	0
$259$	33876.1i	0.505002i
$260$	0	0
$261$	−40512.7	−0.594717
$262$	0	0
$263$	32379.8i	0.468126i	0.972221	+	0.234063i	$0.0752022\pi$
$263$	32379.8i	0.468126i	−0.972221	+	0.234063i	$0.924798\pi$
$264$	0	0
$265$	6371.92	0.0907358
$266$	0	0
$267$	− 10740.9i	− 0.150668i
$268$	0	0
$269$	31972.8	0.441851	0.220926	−	0.975291i	$-0.429092\pi$
$269$	31972.8	0.441851	0.220926	+	0.975291i	$0.429092\pi$
$270$	0	0
$271$	− 27060.2i	− 0.368461i	−0.982883	−	0.184231i	$-0.941021\pi$
$271$	− 27060.2i	− 0.368461i	0.982883	−	0.184231i	$-0.0589793\pi$
$272$	0	0
$273$	52046.6	0.698340
$274$	0	0
$275$	− 5485.40i	− 0.0725343i
$276$	0	0
$277$	−23125.8	−0.301396	−0.150698	−	0.988580i	$-0.548152\pi$
$277$	−23125.8	−0.301396	−0.150698	+	0.988580i	$0.548152\pi$
$278$	0	0
$279$	− 78868.8i	− 1.01320i
$280$	0	0
$281$	47541.8	0.602092	0.301046	−	0.953610i	$-0.402664\pi$
$281$	47541.8	0.602092	0.301046	+	0.953610i	$0.402664\pi$
$282$	0	0
$283$	− 3731.85i	− 0.0465963i	−0.999729	−	0.0232981i	$-0.992583\pi$
$283$	− 3731.85i	− 0.0465963i	0.999729	−	0.0232981i	$-0.00741670\pi$
$284$	0	0
$285$	−73815.3	−0.908775
$286$	0	0
$287$	− 146245.i	− 1.77548i
$288$	0	0
$289$	90230.4	1.08033
$290$	0	0
$291$	− 11936.9i	− 0.140963i
$292$	0	0
$293$	−93014.8	−1.08347	−0.541735	−	0.840549i	$-0.682232\pi$
$293$	−93014.8	−1.08347	−0.541735	+	0.840549i	$0.682232\pi$
$294$	0	0
$295$	114533.i	1.31609i
$296$	0	0
$297$	39144.8	0.443774
$298$	0	0
$299$	2543.15i	0.0284465i
$300$	0	0
$301$	130483.	1.44019
$302$	0	0
$303$	− 17216.1i	− 0.187520i
$304$	0	0
$305$	65559.1	0.704747
$306$	0	0
$307$	− 107545.i	− 1.14107i	−0.821274	−	0.570534i	$-0.806736\pi$
$307$	− 107545.i	− 1.14107i	0.821274	−	0.570534i	$-0.193264\pi$
$308$	0	0
$309$	−39449.1	−0.413162
$310$	0	0
$311$	13540.8i	0.139999i	0.997547	+	0.0699994i	$0.0222997\pi$
$311$	13540.8i	0.139999i	−0.997547	+	0.0699994i	$0.977700\pi$
$312$	0	0
$313$	74861.7	0.764137	0.382068	−	0.924134i	$-0.375212\pi$
$313$	74861.7	0.764137	0.382068	+	0.924134i	$0.375212\pi$
$314$	0	0
$315$	− 48461.4i	− 0.488399i
$316$	0	0
$317$	−149789.	−1.49060	−0.745300	−	0.666729i	$-0.767694\pi$
$317$	−149789.	−1.49060	−0.745300	+	0.666729i	$0.767694\pi$
$318$	0	0
$319$	− 49760.3i	− 0.488992i
$320$	0	0
$321$	−51185.4	−0.496748
$322$	0	0
$323$	− 214950.i	− 2.06031i
$324$	0	0
$325$	17325.1	0.164024
$326$	0	0
$327$	− 118063.i	− 1.10412i
$328$	0	0
$329$	−48433.6	−0.447461
$330$	0	0
$331$	106092.i	0.968339i	0.874974	+	0.484169i	$0.160878\pi$
$331$	106092.i	0.968339i	−0.874974	+	0.484169i	$0.839122\pi$
$332$	0	0
$333$	−27205.8	−0.245342
$334$	0	0
$335$	− 32414.2i	− 0.288832i
$336$	0	0
$337$	−87333.9	−0.768994	−0.384497	−	0.923126i	$-0.625625\pi$
$337$	−87333.9	−0.768994	−0.384497	+	0.923126i	$0.625625\pi$
$338$	0	0
$339$	− 30919.5i	− 0.269050i
$340$	0	0
$341$	96871.6	0.833082
$342$	0	0
$343$	− 110666.i	− 0.940644i
$344$	0	0
$345$	−2268.23	−0.0190567
$346$	0	0
$347$	− 79751.5i	− 0.662338i	−0.943571	−	0.331169i	$-0.892557\pi$
$347$	− 79751.5i	− 0.662338i	0.943571	−	0.331169i	$-0.107443\pi$
$348$	0	0
$349$	−91066.9	−0.747670	−0.373835	−	0.927495i	$-0.621957\pi$
$349$	−91066.9	−0.747670	−0.373835	+	0.927495i	$0.621957\pi$
$350$	0	0
$351$	123635.i	1.00352i
$352$	0	0
$353$	30128.6	0.241785	0.120893	−	0.992666i	$-0.461424\pi$
$353$	30128.6	0.241785	0.120893	+	0.992666i	$0.461424\pi$
$354$	0	0
$355$	202971.i	1.61056i
$356$	0	0
$357$	−135176.	−1.06063
$358$	0	0
$359$	30276.4i	0.234917i	0.993078	+	0.117459i	$0.0374747\pi$
$359$	30276.4i	0.234917i	−0.993078	+	0.117459i	$0.962525\pi$
$360$	0	0
$361$	−135598.	−1.04049
$362$	0	0
$363$	− 75912.2i	− 0.576100i
$364$	0	0
$365$	−101860.	−0.764571
$366$	0	0
$367$	176253.i	1.30859i	0.756240	+	0.654295i	$0.227034\pi$
$367$	176253.i	1.30859i	−0.756240	+	0.654295i	$0.772966\pi$
$368$	0	0
$369$	117449.	0.862573
$370$	0	0
$371$	− 14435.6i	− 0.104879i
$372$	0	0
$373$	73088.4	0.525328	0.262664	−	0.964887i	$-0.415399\pi$
$373$	73088.4	0.525328	0.262664	+	0.964887i	$0.415399\pi$
$374$	0	0
$375$	104917.i	0.746076i
$376$	0	0
$377$	157163.	1.10577
$378$	0	0
$379$	− 50562.7i	− 0.352008i	−0.984389	−	0.176004i	$-0.943683\pi$
$379$	− 50562.7i	− 0.352008i	0.984389	−	0.176004i	$-0.0563171\pi$
$380$	0	0
$381$	139202.	0.958952
$382$	0	0
$383$	− 158124.i	− 1.07795i	−0.842321	−	0.538977i	$-0.818811\pi$
$383$	− 158124.i	− 1.07795i	0.842321	−	0.538977i	$-0.181189\pi$
$384$	0	0
$385$	59523.4	0.401574
$386$	0	0
$387$	104790.i	0.699679i
$388$	0	0
$389$	213200.	1.40892	0.704462	−	0.709742i	$-0.251189\pi$
$389$	213200.	1.40892	0.704462	+	0.709742i	$0.251189\pi$
$390$	0	0
$391$	− 6605.10i	− 0.0432042i
$392$	0	0
$393$	−48938.5	−0.316859
$394$	0	0
$395$	155440.i	0.996247i
$396$	0	0
$397$	77201.5	0.489829	0.244915	−	0.969545i	$-0.421240\pi$
$397$	77201.5	0.489829	0.244915	+	0.969545i	$0.421240\pi$
$398$	0	0
$399$	167229.i	1.05042i
$400$	0	0
$401$	−134093.	−0.833909	−0.416955	−	0.908927i	$-0.636903\pi$
$401$	−134093.	−0.833909	−0.416955	+	0.908927i	$0.636903\pi$
$402$	0	0
$403$	305959.i	1.88388i
$404$	0	0
$405$	−34070.6	−0.207716
$406$	0	0
$407$	− 33415.8i	− 0.201727i
$408$	0	0
$409$	−261043.	−1.56051	−0.780253	−	0.625465i	$-0.784909\pi$
$409$	−261043.	−1.56051	−0.780253	+	0.625465i	$0.784909\pi$
$410$	0	0
$411$	− 67850.1i	− 0.401668i
$412$	0	0
$413$	259475.	1.52123
$414$	0	0
$415$	108906.i	0.632348i
$416$	0	0
$417$	−9683.69	−0.0556889
$418$	0	0
$419$	50041.4i	0.285037i	0.989792	+	0.142518i	$0.0455200\pi$
$419$	50041.4i	0.285037i	−0.989792	+	0.142518i	$0.954480\pi$
$420$	0	0
$421$	43682.4	0.246458	0.123229	−	0.992378i	$-0.460675\pi$
$421$	43682.4	0.246458	0.123229	+	0.992378i	$0.460675\pi$
$422$	0	0
$423$	− 38896.9i	− 0.217387i
$424$	0	0
$425$	−44996.9	−0.249118
$426$	0	0
$427$	− 148524.i	− 0.814594i
$428$	0	0
$429$	−51339.5	−0.278957
$430$	0	0
$431$	52296.9i	0.281528i	0.990043	+	0.140764i	$0.0449558\pi$
$431$	52296.9i	0.281528i	−0.990043	+	0.140764i	$0.955044\pi$
$432$	0	0
$433$	−71215.7	−0.379839	−0.189920	−	0.981800i	$-0.560823\pi$
$433$	−71215.7	−0.379839	−0.189920	+	0.981800i	$0.560823\pi$
$434$	0	0
$435$	140173.i	0.740775i
$436$	0	0
$437$	−8171.26	−0.0427884
$438$	0	0
$439$	− 244384.i	− 1.26807i	−0.773304	−	0.634036i	$-0.781397\pi$
$439$	− 244384.i	− 1.26807i	0.773304	−	0.634036i	$-0.218603\pi$
$440$	0	0
$441$	−10457.0	−0.0537687
$442$	0	0
$443$	135522.i	0.690561i	0.938500	+	0.345280i	$0.112216\pi$
$443$	135522.i	0.690561i	−0.938500	+	0.345280i	$0.887784\pi$
$444$	0	0
$445$	38797.5	0.195922
$446$	0	0
$447$	58552.4i	0.293042i
$448$	0	0
$449$	−199431.	−0.989234	−0.494617	−	0.869111i	$-0.664692\pi$
$449$	−199431.	−0.989234	−0.494617	+	0.869111i	$0.664692\pi$
$450$	0	0
$451$	144258.i	0.709230i
$452$	0	0
$453$	−15406.7	−0.0750781
$454$	0	0
$455$	187998.i	0.908095i
$456$	0	0
$457$	272609.	1.30529	0.652646	−	0.757663i	$-0.273659\pi$
$457$	272609.	1.30529	0.652646	+	0.757663i	$0.273659\pi$
$458$	0	0
$459$	− 321106.i	− 1.52413i
$460$	0	0
$461$	107580.	0.506207	0.253104	−	0.967439i	$-0.418549\pi$
$461$	107580.	0.506207	0.253104	+	0.967439i	$0.418549\pi$
$462$	0	0
$463$	237488.i	1.10785i	0.832567	+	0.553924i	$0.186870\pi$
$463$	237488.i	1.10785i	−0.832567	+	0.553924i	$0.813130\pi$
$464$	0	0
$465$	−272884.	−1.26204
$466$	0	0
$467$	− 91763.7i	− 0.420763i	−0.977619	−	0.210381i	$-0.932529\pi$
$467$	− 91763.7i	− 0.420763i	0.977619	−	0.210381i	$-0.0674705\pi$
$468$	0	0
$469$	−73434.3	−0.333851
$470$	0	0
$471$	152500.i	0.687430i
$472$	0	0
$473$	−128710.	−0.575294
$474$	0	0
$475$	55666.4i	0.246721i
$476$	0	0
$477$	11593.2	0.0509525
$478$	0	0
$479$	174561.i	0.760809i	0.924820	+	0.380405i	$0.124215\pi$
$479$	174561.i	0.760809i	−0.924820	+	0.380405i	$0.875785\pi$
$480$	0	0
$481$	105540.	0.456172
$482$	0	0
$483$	5138.67i	0.0220271i
$484$	0	0
$485$	43117.5	0.183303
$486$	0	0
$487$	− 310539.i	− 1.30936i	−0.755907	−	0.654679i	$-0.772804\pi$
$487$	− 310539.i	− 1.30936i	0.755907	−	0.654679i	$-0.227196\pi$
$488$	0	0
$489$	242496.	1.01411
$490$	0	0
$491$	112453.i	0.466452i	0.972423	+	0.233226i	$0.0749282\pi$
$491$	112453.i	0.466452i	−0.972423	+	0.233226i	$0.925072\pi$
$492$	0	0
$493$	−408185.	−1.67944
$494$	0	0
$495$	47803.0i	0.195094i
$496$	0	0
$497$	459830.	1.86159
$498$	0	0
$499$	− 158244.i	− 0.635514i	−0.948172	−	0.317757i	$-0.897070\pi$
$499$	− 158244.i	− 0.635514i	0.948172	−	0.317757i	$-0.102930\pi$
$500$	0	0
$501$	−96379.7	−0.383982
$502$	0	0
$503$	− 23328.5i	− 0.0922040i	−0.998937	−	0.0461020i	$-0.985320\pi$
$503$	− 23328.5i	− 0.0922040i	0.998937	−	0.0461020i	$-0.0146799\pi$
$504$	0	0
$505$	62186.3	0.243844
$506$	0	0
$507$	17645.0i	0.0686447i
$508$	0	0
$509$	−87962.0	−0.339515	−0.169758	−	0.985486i	$-0.554299\pi$
$509$	−87962.0	−0.339515	−0.169758	+	0.985486i	$0.554299\pi$
$510$	0	0
$511$	230764.i	0.883743i
$512$	0	0
$513$	−397245.	−1.50947
$514$	0	0
$515$	− 142495.i	− 0.537259i
$516$	0	0
$517$	47775.6	0.178741
$518$	0	0
$519$	− 348750.i	− 1.29473i
$520$	0	0
$521$	−317864.	−1.17102	−0.585512	−	0.810664i	$-0.699106\pi$
$521$	−317864.	−1.17102	−0.585512	+	0.810664i	$0.699106\pi$
$522$	0	0
$523$	− 232286.i	− 0.849219i	−0.905377	−	0.424610i	$-0.860411\pi$
$523$	− 232286.i	− 0.849219i	0.905377	−	0.424610i	$-0.139589\pi$
$524$	0	0
$525$	35007.0	0.127009
$526$	0	0
$527$	− 794640.i	− 2.86121i
$528$	0	0
$529$	279590.	0.999103
$530$	0	0
$531$	208383.i	0.739050i
$532$	0	0
$533$	−455624.	−1.60381
$534$	0	0
$535$	− 184888.i	− 0.645952i
$536$	0	0
$537$	−26146.7	−0.0906709
$538$	0	0
$539$	− 12843.9i	− 0.0442100i
$540$	0	0
$541$	534465.	1.82610	0.913051	−	0.407846i	$-0.133720\pi$
$541$	534465.	1.82610	0.913051	+	0.407846i	$0.133720\pi$
$542$	0	0
$543$	− 315059.i	− 1.06854i
$544$	0	0
$545$	426456.	1.43576
$546$	0	0
$547$	− 538847.i	− 1.80091i	−0.434954	−	0.900453i	$-0.643235\pi$
$547$	− 538847.i	− 1.80091i	0.434954	−	0.900453i	$-0.356765\pi$
$548$	0	0
$549$	119279.	0.395749
$550$	0	0
$551$	504972.i	1.66328i
$552$	0	0
$553$	352148.	1.15153
$554$	0	0
$555$	94131.3i	0.305596i
$556$	0	0
$557$	−505360.	−1.62889	−0.814443	−	0.580243i	$-0.802957\pi$
$557$	−505360.	−1.62889	−0.814443	+	0.580243i	$0.802957\pi$
$558$	0	0
$559$	− 406517.i	− 1.30093i
$560$	0	0
$561$	133340.	0.423676
$562$	0	0
$563$	− 26214.2i	− 0.0827026i	−0.999145	−	0.0413513i	$-0.986834\pi$
$563$	− 26214.2i	− 0.0827026i	0.999145	−	0.0413513i	$-0.0131663\pi$
$564$	0	0
$565$	111685.	0.349862
$566$	0	0
$567$	77186.9i	0.240092i
$568$	0	0
$569$	207163.	0.639864	0.319932	−	0.947440i	$-0.396340\pi$
$569$	207163.	0.639864	0.319932	+	0.947440i	$0.396340\pi$
$570$	0	0
$571$	− 266544.i	− 0.817515i	−0.912643	−	0.408758i	$-0.865962\pi$
$571$	− 266544.i	− 0.817515i	0.912643	−	0.408758i	$-0.134038\pi$
$572$	0	0
$573$	185291.	0.564345
$574$	0	0
$575$	1710.54i	0.00517366i
$576$	0	0
$577$	167147.	0.502049	0.251025	−	0.967981i	$-0.419233\pi$
$577$	167147.	0.502049	0.251025	+	0.967981i	$0.419233\pi$
$578$	0	0
$579$	219954.i	0.656108i
$580$	0	0
$581$	246727.	0.730910
$582$	0	0
$583$	14239.5i	0.0418945i
$584$	0	0
$585$	−150981.	−0.441174
$586$	0	0
$587$	351724.i	1.02076i	0.859948	+	0.510382i	$0.170496\pi$
$587$	351724.i	1.02076i	−0.859948	+	0.510382i	$0.829504\pi$
$588$	0	0
$589$	−983062.	−2.83368
$590$	0	0
$591$	346744.i	0.992736i
$592$	0	0
$593$	−274612.	−0.780925	−0.390463	−	0.920619i	$-0.627685\pi$
$593$	−274612.	−0.780925	−0.390463	+	0.920619i	$0.627685\pi$
$594$	0	0
$595$	− 488272.i	− 1.37920i
$596$	0	0
$597$	118041.	0.331196
$598$	0	0
$599$	505626.i	1.40921i	0.709600	+	0.704605i	$0.248876\pi$
$599$	505626.i	1.40921i	−0.709600	+	0.704605i	$0.751124\pi$
$600$	0	0
$601$	−239547.	−0.663195	−0.331598	−	0.943421i	$-0.607588\pi$
$601$	−239547.	−0.663195	−0.331598	+	0.943421i	$0.607588\pi$
$602$	0	0
$603$	− 58974.9i	− 0.162193i
$604$	0	0
$605$	274203.	0.749139
$606$	0	0
$607$	− 434687.i	− 1.17977i	−0.807486	−	0.589887i	$-0.799172\pi$
$607$	− 434687.i	− 1.17977i	0.807486	−	0.589887i	$-0.200828\pi$
$608$	0	0
$609$	317562.	0.856238
$610$	0	0
$611$	150894.i	0.404194i
$612$	0	0
$613$	−381608.	−1.01554	−0.507770	−	0.861493i	$-0.669530\pi$
$613$	−381608.	−1.01554	−0.507770	+	0.861493i	$0.669530\pi$
$614$	0	0
$615$	− 406370.i	− 1.07441i
$616$	0	0
$617$	104794.	0.275276	0.137638	−	0.990483i	$-0.456049\pi$
$617$	104794.	0.275276	0.137638	+	0.990483i	$0.456049\pi$
$618$	0	0
$619$	66782.1i	0.174292i	0.996196	+	0.0871462i	$0.0277747\pi$
$619$	66782.1i	0.174292i	−0.996196	+	0.0871462i	$0.972225\pi$
$620$	0	0
$621$	−12206.7	−0.0316531
$622$	0	0
$623$	− 87895.7i	− 0.226460i
$624$	0	0
$625$	−311504.	−0.797450
$626$	0	0
$627$	− 164957.i	− 0.419599i
$628$	0	0
$629$	−274111.	−0.692827
$630$	0	0
$631$	92797.8i	0.233066i	0.993187	+	0.116533i	$0.0371781\pi$
$631$	92797.8i	0.233066i	−0.993187	+	0.116533i	$0.962822\pi$
$632$	0	0
$633$	353920.	0.883279
$634$	0	0
$635$	502815.i	1.24698i
$636$	0	0
$637$	40566.2	0.0999736
$638$	0	0
$639$	369288.i	0.904406i
$640$	0	0
$641$	−752228.	−1.83077	−0.915385	−	0.402581i	$-0.868113\pi$
$641$	−752228.	−1.83077	−0.915385	+	0.402581i	$0.868113\pi$
$642$	0	0
$643$	− 202173.i	− 0.488992i	−0.969650	−	0.244496i	$-0.921377\pi$
$643$	− 202173.i	− 0.488992i	0.969650	−	0.244496i	$-0.0786225\pi$
$644$	0	0
$645$	362572.	0.871515
$646$	0	0
$647$	615031.i	1.46923i	0.678487	+	0.734613i	$0.262636\pi$
$647$	615031.i	1.46923i	−0.678487	+	0.734613i	$0.737364\pi$
$648$	0	0
$649$	−255949.	−0.607666
$650$	0	0
$651$	618219.i	1.45875i
$652$	0	0
$653$	419103.	0.982866	0.491433	−	0.870915i	$-0.336473\pi$
$653$	419103.	0.982866	0.491433	+	0.870915i	$0.336473\pi$
$654$	0	0
$655$	− 176771.i	− 0.412031i
$656$	0	0
$657$	−185326.	−0.429344
$658$	0	0
$659$	− 737578.i	− 1.69839i	−0.528081	−	0.849194i	$-0.677088\pi$
$659$	− 737578.i	− 1.69839i	0.528081	−	0.849194i	$-0.322912\pi$
$660$	0	0
$661$	−10114.0	−0.0231485	−0.0115742	−	0.999933i	$-0.503684\pi$
$661$	−10114.0	−0.0231485	−0.0115742	+	0.999933i	$0.503684\pi$
$662$	0	0
$663$	421139.i	0.958073i
$664$	0	0
$665$	−604048.	−1.36593
$666$	0	0
$667$	15517.0i	0.0348784i
$668$	0	0
$669$	464771.	1.03845
$670$	0	0
$671$	146506.i	0.325395i
$672$	0	0
$673$	347749.	0.767779	0.383889	−	0.923379i	$-0.374584\pi$
$673$	347749.	0.767779	0.383889	+	0.923379i	$0.374584\pi$
$674$	0	0
$675$	83157.8i	0.182514i
$676$	0	0
$677$	412391.	0.899772	0.449886	−	0.893086i	$-0.351465\pi$
$677$	412391.	0.899772	0.449886	+	0.893086i	$0.351465\pi$
$678$	0	0
$679$	− 97682.7i	− 0.211874i
$680$	0	0
$681$	−582022.	−1.25501
$682$	0	0
$683$	234274.i	0.502207i	0.967960	+	0.251103i	$0.0807934\pi$
$683$	234274.i	0.502207i	−0.967960	+	0.251103i	$0.919207\pi$
$684$	0	0
$685$	245082.	0.522313
$686$	0	0
$687$	− 330137.i	− 0.699489i
$688$	0	0
$689$	−44973.9	−0.0947375
$690$	0	0
$691$	908475.i	1.90264i	0.308202	+	0.951321i	$0.400273\pi$
$691$	908475.i	1.90264i	−0.308202	+	0.951321i	$0.599727\pi$
$692$	0	0
$693$	108298.	0.225503
$694$	0	0
$695$	− 34978.6i	− 0.0724157i
$696$	0	0
$697$	1.18335e6	2.43584
$698$	0	0
$699$	− 202530.i	− 0.414511i
$700$	0	0
$701$	835919.	1.70109	0.850547	−	0.525899i	$-0.176271\pi$
$701$	835919.	1.70109	0.850547	+	0.525899i	$0.176271\pi$
$702$	0	0
$703$	339107.i	0.686161i
$704$	0	0
$705$	−134582.	−0.270776
$706$	0	0
$707$	− 140883.i	− 0.281851i
$708$	0	0
$709$	−496492.	−0.987689	−0.493844	−	0.869550i	$-0.664409\pi$
$709$	−496492.	−0.987689	−0.493844	+	0.869550i	$0.664409\pi$
$710$	0	0
$711$	282809.i	0.559441i
$712$	0	0
$713$	−30208.0	−0.0594213
$714$	0	0
$715$	− 185444.i	− 0.362745i
$716$	0	0
$717$	−353136.	−0.686916
$718$	0	0
$719$	345833.i	0.668973i	0.942401	+	0.334486i	$0.108563\pi$
$719$	345833.i	0.668973i	−0.942401	+	0.334486i	$0.891437\pi$
$720$	0	0
$721$	−322822.	−0.621001
$722$	0	0
$723$	602250.i	1.15213i
$724$	0	0
$725$	105709.	0.201111
$726$	0	0
$727$	− 350951.i	− 0.664015i	−0.943277	−	0.332008i	$-0.892274\pi$
$727$	− 350951.i	− 0.664015i	0.943277	−	0.332008i	$-0.107726\pi$
$728$	0	0
$729$	−454792.	−0.855771
$730$	0	0
$731$	1.05581e6i	1.97584i
$732$	0	0
$733$	−254781.	−0.474197	−0.237098	−	0.971486i	$-0.576196\pi$
$733$	−254781.	−0.474197	−0.237098	+	0.971486i	$0.576196\pi$
$734$	0	0
$735$	36180.9i	0.0669738i
$736$	0	0
$737$	72436.6	0.133359
$738$	0	0
$739$	455196.i	0.833507i	0.909019	+	0.416754i	$0.136832\pi$
$739$	455196.i	0.833507i	−0.909019	+	0.416754i	$0.863168\pi$
$740$	0	0
$741$	520998.	0.948854
$742$	0	0
$743$	46222.1i	0.0837282i	0.999123	+	0.0418641i	$0.0133296\pi$
$743$	46222.1i	0.0837282i	−0.999123	+	0.0418641i	$0.986670\pi$
$744$	0	0
$745$	−211498.	−0.381060
$746$	0	0
$747$	198145.i	0.355094i
$748$	0	0
$749$	−418863.	−0.746635
$750$	0	0
$751$	166368.i	0.294979i	0.989064	+	0.147489i	$0.0471192\pi$
$751$	166368.i	0.294979i	−0.989064	+	0.147489i	$0.952881\pi$
$752$	0	0
$753$	584983.	1.03170
$754$	0	0
$755$	− 55650.8i	− 0.0976287i
$756$	0	0
$757$	−736728.	−1.28563	−0.642814	−	0.766022i	$-0.722233\pi$
$757$	−736728.	−1.28563	−0.642814	+	0.766022i	$0.722233\pi$
$758$	0	0
$759$	− 5068.86i	− 0.00879886i
$760$	0	0
$761$	225460.	0.389315	0.194657	−	0.980871i	$-0.437641\pi$
$761$	225460.	0.389315	0.194657	+	0.980871i	$0.437641\pi$
$762$	0	0
$763$	− 966137.i	− 1.65955i
$764$	0	0
$765$	392129.	0.670049
$766$	0	0
$767$	− 808389.i	− 1.37414i
$768$	0	0
$769$	−355916.	−0.601858	−0.300929	−	0.953646i	$-0.597297\pi$
$769$	−355916.	−0.601858	−0.300929	+	0.953646i	$0.597297\pi$
$770$	0	0
$771$	89053.2i	0.149810i
$772$	0	0
$773$	859376.	1.43822	0.719108	−	0.694898i	$-0.244551\pi$
$773$	859376.	1.43822	0.719108	+	0.694898i	$0.244551\pi$
$774$	0	0
$775$	205790.i	0.342627i
$776$	0	0
$777$	213254.	0.353229
$778$	0	0
$779$	− 1.46394e6i	− 2.41240i
$780$	0	0
$781$	−453583.	−0.743626
$782$	0	0
$783$	754358.i	1.23042i
$784$	0	0
$785$	−550848.	−0.893908
$786$	0	0
$787$	− 253820.i	− 0.409804i	−0.978782	−	0.204902i	$-0.934312\pi$
$787$	− 253820.i	− 0.409804i	0.978782	−	0.204902i	$-0.0656876\pi$
$788$	0	0
$789$	203835.	0.327435
$790$	0	0
$791$	− 253022.i	− 0.404394i
$792$	0	0
$793$	−462725.	−0.735828
$794$	0	0
$795$	− 40112.1i	− 0.0634660i
$796$	0	0
$797$	−626038.	−0.985563	−0.492781	−	0.870153i	$-0.664020\pi$
$797$	−626038.	−0.985563	−0.492781	+	0.870153i	$0.664020\pi$
$798$	0	0
$799$	− 391905.i	− 0.613885i
$800$	0	0
$801$	70588.8	0.110020
$802$	0	0
$803$	− 227629.i	− 0.353017i
$804$	0	0
$805$	−18561.5	−0.0286431
$806$	0	0
$807$	− 201273.i	− 0.309057i
$808$	0	0
$809$	528309.	0.807218	0.403609	−	0.914932i	$-0.367756\pi$
$809$	528309.	0.807218	0.403609	+	0.914932i	$0.367756\pi$
$810$	0	0
$811$	− 619684.i	− 0.942168i	−0.882088	−	0.471084i	$-0.843863\pi$
$811$	− 619684.i	− 0.942168i	0.882088	−	0.471084i	$-0.156137\pi$
$812$	0	0
$813$	−170347.	−0.257724
$814$	0	0
$815$	875922.i	1.31871i
$816$	0	0
$817$	1.30616e6	1.95683
$818$	0	0
$819$	342047.i	0.509939i
$820$	0	0
$821$	−94972.1	−0.140900	−0.0704498	−	0.997515i	$-0.522443\pi$
$821$	−94972.1	−0.140900	−0.0704498	+	0.997515i	$0.522443\pi$
$822$	0	0
$823$	− 936215.i	− 1.38222i	−0.722751	−	0.691108i	$-0.757123\pi$
$823$	− 936215.i	− 1.38222i	0.722751	−	0.691108i	$-0.242877\pi$
$824$	0	0
$825$	−34531.4	−0.0507348
$826$	0	0
$827$	29165.9i	0.0426446i	0.999773	+	0.0213223i	$0.00678761\pi$
$827$	29165.9i	0.0426446i	−0.999773	+	0.0213223i	$0.993212\pi$
$828$	0	0
$829$	61369.7	0.0892987	0.0446494	−	0.999003i	$-0.485783\pi$
$829$	61369.7	0.0892987	0.0446494	+	0.999003i	$0.485783\pi$
$830$	0	0
$831$	145580.i	0.210814i
$832$	0	0
$833$	−105359.	−0.151839
$834$	0	0
$835$	− 348135.i	− 0.499315i
$836$	0	0
$837$	−1.46856e6	−2.09624
$838$	0	0
$839$	− 1.34682e6i	− 1.91331i	−0.291217	−	0.956657i	$-0.594060\pi$
$839$	− 1.34682e6i	− 1.91331i	0.291217	−	0.956657i	$-0.405940\pi$
$840$	0	0
$841$	251647.	0.355795
$842$	0	0
$843$	− 299282.i	− 0.421139i
$844$	0	0
$845$	−63735.9	−0.0892628
$846$	0	0
$847$	− 621208.i	− 0.865905i
$848$	0	0
$849$	−23492.5	−0.0325922
$850$	0	0
$851$	10420.2i	0.0143886i
$852$	0	0
$853$	−165126.	−0.226943	−0.113472	−	0.993541i	$-0.536197\pi$
$853$	−165126.	−0.226943	−0.113472	+	0.993541i	$0.536197\pi$
$854$	0	0
$855$	− 485109.i	− 0.663602i
$856$	0	0
$857$	−1.03219e6	−1.40539	−0.702697	−	0.711489i	$-0.748021\pi$
$857$	−1.03219e6	−1.40539	−0.702697	+	0.711489i	$0.748021\pi$
$858$	0	0
$859$	− 298272.i	− 0.404228i	−0.979362	−	0.202114i	$-0.935219\pi$
$859$	− 298272.i	− 0.404228i	0.979362	−	0.202114i	$-0.0647812\pi$
$860$	0	0
$861$	−920632.	−1.24188
$862$	0	0
$863$	539155.i	0.723923i	0.932193	+	0.361961i	$0.117893\pi$
$863$	539155.i	0.723923i	−0.932193	+	0.361961i	$0.882107\pi$
$864$	0	0
$865$	1.25972e6	1.68362
$866$	0	0
$867$	− 568013.i	− 0.755649i
$868$	0	0
$869$	−347364.	−0.459987
$870$	0	0
$871$	228783.i	0.301570i
$872$	0	0
$873$	78448.6	0.102934
$874$	0	0
$875$	858561.i	1.12139i
$876$	0	0
$877$	859143.	1.11703	0.558517	−	0.829493i	$-0.311371\pi$
$877$	859143.	1.11703	0.558517	+	0.829493i	$0.311371\pi$
$878$	0	0
$879$	585541.i	0.757844i
$880$	0	0
$881$	35429.1	0.0456466	0.0228233	−	0.999740i	$-0.492734\pi$
$881$	35429.1	0.0456466	0.0228233	+	0.999740i	$0.492734\pi$
$882$	0	0
$883$	246369.i	0.315984i	0.987440	+	0.157992i	$0.0505021\pi$
$883$	246369.i	0.315984i	−0.987440	+	0.157992i	$0.949498\pi$
$884$	0	0
$885$	721001.	0.920554
$886$	0	0
$887$	− 51297.1i	− 0.0651997i	−0.999468	−	0.0325999i	$-0.989621\pi$
$887$	− 51297.1i	− 0.0651997i	0.999468	−	0.0325999i	$-0.0103787\pi$
$888$	0	0
$889$	1.13913e6	1.44135
$890$	0	0
$891$	− 76138.2i	− 0.0959064i
$892$	0	0
$893$	−484831.	−0.607978
$894$	0	0
$895$	− 94444.8i	− 0.117905i
$896$	0	0
$897$	16009.5	0.0198972
$898$	0	0
$899$	1.86681e6i	2.30983i
$900$	0	0
$901$	116807.	0.143886
$902$	0	0
$903$	− 821407.i	− 1.00736i
$904$	0	0
$905$	1.13803e6	1.38949
$906$	0	0
$907$	− 569783.i	− 0.692620i	−0.938120	−	0.346310i	$-0.887435\pi$
$907$	− 569783.i	− 0.692620i	0.938120	−	0.346310i	$-0.112565\pi$
$908$	0	0
$909$	113143.	0.136930
$910$	0	0
$911$	− 176538.i	− 0.212717i	−0.994328	−	0.106358i	$-0.966081\pi$
$911$	− 176538.i	− 0.212717i	0.994328	−	0.106358i	$-0.0339191\pi$
$912$	0	0
$913$	−243375.	−0.291967
$914$	0	0
$915$	− 412704.i	− 0.492942i
$916$	0	0
$917$	−400476.	−0.476253
$918$	0	0
$919$	− 423503.i	− 0.501448i	−0.968059	−	0.250724i	$-0.919331\pi$
$919$	− 423503.i	− 0.501448i	0.968059	−	0.250724i	$-0.0806686\pi$
$920$	0	0
$921$	−677008.	−0.798132
$922$	0	0
$923$	− 1.43259e6i	− 1.68159i
$924$	0	0
$925$	70987.3	0.0829654
$926$	0	0
$927$	− 259257.i	− 0.301697i
$928$	0	0
$929$	277042.	0.321007	0.160503	−	0.987035i	$-0.448688\pi$
$929$	277042.	0.321007	0.160503	+	0.987035i	$0.448688\pi$
$930$	0	0
$931$	130341.i	0.150377i
$932$	0	0
$933$	85241.4	0.0979235
$934$	0	0
$935$	481638.i	0.550932i
$936$	0	0
$937$	669529.	0.762588	0.381294	−	0.924454i	$-0.375479\pi$
$937$	669529.	0.762588	0.381294	+	0.924454i	$0.375479\pi$
$938$	0	0
$939$	− 471265.i	− 0.534483i
$940$	0	0
$941$	1.53628e6	1.73497	0.867485	−	0.497463i	$-0.165735\pi$
$941$	1.53628e6	1.73497	0.867485	+	0.497463i	$0.165735\pi$
$942$	0	0
$943$	− 44984.7i	− 0.0505873i
$944$	0	0
$945$	−902365.	−1.01046
$946$	0	0
$947$	− 287770.i	− 0.320882i	−0.987045	−	0.160441i	$-0.948708\pi$
$947$	− 287770.i	− 0.320882i	0.987045	−	0.160441i	$-0.0512916\pi$
$948$	0	0
$949$	718941.	0.798291
$950$	0	0
$951$	942942.i	1.04262i
$952$	0	0
$953$	886634.	0.976244	0.488122	−	0.872775i	$-0.337682\pi$
$953$	886634.	0.976244	0.488122	+	0.872775i	$0.337682\pi$
$954$	0	0
$955$	669292.i	0.733853i
$956$	0	0
$957$	−313248.	−0.342030
$958$	0	0
$959$	− 555234.i	− 0.603725i
$960$	0	0
$961$	−2.71071e6	−2.93519
$962$	0	0
$963$	− 336387.i	− 0.362733i
$964$	0	0
$965$	−794500.	−0.853177
$966$	0	0
$967$	686857.i	0.734536i	0.930115	+	0.367268i	$0.119707\pi$
$967$	686857.i	0.734536i	−0.930115	+	0.367268i	$0.880293\pi$
$968$	0	0
$969$	−1.35314e6	−1.44111
$970$	0	0
$971$	892898.i	0.947029i	0.880786	+	0.473515i	$0.157015\pi$
$971$	892898.i	0.947029i	−0.880786	+	0.473515i	$0.842985\pi$
$972$	0	0
$973$	−79244.0	−0.0837029
$974$	0	0
$975$	− 109064.i	− 0.114728i
$976$	0	0
$977$	11191.9	0.0117251	0.00586255	−	0.999983i	$-0.498134\pi$
$977$	11191.9	0.0117251	0.00586255	+	0.999983i	$0.498134\pi$
$978$	0	0
$979$	86701.6i	0.0904610i
$980$	0	0
$981$	775902.	0.806248
$982$	0	0
$983$	136809.i	0.141582i	0.997491	+	0.0707908i	$0.0225523\pi$
$983$	136809.i	0.141582i	−0.997491	+	0.0707908i	$0.977448\pi$
$984$	0	0
$985$	−1.25248e6	−1.29092
$986$	0	0
$987$	304896.i	0.312981i
$988$	0	0
$989$	40136.3	0.0410341
$990$	0	0
$991$	1.02291e6i	1.04157i	0.853688	+	0.520785i	$0.174361\pi$
$991$	1.02291e6i	1.04157i	−0.853688	+	0.520785i	$0.825639\pi$
$992$	0	0
$993$	667865.	0.677314
$994$	0	0
$995$	426379.i	0.430675i
$996$	0	0
$997$	396177.	0.398565	0.199283	−	0.979942i	$-0.436139\pi$
$997$	396177.	0.398565	0.199283	+	0.979942i	$0.436139\pi$
$998$	0	0
$999$	506578.i	0.507593i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.c.a.127.4	✓	8
3.2	odd	2		1152.5.g.b.127.1		8
4.3	odd	2	inner	128.5.c.a.127.5	yes	8
8.3	odd	2		128.5.c.b.127.4	yes	8
8.5	even	2		128.5.c.b.127.5	yes	8
12.11	even	2		1152.5.g.b.127.2		8
16.3	odd	4		256.5.d.g.127.3		8
16.5	even	4		256.5.d.g.127.4		8
16.11	odd	4		256.5.d.h.127.6		8
16.13	even	4		256.5.d.h.127.5		8
24.5	odd	2		1152.5.g.a.127.7		8
24.11	even	2		1152.5.g.a.127.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.c.a.127.4	✓	8	1.1	even	1	trivial
128.5.c.a.127.5	yes	8	4.3	odd	2	inner
128.5.c.b.127.4	yes	8	8.3	odd	2
128.5.c.b.127.5	yes	8	8.5	even	2
256.5.d.g.127.3		8	16.3	odd	4
256.5.d.g.127.4		8	16.5	even	4
256.5.d.h.127.5		8	16.13	even	4
256.5.d.h.127.6		8	16.11	odd	4
1152.5.g.a.127.7		8	24.5	odd	2
1152.5.g.a.127.8		8	24.11	even	2
1152.5.g.b.127.1		8	3.2	odd	2
1152.5.g.b.127.2		8	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.29514i q^{3} +22.7388 q^{5} -51.5147i q^{7} +41.3713 q^{9} +O(q^{10})\)	\(q-6.29514i q^{3} +22.7388 q^{5} -51.5147i q^{7} +41.3713 q^{9} +50.8148i q^{11} -160.493 q^{13} -143.144i q^{15} +416.835 q^{17} -515.673i q^{19} -324.292 q^{21} -15.8458i q^{23} -107.949 q^{25} -770.344i q^{27} -979.249 q^{29} -1906.37i q^{31} +319.886 q^{33} -1171.38i q^{35} -657.600 q^{37} +1010.33i q^{39} +2838.90 q^{41} +2532.93i q^{43} +940.731 q^{45} -940.191i q^{47} -252.760 q^{49} -2624.03i q^{51} +280.223 q^{53} +1155.46i q^{55} -3246.23 q^{57} +5036.91i q^{59} +2883.14 q^{61} -2131.23i q^{63} -3649.41 q^{65} -1425.50i q^{67} -99.7517 q^{69} +8926.20i q^{71} -4479.58 q^{73} +679.554i q^{75} +2617.71 q^{77} +6835.88i q^{79} -1498.35 q^{81} +4789.45i q^{83} +9478.31 q^{85} +6164.50i q^{87} +1706.23 q^{89} +8267.75i q^{91} -12000.8 q^{93} -11725.8i q^{95} +1896.21 q^{97} +2102.27i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 48 q^{5} - 216 q^{9}+O(q^{10}) \) 8 * q - 48 * q^5 - 216 * q^9	\( 8 q - 48 q^{5} - 216 q^{9} - 240 q^{13} + 240 q^{17} + 1216 q^{21} + 664 q^{25} - 432 q^{29} + 992 q^{33} - 2800 q^{37} - 2928 q^{41} + 4880 q^{45} - 5752 q^{49} - 1776 q^{53} + 8608 q^{57} - 12656 q^{61} + 672 q^{65} + 4416 q^{69} + 560 q^{73} + 31296 q^{77} + 14696 q^{81} + 14432 q^{85} - 22992 q^{89} - 56320 q^{93} - 3728 q^{97}+O(q^{100}) \) 8 * q - 48 * q^5 - 216 * q^9 - 240 * q^13 + 240 * q^17 + 1216 * q^21 + 664 * q^25 - 432 * q^29 + 992 * q^33 - 2800 * q^37 - 2928 * q^41 + 4880 * q^45 - 5752 * q^49 - 1776 * q^53 + 8608 * q^57 - 12656 * q^61 + 672 * q^65 + 4416 * q^69 + 560 * q^73 + 31296 * q^77 + 14696 * q^81 + 14432 * q^85 - 22992 * q^89 - 56320 * q^93 - 3728 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more