LMFDB - Embedded newform 128.4.b.e.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,4,Mod(65,128)]

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, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("128.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	128.b (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:	$7.55224448073$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 9 $ x^4 + 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$-0.707107 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.4.b.e.65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.32456i q^{3} +17.8885i q^{5} -22.6274 q^{7} -13.0000 q^{9} +O(q^{10})$	$q+6.32456i q^{3} +17.8885i q^{5} -22.6274 q^{7} -13.0000 q^{9} -44.2719i q^{11} -17.8885i q^{13} -113.137 q^{15} +70.0000 q^{17} +82.2192i q^{19} -143.108i q^{21} -158.392 q^{23} -195.000 q^{25} +88.5438i q^{27} +125.220i q^{29} +280.000 q^{33} -404.772i q^{35} +375.659i q^{37} +113.137 q^{39} -182.000 q^{41} +132.816i q^{43} -232.551i q^{45} +316.784 q^{47} +169.000 q^{49} +442.719i q^{51} -125.220i q^{53} +791.960 q^{55} -520.000 q^{57} +82.2192i q^{59} -232.551i q^{61} +294.156 q^{63} +320.000 q^{65} -221.359i q^{67} -1001.76i q^{69} -113.137 q^{71} +910.000 q^{73} -1233.29i q^{75} +1001.76i q^{77} +678.823 q^{79} -911.000 q^{81} +714.675i q^{83} +1252.20i q^{85} -791.960 q^{87} -546.000 q^{89} +404.772i q^{91} -1470.78 q^{95} -490.000 q^{97} +575.535i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 52 q^{9}+O(q^{10}) $ 4 * q - 52 * q^9	$ 4 q - 52 q^{9} + 280 q^{17} - 780 q^{25} + 1120 q^{33} - 728 q^{41} + 676 q^{49} - 2080 q^{57} + 1280 q^{65} + 3640 q^{73} - 3644 q^{81} - 2184 q^{89} - 1960 q^{97}+O(q^{100}) $ 4 * q - 52 * q^9 + 280 * q^17 - 780 * q^25 + 1120 * q^33 - 728 * q^41 + 676 * q^49 - 2080 * q^57 + 1280 * q^65 + 3640 * q^73 - 3644 * q^81 - 2184 * q^89 - 1960 * 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
$3$	6.32456i	1.21716i	0.793492	+	0.608581i	$0.208261\pi$
$3$	6.32456i	1.21716i	−0.793492	+	0.608581i	$0.791739\pi$
$4$	0	0
$5$	17.8885i	1.60000i	0.600000	+	0.800000i	$0.295167\pi$
$5$	17.8885i	1.60000i	−0.600000	+	0.800000i	$0.704833\pi$
$6$	0	0
$7$	−22.6274	−1.22177	−0.610883	−	0.791721i	$-0.709185\pi$
$7$	−22.6274	−1.22177	−0.610883	+	0.791721i	$0.709185\pi$
$8$	0	0
$9$	−13.0000	−0.481481
$10$	0	0
$11$	− 44.2719i	− 1.21350i	−0.794894	−	0.606749i	$-0.792473\pi$
$11$	− 44.2719i	− 1.21350i	0.794894	−	0.606749i	$-0.207527\pi$
$12$	0	0
$13$	− 17.8885i	− 0.381645i	−0.981625	−	0.190823i	$-0.938884\pi$
$13$	− 17.8885i	− 0.381645i	0.981625	−	0.190823i	$-0.0611155\pi$
$14$	0	0
$15$	−113.137	−1.94746
$16$	0	0
$17$	70.0000	0.998676	0.499338	−	0.866407i	$-0.333577\pi$
$17$	70.0000	0.998676	0.499338	+	0.866407i	$0.333577\pi$
$18$	0	0
$19$	82.2192i	0.992757i	0.868106	+	0.496378i	$0.165337\pi$
$19$	82.2192i	0.992757i	−0.868106	+	0.496378i	$0.834663\pi$
$20$	0	0
$21$	− 143.108i	− 1.48709i
$22$	0	0
$23$	−158.392	−1.43596	−0.717978	−	0.696066i	$-0.754932\pi$
$23$	−158.392	−1.43596	−0.717978	+	0.696066i	$0.754932\pi$
$24$	0	0
$25$	−195.000	−1.56000
$26$	0	0
$27$	88.5438i	0.631121i
$28$	0	0
$29$	125.220i	0.801818i	0.916118	+	0.400909i	$0.131306\pi$
$29$	125.220i	0.801818i	−0.916118	+	0.400909i	$0.868694\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	280.000	1.47702
$34$	0	0
$35$	− 404.772i	− 1.95483i
$36$	0	0
$37$	375.659i	1.66914i	0.550905	+	0.834568i	$0.314283\pi$
$37$	375.659i	1.66914i	−0.550905	+	0.834568i	$0.685717\pi$
$38$	0	0
$39$	113.137	0.464524
$40$	0	0
$41$	−182.000	−0.693259	−0.346630	−	0.938002i	$-0.612674\pi$
$41$	−182.000	−0.693259	−0.346630	+	0.938002i	$0.612674\pi$
$42$	0	0
$43$	132.816i	0.471028i	0.971871	+	0.235514i	$0.0756773\pi$
$43$	132.816i	0.471028i	−0.971871	+	0.235514i	$0.924323\pi$
$44$	0	0
$45$	− 232.551i	− 0.770370i
$46$	0	0
$47$	316.784	0.983142	0.491571	−	0.870838i	$-0.336423\pi$
$47$	316.784	0.983142	0.491571	+	0.870838i	$0.336423\pi$
$48$	0	0
$49$	169.000	0.492711
$50$	0	0
$51$	442.719i	1.21555i
$52$	0	0
$53$	− 125.220i	− 0.324533i	−0.986747	−	0.162267i	$-0.948120\pi$
$53$	− 125.220i	− 0.324533i	0.986747	−	0.162267i	$-0.0518805\pi$
$54$	0	0
$55$	791.960	1.94160
$56$	0	0
$57$	−520.000	−1.20835
$58$	0	0
$59$	82.2192i	0.181424i	0.995877	+	0.0907121i	$0.0289143\pi$
$59$	82.2192i	0.181424i	−0.995877	+	0.0907121i	$0.971086\pi$
$60$	0	0
$61$	− 232.551i	− 0.488117i	−0.969761	−	0.244058i	$-0.921521\pi$
$61$	− 232.551i	− 0.488117i	0.969761	−	0.244058i	$-0.0784788\pi$
$62$	0	0
$63$	294.156	0.588258
$64$	0	0
$65$	320.000	0.610633
$66$	0	0
$67$	− 221.359i	− 0.403632i	−0.979423	−	0.201816i	$-0.935316\pi$
$67$	− 221.359i	− 0.403632i	0.979423	−	0.201816i	$-0.0646843\pi$
$68$	0	0
$69$	− 1001.76i	− 1.74779i
$70$	0	0
$71$	−113.137	−0.189111	−0.0945556	−	0.995520i	$-0.530143\pi$
$71$	−113.137	−0.189111	−0.0945556	+	0.995520i	$0.530143\pi$
$72$	0	0
$73$	910.000	1.45901	0.729503	−	0.683978i	$-0.239751\pi$
$73$	910.000	1.45901	0.729503	+	0.683978i	$0.239751\pi$
$74$	0	0
$75$	− 1233.29i	− 1.89877i
$76$	0	0
$77$	1001.76i	1.48261i
$78$	0	0
$79$	678.823	0.966753	0.483377	−	0.875413i	$-0.339410\pi$
$79$	678.823	0.966753	0.483377	+	0.875413i	$0.339410\pi$
$80$	0	0
$81$	−911.000	−1.24966
$82$	0	0
$83$	714.675i	0.945129i	0.881296	+	0.472565i	$0.156672\pi$
$83$	714.675i	0.945129i	−0.881296	+	0.472565i	$0.843328\pi$
$84$	0	0
$85$	1252.20i	1.59788i
$86$	0	0
$87$	−791.960	−0.975942
$88$	0	0
$89$	−546.000	−0.650291	−0.325145	−	0.945664i	$-0.605413\pi$
$89$	−546.000	−0.650291	−0.325145	+	0.945664i	$0.605413\pi$
$90$	0	0
$91$	404.772i	0.466281i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1470.78	−1.58841
$96$	0	0
$97$	−490.000	−0.512907	−0.256453	−	0.966557i	$-0.582554\pi$
$97$	−490.000	−0.512907	−0.256453	+	0.966557i	$0.582554\pi$
$98$	0	0
$99$	575.535i	0.584277i
$100$	0	0
$101$	232.551i	0.229106i	0.993417	+	0.114553i	$0.0365436\pi$
$101$	232.551i	0.229106i	−0.993417	+	0.114553i	$0.963456\pi$
$102$	0	0
$103$	158.392	0.151523	0.0757613	−	0.997126i	$-0.475861\pi$
$103$	158.392	0.151523	0.0757613	+	0.997126i	$0.475861\pi$
$104$	0	0
$105$	2560.00	2.37934
$106$	0	0
$107$	1726.60i	1.55997i	0.625797	+	0.779986i	$0.284774\pi$
$107$	1726.60i	1.55997i	−0.625797	+	0.779986i	$0.715226\pi$
$108$	0	0
$109$	− 1377.42i	− 1.21039i	−0.796077	−	0.605196i	$-0.793095\pi$
$109$	− 1377.42i	− 1.21039i	0.796077	−	0.605196i	$-0.206905\pi$
$110$	0	0
$111$	−2375.88	−2.03161
$112$	0	0
$113$	−910.000	−0.757572	−0.378786	−	0.925484i	$-0.623658\pi$
$113$	−910.000	−0.757572	−0.378786	+	0.925484i	$0.623658\pi$
$114$	0	0
$115$	− 2833.40i	− 2.29753i
$116$	0	0
$117$	232.551i	0.183755i
$118$	0	0
$119$	−1583.92	−1.22015
$120$	0	0
$121$	−629.000	−0.472577
$122$	0	0
$123$	− 1151.07i	− 0.843808i
$124$	0	0
$125$	− 1252.20i	− 0.896000i
$126$	0	0
$127$	1900.70	1.32803	0.664016	−	0.747718i	$-0.268851\pi$
$127$	1900.70	1.32803	0.664016	+	0.747718i	$0.268851\pi$
$128$	0	0
$129$	−840.000	−0.573317
$130$	0	0
$131$	− 170.763i	− 0.113890i	−0.998377	−	0.0569452i	$-0.981864\pi$
$131$	− 170.763i	− 0.113890i	0.998377	−	0.0569452i	$-0.0181360\pi$
$132$	0	0
$133$	− 1860.41i	− 1.21292i
$134$	0	0
$135$	−1583.92	−1.00979
$136$	0	0
$137$	1930.00	1.20358	0.601792	−	0.798653i	$-0.294454\pi$
$137$	1930.00	1.20358	0.601792	+	0.798653i	$0.294454\pi$
$138$	0	0
$139$	1144.74i	0.698532i	0.937024	+	0.349266i	$0.113569\pi$
$139$	1144.74i	0.698532i	−0.937024	+	0.349266i	$0.886431\pi$
$140$	0	0
$141$	2003.52i	1.19664i
$142$	0	0
$143$	−791.960	−0.463126
$144$	0	0
$145$	−2240.00	−1.28291
$146$	0	0
$147$	1068.85i	0.599709i
$148$	0	0
$149$	− 1627.86i	− 0.895029i	−0.894277	−	0.447514i	$-0.852309\pi$
$149$	− 1627.86i	− 0.895029i	0.894277	−	0.447514i	$-0.147691\pi$
$150$	0	0
$151$	2375.88	1.28044	0.640219	−	0.768192i	$-0.278843\pi$
$151$	2375.88	1.28044	0.640219	+	0.768192i	$0.278843\pi$
$152$	0	0
$153$	−910.000	−0.480844
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3488.27i	1.77321i	0.462528	+	0.886605i	$0.346943\pi$
$157$	3488.27i	1.77321i	−0.462528	+	0.886605i	$0.653057\pi$
$158$	0	0
$159$	791.960	0.395009
$160$	0	0
$161$	3584.00	1.75440
$162$	0	0
$163$	3497.48i	1.68064i	0.542094	+	0.840318i	$0.317632\pi$
$163$	3497.48i	1.68064i	−0.542094	+	0.840318i	$0.682368\pi$
$164$	0	0
$165$	5008.79i	2.36324i
$166$	0	0
$167$	2059.09	0.954117	0.477059	−	0.878872i	$-0.341703\pi$
$167$	2059.09	0.954117	0.477059	+	0.878872i	$0.341703\pi$
$168$	0	0
$169$	1877.00	0.854347
$170$	0	0
$171$	− 1068.85i	− 0.477994i
$172$	0	0
$173$	− 2021.41i	− 0.888350i	−0.895940	−	0.444175i	$-0.853497\pi$
$173$	− 2021.41i	− 0.888350i	0.895940	−	0.444175i	$-0.146503\pi$
$174$	0	0
$175$	4412.35	1.90595
$176$	0	0
$177$	−520.000	−0.220823
$178$	0	0
$179$	− 3586.02i	− 1.49739i	−0.662917	−	0.748693i	$-0.730682\pi$
$179$	− 3586.02i	− 1.49739i	0.662917	−	0.748693i	$-0.269318\pi$
$180$	0	0
$181$	− 2486.51i	− 1.02111i	−0.859846	−	0.510554i	$-0.829440\pi$
$181$	− 2486.51i	− 1.02111i	0.859846	−	0.510554i	$-0.170560\pi$
$182$	0	0
$183$	1470.78	0.594117
$184$	0	0
$185$	−6720.00	−2.67062
$186$	0	0
$187$	− 3099.03i	− 1.21189i
$188$	0	0
$189$	− 2003.52i	− 0.771082i
$190$	0	0
$191$	−2262.74	−0.857205	−0.428603	−	0.903493i	$-0.640994\pi$
$191$	−2262.74	−0.857205	−0.428603	+	0.903493i	$0.640994\pi$
$192$	0	0
$193$	630.000	0.234966	0.117483	−	0.993075i	$-0.462517\pi$
$193$	630.000	0.234966	0.117483	+	0.993075i	$0.462517\pi$
$194$	0	0
$195$	2023.86i	0.743238i
$196$	0	0
$197$	1878.30i	0.679305i	0.940551	+	0.339653i	$0.110310\pi$
$197$	1878.30i	0.679305i	−0.940551	+	0.339653i	$0.889690\pi$
$198$	0	0
$199$	3959.80	1.41057	0.705283	−	0.708926i	$-0.250820\pi$
$199$	3959.80	1.41057	0.705283	+	0.708926i	$0.250820\pi$
$200$	0	0
$201$	1400.00	0.491286
$202$	0	0
$203$	− 2833.40i	− 0.979634i
$204$	0	0
$205$	− 3255.71i	− 1.10921i
$206$	0	0
$207$	2059.09	0.691386
$208$	0	0
$209$	3640.00	1.20471
$210$	0	0
$211$	2789.13i	0.910007i	0.890490	+	0.455004i	$0.150362\pi$
$211$	2789.13i	0.910007i	−0.890490	+	0.455004i	$0.849638\pi$
$212$	0	0
$213$	− 715.542i	− 0.230179i
$214$	0	0
$215$	−2375.88	−0.753645
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5755.35i	1.77585i
$220$	0	0
$221$	− 1252.20i	− 0.381140i
$222$	0	0
$223$	2534.27	0.761019	0.380510	−	0.924777i	$-0.375749\pi$
$223$	2534.27	0.761019	0.380510	+	0.924777i	$0.375749\pi$
$224$	0	0
$225$	2535.00	0.751111
$226$	0	0
$227$	− 2219.92i	− 0.649080i	−0.945872	−	0.324540i	$-0.894790\pi$
$227$	− 2219.92i	− 0.649080i	0.945872	−	0.324540i	$-0.105210\pi$
$228$	0	0
$229$	− 4275.36i	− 1.23373i	−0.787069	−	0.616864i	$-0.788403\pi$
$229$	− 4275.36i	− 1.23373i	0.787069	−	0.616864i	$-0.211597\pi$
$230$	0	0
$231$	−6335.68	−1.80458
$232$	0	0
$233$	−5010.00	−1.40865	−0.704326	−	0.709876i	$-0.748751\pi$
$233$	−5010.00	−1.40865	−0.704326	+	0.709876i	$0.748751\pi$
$234$	0	0
$235$	5666.80i	1.57303i
$236$	0	0
$237$	4293.25i	1.17669i
$238$	0	0
$239$	−1583.92	−0.428683	−0.214341	−	0.976759i	$-0.568761\pi$
$239$	−1583.92	−0.428683	−0.214341	+	0.976759i	$0.568761\pi$
$240$	0	0
$241$	1638.00	0.437813	0.218906	−	0.975746i	$-0.429751\pi$
$241$	1638.00	0.437813	0.218906	+	0.975746i	$0.429751\pi$
$242$	0	0
$243$	− 3370.99i	− 0.889913i
$244$	0	0
$245$	3023.16i	0.788338i
$246$	0	0
$247$	1470.78	0.378881
$248$	0	0
$249$	−4520.00	−1.15037
$250$	0	0
$251$	− 1233.29i	− 0.310137i	−0.987904	−	0.155069i	$-0.950440\pi$
$251$	− 1233.29i	− 0.310137i	0.987904	−	0.155069i	$-0.0495599\pi$
$252$	0	0
$253$	7012.31i	1.74253i
$254$	0	0
$255$	−7919.60	−1.94488
$256$	0	0
$257$	1890.00	0.458735	0.229368	−	0.973340i	$-0.426334\pi$
$257$	1890.00	0.458735	0.229368	+	0.973340i	$0.426334\pi$
$258$	0	0
$259$	− 8500.20i	− 2.03929i
$260$	0	0
$261$	− 1627.86i	− 0.386061i
$262$	0	0
$263$	−2647.41	−0.620708	−0.310354	−	0.950621i	$-0.600448\pi$
$263$	−2647.41	−0.620708	−0.310354	+	0.950621i	$0.600448\pi$
$264$	0	0
$265$	2240.00	0.519253
$266$	0	0
$267$	− 3453.21i	− 0.791509i
$268$	0	0
$269$	3488.27i	0.790644i	0.918543	+	0.395322i	$0.129367\pi$
$269$	3488.27i	0.790644i	−0.918543	+	0.395322i	$0.870633\pi$
$270$	0	0
$271$	−7919.60	−1.77521	−0.887604	−	0.460608i	$-0.847631\pi$
$271$	−7919.60	−1.77521	−0.887604	+	0.460608i	$0.847631\pi$
$272$	0	0
$273$	−2560.00	−0.567539
$274$	0	0
$275$	8633.02i	1.89306i
$276$	0	0
$277$	4883.57i	1.05930i	0.848217	+	0.529649i	$0.177676\pi$
$277$	4883.57i	1.05930i	−0.848217	+	0.529649i	$0.822324\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1118.00	0.237346	0.118673	−	0.992933i	$-0.462136\pi$
$281$	1118.00	0.237346	0.118673	+	0.992933i	$0.462136\pi$
$282$	0	0
$283$	− 2118.73i	− 0.445036i	−0.974929	−	0.222518i	$-0.928572\pi$
$283$	− 2118.73i	− 0.445036i	0.974929	−	0.222518i	$-0.0714276\pi$
$284$	0	0
$285$	− 9302.04i	− 1.93335i
$286$	0	0
$287$	4118.19	0.847000
$288$	0	0
$289$	−13.0000	−0.00264604
$290$	0	0
$291$	− 3099.03i	− 0.624290i
$292$	0	0
$293$	− 4275.36i	− 0.852455i	−0.904616	−	0.426227i	$-0.859842\pi$
$293$	− 4275.36i	− 0.852455i	0.904616	−	0.426227i	$-0.140158\pi$
$294$	0	0
$295$	−1470.78	−0.290279
$296$	0	0
$297$	3920.00	0.765864
$298$	0	0
$299$	2833.40i	0.548026i
$300$	0	0
$301$	− 3005.28i	− 0.575486i
$302$	0	0
$303$	−1470.78	−0.278859
$304$	0	0
$305$	4160.00	0.780987
$306$	0	0
$307$	7975.26i	1.48265i	0.671148	+	0.741323i	$0.265801\pi$
$307$	7975.26i	1.48265i	−0.671148	+	0.741323i	$0.734199\pi$
$308$	0	0
$309$	1001.76i	0.184427i
$310$	0	0
$311$	−10295.5	−1.87718	−0.938590	−	0.345035i	$-0.887867\pi$
$311$	−10295.5	−1.87718	−0.938590	+	0.345035i	$0.887867\pi$
$312$	0	0
$313$	2170.00	0.391871	0.195936	−	0.980617i	$-0.437226\pi$
$313$	2170.00	0.391871	0.195936	+	0.980617i	$0.437226\pi$
$314$	0	0
$315$	5262.03i	0.941212i
$316$	0	0
$317$	− 4883.57i	− 0.865264i	−0.901571	−	0.432632i	$-0.857585\pi$
$317$	− 4883.57i	− 0.865264i	0.901571	−	0.432632i	$-0.142415\pi$
$318$	0	0
$319$	5543.72	0.973005
$320$	0	0
$321$	−10920.0	−1.89874
$322$	0	0
$323$	5755.35i	0.991443i
$324$	0	0
$325$	3488.27i	0.595367i
$326$	0	0
$327$	8711.56	1.47324
$328$	0	0
$329$	−7168.00	−1.20117
$330$	0	0
$331$	− 11732.1i	− 1.94819i	−0.226133	−	0.974096i	$-0.572608\pi$
$331$	− 11732.1i	− 1.94819i	0.226133	−	0.974096i	$-0.427392\pi$
$332$	0	0
$333$	− 4883.57i	− 0.803658i
$334$	0	0
$335$	3959.80	0.645812
$336$	0	0
$337$	−10990.0	−1.77645	−0.888225	−	0.459409i	$-0.848061\pi$
$337$	−10990.0	−1.77645	−0.888225	+	0.459409i	$0.848061\pi$
$338$	0	0
$339$	− 5755.35i	− 0.922087i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	3937.17	0.619788
$344$	0	0
$345$	17920.0	2.79646
$346$	0	0
$347$	− 4117.29i	− 0.636967i	−0.947928	−	0.318483i	$-0.896826\pi$
$347$	− 4117.29i	− 0.636967i	0.947928	−	0.318483i	$-0.103174\pi$
$348$	0	0
$349$	4275.36i	0.655745i	0.944722	+	0.327872i	$0.106332\pi$
$349$	4275.36i	0.655745i	−0.944722	+	0.327872i	$0.893668\pi$
$350$	0	0
$351$	1583.92	0.240864
$352$	0	0
$353$	8610.00	1.29820	0.649099	−	0.760704i	$-0.275146\pi$
$353$	8610.00	1.29820	0.649099	+	0.760704i	$0.275146\pi$
$354$	0	0
$355$	− 2023.86i	− 0.302578i
$356$	0	0
$357$	− 10017.6i	− 1.48512i
$358$	0	0
$359$	10295.5	1.51358	0.756789	−	0.653659i	$-0.226767\pi$
$359$	10295.5	1.51358	0.756789	+	0.653659i	$0.226767\pi$
$360$	0	0
$361$	99.0000	0.0144336
$362$	0	0
$363$	− 3978.15i	− 0.575202i
$364$	0	0
$365$	16278.6i	2.33441i
$366$	0	0
$367$	2851.05	0.405515	0.202757	−	0.979229i	$-0.435010\pi$
$367$	2851.05	0.405515	0.202757	+	0.979229i	$0.435010\pi$
$368$	0	0
$369$	2366.00	0.333791
$370$	0	0
$371$	2833.40i	0.396504i
$372$	0	0
$373$	4883.57i	0.677914i	0.940802	+	0.338957i	$0.110074\pi$
$373$	4883.57i	0.677914i	−0.940802	+	0.338957i	$0.889926\pi$
$374$	0	0
$375$	7919.60	1.09058
$376$	0	0
$377$	2240.00	0.306010
$378$	0	0
$379$	3674.57i	0.498021i	0.968501	+	0.249010i	$0.0801053\pi$
$379$	3674.57i	0.498021i	−0.968501	+	0.249010i	$0.919895\pi$
$380$	0	0
$381$	12021.1i	1.61643i
$382$	0	0
$383$	2534.27	0.338108	0.169054	−	0.985607i	$-0.445929\pi$
$383$	2534.27	0.338108	0.169054	+	0.985607i	$0.445929\pi$
$384$	0	0
$385$	−17920.0	−2.37218
$386$	0	0
$387$	− 1726.60i	− 0.226791i
$388$	0	0
$389$	11395.0i	1.48522i	0.669726	+	0.742609i	$0.266412\pi$
$389$	11395.0i	1.48522i	−0.669726	+	0.742609i	$0.733588\pi$
$390$	0	0
$391$	−11087.4	−1.43406
$392$	0	0
$393$	1080.00	0.138623
$394$	0	0
$395$	12143.1i	1.54681i
$396$	0	0
$397$	− 13255.4i	− 1.67574i	−0.545867	−	0.837872i	$-0.683800\pi$
$397$	− 13255.4i	− 1.67574i	0.545867	−	0.837872i	$-0.316200\pi$
$398$	0	0
$399$	11766.3	1.47631
$400$	0	0
$401$	−1722.00	−0.214445	−0.107223	−	0.994235i	$-0.534196\pi$
$401$	−1722.00	−0.214445	−0.107223	+	0.994235i	$0.534196\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 16296.5i	− 1.99945i
$406$	0	0
$407$	16631.2	2.02549
$408$	0	0
$409$	13594.0	1.64347	0.821736	−	0.569868i	$-0.193006\pi$
$409$	13594.0	1.64347	0.821736	+	0.569868i	$0.193006\pi$
$410$	0	0
$411$	12206.4i	1.46496i
$412$	0	0
$413$	− 1860.41i	− 0.221658i
$414$	0	0
$415$	−12784.5	−1.51221
$416$	0	0
$417$	−7240.00	−0.850226
$418$	0	0
$419$	− 4775.04i	− 0.556744i	−0.960473	−	0.278372i	$-0.910205\pi$
$419$	− 4775.04i	− 0.556744i	0.960473	−	0.278372i	$-0.0897949\pi$
$420$	0	0
$421$	7888.85i	0.913252i	0.889659	+	0.456626i	$0.150942\pi$
$421$	7888.85i	0.913252i	−0.889659	+	0.456626i	$0.849058\pi$
$422$	0	0
$423$	−4118.19	−0.473365
$424$	0	0
$425$	−13650.0	−1.55793
$426$	0	0
$427$	5262.03i	0.596364i
$428$	0	0
$429$	− 5008.79i	− 0.563699i
$430$	0	0
$431$	−1583.92	−0.177018	−0.0885089	−	0.996075i	$-0.528210\pi$
$431$	−1583.92	−0.177018	−0.0885089	+	0.996075i	$0.528210\pi$
$432$	0	0
$433$	14630.0	1.62373	0.811863	−	0.583849i	$-0.198454\pi$
$433$	14630.0	1.62373	0.811863	+	0.583849i	$0.198454\pi$
$434$	0	0
$435$	− 14167.0i	− 1.56151i
$436$	0	0
$437$	− 13022.9i	− 1.42556i
$438$	0	0
$439$	−10295.5	−1.11931	−0.559654	−	0.828726i	$-0.689066\pi$
$439$	−10295.5	−1.11931	−0.559654	+	0.828726i	$0.689066\pi$
$440$	0	0
$441$	−2197.00	−0.237231
$442$	0	0
$443$	− 17753.0i	− 1.90400i	−0.306099	−	0.952000i	$-0.599024\pi$
$443$	− 17753.0i	− 1.90400i	0.306099	−	0.952000i	$-0.400976\pi$
$444$	0	0
$445$	− 9767.14i	− 1.04047i
$446$	0	0
$447$	10295.5	1.08939
$448$	0	0
$449$	3894.00	0.409286	0.204643	−	0.978837i	$-0.434397\pi$
$449$	3894.00	0.409286	0.204643	+	0.978837i	$0.434397\pi$
$450$	0	0
$451$	8057.48i	0.841268i
$452$	0	0
$453$	15026.4i	1.55850i
$454$	0	0
$455$	−7240.77	−0.746050
$456$	0	0
$457$	2730.00	0.279440	0.139720	−	0.990191i	$-0.455380\pi$
$457$	2730.00	0.279440	0.139720	+	0.990191i	$0.455380\pi$
$458$	0	0
$459$	6198.06i	0.630285i
$460$	0	0
$461$	− 10250.1i	− 1.03557i	−0.855512	−	0.517784i	$-0.826757\pi$
$461$	− 10250.1i	− 1.03557i	0.855512	−	0.517784i	$-0.173243\pi$
$462$	0	0
$463$	−7648.07	−0.767680	−0.383840	−	0.923400i	$-0.625399\pi$
$463$	−7648.07	−0.767680	−0.383840	+	0.923400i	$0.625399\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 13730.6i	− 1.36055i	−0.732957	−	0.680275i	$-0.761860\pi$
$467$	− 13730.6i	− 1.36055i	0.732957	−	0.680275i	$-0.238140\pi$
$468$	0	0
$469$	5008.79i	0.493144i
$470$	0	0
$471$	−22061.7	−2.15828
$472$	0	0
$473$	5880.00	0.571591
$474$	0	0
$475$	− 16032.7i	− 1.54870i
$476$	0	0
$477$	1627.86i	0.156257i
$478$	0	0
$479$	−12671.4	−1.20870	−0.604352	−	0.796718i	$-0.706568\pi$
$479$	−12671.4	−1.20870	−0.604352	+	0.796718i	$0.706568\pi$
$480$	0	0
$481$	6720.00	0.637018
$482$	0	0
$483$	22667.2i	2.13539i
$484$	0	0
$485$	− 8765.39i	− 0.820651i
$486$	0	0
$487$	2059.09	0.191594	0.0957972	−	0.995401i	$-0.469460\pi$
$487$	2059.09	0.191594	0.0957972	+	0.995401i	$0.469460\pi$
$488$	0	0
$489$	−22120.0	−2.04561
$490$	0	0
$491$	− 5888.16i	− 0.541200i	−0.962692	−	0.270600i	$-0.912778\pi$
$491$	− 5888.16i	− 0.541200i	0.962692	−	0.270600i	$-0.0872220\pi$
$492$	0	0
$493$	8765.39i	0.800757i
$494$	0	0
$495$	−10295.5	−0.934843
$496$	0	0
$497$	2560.00	0.231050
$498$	0	0
$499$	10935.2i	0.981012i	0.871438	+	0.490506i	$0.163188\pi$
$499$	10935.2i	0.981012i	−0.871438	+	0.490506i	$0.836812\pi$
$500$	0	0
$501$	13022.9i	1.16131i
$502$	0	0
$503$	14413.7	1.27768	0.638840	−	0.769339i	$-0.279414\pi$
$503$	14413.7	1.27768	0.638840	+	0.769339i	$0.279414\pi$
$504$	0	0
$505$	−4160.00	−0.366569
$506$	0	0
$507$	11871.2i	1.03988i
$508$	0	0
$509$	12790.3i	1.11379i	0.830582	+	0.556896i	$0.188008\pi$
$509$	12790.3i	1.11379i	−0.830582	+	0.556896i	$0.811992\pi$
$510$	0	0
$511$	−20590.9	−1.78256
$512$	0	0
$513$	−7280.00	−0.626549
$514$	0	0
$515$	2833.40i	0.242436i
$516$	0	0
$517$	− 14024.6i	− 1.19304i
$518$	0	0
$519$	12784.5	1.08127
$520$	0	0
$521$	−16758.0	−1.40918	−0.704589	−	0.709616i	$-0.748868\pi$
$521$	−16758.0	−1.40918	−0.704589	+	0.709616i	$0.748868\pi$
$522$	0	0
$523$	16197.2i	1.35421i	0.735885	+	0.677107i	$0.236766\pi$
$523$	16197.2i	1.35421i	−0.735885	+	0.677107i	$0.763234\pi$
$524$	0	0
$525$	27906.1i	2.31985i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	12921.0	1.06197
$530$	0	0
$531$	− 1068.85i	− 0.0873524i
$532$	0	0
$533$	3255.71i	0.264579i
$534$	0	0
$535$	−30886.4	−2.49596
$536$	0	0
$537$	22680.0	1.82256
$538$	0	0
$539$	− 7481.95i	− 0.597904i
$540$	0	0
$541$	− 375.659i	− 0.0298537i	−0.999889	−	0.0149269i	$-0.995248\pi$
$541$	− 375.659i	− 0.0298537i	0.999889	−	0.0149269i	$-0.00475154\pi$
$542$	0	0
$543$	15726.1	1.24285
$544$	0	0
$545$	24640.0	1.93663
$546$	0	0
$547$	− 1638.06i	− 0.128041i	−0.997949	−	0.0640205i	$-0.979608\pi$
$547$	− 1638.06i	− 0.128041i	0.997949	−	0.0640205i	$-0.0203923\pi$
$548$	0	0
$549$	3023.16i	0.235019i
$550$	0	0
$551$	−10295.5	−0.796011
$552$	0	0
$553$	−15360.0	−1.18115
$554$	0	0
$555$	− 42501.0i	− 3.25057i
$556$	0	0
$557$	− 19909.9i	− 1.51456i	−0.653089	−	0.757282i	$-0.726527\pi$
$557$	− 19909.9i	− 1.51456i	0.653089	−	0.757282i	$-0.273473\pi$
$558$	0	0
$559$	2375.88	0.179766
$560$	0	0
$561$	19600.0	1.47507
$562$	0	0
$563$	790.569i	0.0591803i	0.999562	+	0.0295902i	$0.00942022\pi$
$563$	790.569i	0.0591803i	−0.999562	+	0.0295902i	$0.990580\pi$
$564$	0	0
$565$	− 16278.6i	− 1.21211i
$566$	0	0
$567$	20613.6	1.52679
$568$	0	0
$569$	−20454.0	−1.50699	−0.753494	−	0.657455i	$-0.771633\pi$
$569$	−20454.0	−1.50699	−0.753494	+	0.657455i	$0.771633\pi$
$570$	0	0
$571$	5622.53i	0.412076i	0.978544	+	0.206038i	$0.0660571\pi$
$571$	5622.53i	0.412076i	−0.978544	+	0.206038i	$0.933943\pi$
$572$	0	0
$573$	− 14310.8i	− 1.04336i
$574$	0	0
$575$	30886.4	2.24009
$576$	0	0
$577$	−22750.0	−1.64141	−0.820706	−	0.571351i	$-0.806420\pi$
$577$	−22750.0	−1.64141	−0.820706	+	0.571351i	$0.806420\pi$
$578$	0	0
$579$	3984.47i	0.285991i
$580$	0	0
$581$	− 16171.2i	− 1.15473i
$582$	0	0
$583$	−5543.72	−0.393820
$584$	0	0
$585$	−4160.00	−0.294008
$586$	0	0
$587$	15843.0i	1.11399i	0.830516	+	0.556994i	$0.188045\pi$
$587$	15843.0i	1.11399i	−0.830516	+	0.556994i	$0.811955\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−11879.4	−0.826824
$592$	0	0
$593$	−12110.0	−0.838614	−0.419307	−	0.907845i	$-0.637727\pi$
$593$	−12110.0	−0.838614	−0.419307	+	0.907845i	$0.637727\pi$
$594$	0	0
$595$	− 28334.0i	− 1.95224i
$596$	0	0
$597$	25044.0i	1.71689i
$598$	0	0
$599$	19120.2	1.30422	0.652111	−	0.758124i	$-0.273884\pi$
$599$	19120.2	1.30422	0.652111	+	0.758124i	$0.273884\pi$
$600$	0	0
$601$	4382.00	0.297413	0.148707	−	0.988881i	$-0.452489\pi$
$601$	4382.00	0.297413	0.148707	+	0.988881i	$0.452489\pi$
$602$	0	0
$603$	2877.67i	0.194341i
$604$	0	0
$605$	− 11251.9i	− 0.756123i
$606$	0	0
$607$	8236.38	0.550749	0.275374	−	0.961337i	$-0.411198\pi$
$607$	8236.38	0.550749	0.275374	+	0.961337i	$0.411198\pi$
$608$	0	0
$609$	17920.0	1.19237
$610$	0	0
$611$	− 5666.80i	− 0.375212i
$612$	0	0
$613$	− 2128.74i	− 0.140259i	−0.997538	−	0.0701296i	$-0.977659\pi$
$613$	− 2128.74i	− 0.140259i	0.997538	−	0.0701296i	$-0.0223413\pi$
$614$	0	0
$615$	20590.9	1.35009
$616$	0	0
$617$	15470.0	1.00940	0.504699	−	0.863295i	$-0.331603\pi$
$617$	15470.0	1.00940	0.504699	+	0.863295i	$0.331603\pi$
$618$	0	0
$619$	19486.0i	1.26528i	0.774447	+	0.632639i	$0.218028\pi$
$619$	19486.0i	1.26528i	−0.774447	+	0.632639i	$0.781972\pi$
$620$	0	0
$621$	− 14024.6i	− 0.906262i
$622$	0	0
$623$	12354.6	0.794503
$624$	0	0
$625$	−1975.00	−0.126400
$626$	0	0
$627$	23021.4i	1.46632i
$628$	0	0
$629$	26296.2i	1.66693i
$630$	0	0
$631$	6448.81	0.406851	0.203426	−	0.979090i	$-0.434792\pi$
$631$	6448.81	0.406851	0.203426	+	0.979090i	$0.434792\pi$
$632$	0	0
$633$	−17640.0	−1.10763
$634$	0	0
$635$	34000.8i	2.12485i
$636$	0	0
$637$	− 3023.16i	− 0.188041i
$638$	0	0
$639$	1470.78	0.0910536
$640$	0	0
$641$	182.000	0.0112146	0.00560731	−	0.999984i	$-0.498215\pi$
$641$	182.000	0.0112146	0.00560731	+	0.999984i	$0.498215\pi$
$642$	0	0
$643$	− 23293.3i	− 1.42862i	−0.699832	−	0.714308i	$-0.746742\pi$
$643$	− 23293.3i	− 1.42862i	0.699832	−	0.714308i	$-0.253258\pi$
$644$	0	0
$645$	− 15026.4i	− 0.917307i
$646$	0	0
$647$	2059.09	0.125118	0.0625590	−	0.998041i	$-0.480074\pi$
$647$	2059.09	0.125118	0.0625590	+	0.998041i	$0.480074\pi$
$648$	0	0
$649$	3640.00	0.220158
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14650.7i	0.877989i	0.898490	+	0.438995i	$0.144665\pi$
$653$	14650.7i	0.877989i	−0.898490	+	0.438995i	$0.855335\pi$
$654$	0	0
$655$	3054.70	0.182225
$656$	0	0
$657$	−11830.0	−0.702484
$658$	0	0
$659$	10935.2i	0.646393i	0.946332	+	0.323197i	$0.104758\pi$
$659$	10935.2i	0.646393i	−0.946332	+	0.323197i	$0.895242\pi$
$660$	0	0
$661$	− 3774.48i	− 0.222103i	−0.993815	−	0.111052i	$-0.964578\pi$
$661$	− 3774.48i	− 0.222103i	0.993815	−	0.111052i	$-0.0354219\pi$
$662$	0	0
$663$	7919.60	0.463909
$664$	0	0
$665$	33280.0	1.94067
$666$	0	0
$667$	− 19833.8i	− 1.15138i
$668$	0	0
$669$	16028.1i	0.926283i
$670$	0	0
$671$	−10295.5	−0.592328
$672$	0	0
$673$	−2410.00	−0.138037	−0.0690183	−	0.997615i	$-0.521987\pi$
$673$	−2410.00	−0.138037	−0.0690183	+	0.997615i	$0.521987\pi$
$674$	0	0
$675$	− 17266.0i	− 0.984548i
$676$	0	0
$677$	27566.2i	1.56493i	0.622695	+	0.782464i	$0.286038\pi$
$677$	27566.2i	1.56493i	−0.622695	+	0.782464i	$0.713962\pi$
$678$	0	0
$679$	11087.4	0.626652
$680$	0	0
$681$	14040.0	0.790035
$682$	0	0
$683$	− 2346.41i	− 0.131454i	−0.997838	−	0.0657269i	$-0.979063\pi$
$683$	− 2346.41i	− 0.131454i	0.997838	−	0.0657269i	$-0.0209366\pi$
$684$	0	0
$685$	34524.9i	1.92573i
$686$	0	0
$687$	27039.8	1.50165
$688$	0	0
$689$	−2240.00	−0.123857
$690$	0	0
$691$	10277.4i	0.565804i	0.959149	+	0.282902i	$0.0912972\pi$
$691$	10277.4i	0.565804i	−0.959149	+	0.282902i	$0.908703\pi$
$692$	0	0
$693$	− 13022.9i	− 0.713849i
$694$	0	0
$695$	−20477.8	−1.11765
$696$	0	0
$697$	−12740.0	−0.692341
$698$	0	0
$699$	− 31686.0i	− 1.71456i
$700$	0	0
$701$	626.099i	0.0337339i	0.999858	+	0.0168669i	$0.00536916\pi$
$701$	626.099i	0.0337339i	−0.999858	+	0.0168669i	$0.994631\pi$
$702$	0	0
$703$	−30886.4	−1.65705
$704$	0	0
$705$	−35840.0	−1.91463
$706$	0	0
$707$	− 5262.03i	− 0.279914i
$708$	0	0
$709$	− 14650.7i	− 0.776050i	−0.921649	−	0.388025i	$-0.873157\pi$
$709$	− 14650.7i	− 0.776050i	0.921649	−	0.388025i	$-0.126843\pi$
$710$	0	0
$711$	−8824.69	−0.465474
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 14167.0i	− 0.741001i
$716$	0	0
$717$	− 10017.6i	− 0.521776i
$718$	0	0
$719$	14255.3	0.739405	0.369702	−	0.929150i	$-0.379460\pi$
$719$	14255.3	0.739405	0.369702	+	0.929150i	$0.379460\pi$
$720$	0	0
$721$	−3584.00	−0.185125
$722$	0	0
$723$	10359.6i	0.532889i
$724$	0	0
$725$	− 24417.9i	− 1.25084i
$726$	0	0
$727$	4276.58	0.218170	0.109085	−	0.994032i	$-0.465208\pi$
$727$	4276.58	0.218170	0.109085	+	0.994032i	$0.465208\pi$
$728$	0	0
$729$	−3277.00	−0.166489
$730$	0	0
$731$	9297.10i	0.470404i
$732$	0	0
$733$	9284.15i	0.467828i	0.972257	+	0.233914i	$0.0751534\pi$
$733$	9284.15i	0.467828i	−0.972257	+	0.233914i	$0.924847\pi$
$734$	0	0
$735$	−19120.2	−0.959535
$736$	0	0
$737$	−9800.00	−0.489807
$738$	0	0
$739$	1726.60i	0.0859461i	0.999076	+	0.0429730i	$0.0136830\pi$
$739$	1726.60i	0.0859461i	−0.999076	+	0.0429730i	$0.986317\pi$
$740$	0	0
$741$	9302.04i	0.461159i
$742$	0	0
$743$	−28352.2	−1.39992	−0.699959	−	0.714183i	$-0.746799\pi$
$743$	−28352.2	−1.39992	−0.699959	+	0.714183i	$0.746799\pi$
$744$	0	0
$745$	29120.0	1.43205
$746$	0	0
$747$	− 9290.77i	− 0.455062i
$748$	0	0
$749$	− 39068.6i	− 1.90592i
$750$	0	0
$751$	−20590.9	−1.00050	−0.500249	−	0.865881i	$-0.666758\pi$
$751$	−20590.9	−1.00050	−0.500249	+	0.865881i	$0.666758\pi$
$752$	0	0
$753$	7800.00	0.377487
$754$	0	0
$755$	42501.0i	2.04870i
$756$	0	0
$757$	− 22664.8i	− 1.08820i	−0.839021	−	0.544099i	$-0.816872\pi$
$757$	− 22664.8i	− 1.08820i	0.839021	−	0.544099i	$-0.183128\pi$
$758$	0	0
$759$	−44349.7	−2.12094
$760$	0	0
$761$	7098.00	0.338111	0.169055	−	0.985607i	$-0.445928\pi$
$761$	7098.00	0.338111	0.169055	+	0.985607i	$0.445928\pi$
$762$	0	0
$763$	31167.4i	1.47882i
$764$	0	0
$765$	− 16278.6i	− 0.769350i
$766$	0	0
$767$	1470.78	0.0692397
$768$	0	0
$769$	17654.0	0.827854	0.413927	−	0.910310i	$-0.364157\pi$
$769$	17654.0	0.827854	0.413927	+	0.910310i	$0.364157\pi$
$770$	0	0
$771$	11953.4i	0.558355i
$772$	0	0
$773$	6529.32i	0.303808i	0.988395	+	0.151904i	$0.0485404\pi$
$773$	6529.32i	0.303808i	−0.988395	+	0.151904i	$0.951460\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	53760.0	2.48215
$778$	0	0
$779$	− 14963.9i	− 0.688238i
$780$	0	0
$781$	5008.79i	0.229486i
$782$	0	0
$783$	−11087.4	−0.506044
$784$	0	0
$785$	−62400.0	−2.83714
$786$	0	0
$787$	2384.36i	0.107996i	0.998541	+	0.0539982i	$0.0171965\pi$
$787$	2384.36i	0.107996i	−0.998541	+	0.0539982i	$0.982803\pi$
$788$	0	0
$789$	− 16743.7i	− 0.755502i
$790$	0	0
$791$	20590.9	0.925575
$792$	0	0
$793$	−4160.00	−0.186287
$794$	0	0
$795$	14167.0i	0.632015i
$796$	0	0
$797$	29534.0i	1.31261i	0.754497	+	0.656303i	$0.227881\pi$
$797$	29534.0i	1.31261i	−0.754497	+	0.656303i	$0.772119\pi$
$798$	0	0
$799$	22174.9	0.981840
$800$	0	0
$801$	7098.00	0.313103
$802$	0	0
$803$	− 40287.4i	− 1.77050i
$804$	0	0
$805$	64112.5i	2.80704i
$806$	0	0
$807$	−22061.7	−0.962342
$808$	0	0
$809$	−10934.0	−0.475178	−0.237589	−	0.971366i	$-0.576357\pi$
$809$	−10934.0	−0.475178	−0.237589	+	0.971366i	$0.576357\pi$
$810$	0	0
$811$	− 17348.3i	− 0.751146i	−0.926793	−	0.375573i	$-0.877446\pi$
$811$	− 17348.3i	− 0.751146i	0.926793	−	0.375573i	$-0.122554\pi$
$812$	0	0
$813$	− 50087.9i	− 2.16071i
$814$	0	0
$815$	−62564.8	−2.68902
$816$	0	0
$817$	−10920.0	−0.467616
$818$	0	0
$819$	− 5262.03i	− 0.224506i
$820$	0	0
$821$	14901.2i	0.633440i	0.948519	+	0.316720i	$0.102581\pi$
$821$	14901.2i	0.633440i	−0.948519	+	0.316720i	$0.897419\pi$
$822$	0	0
$823$	17943.5	0.759991	0.379995	−	0.924988i	$-0.375926\pi$
$823$	17943.5	0.759991	0.379995	+	0.924988i	$0.375926\pi$
$824$	0	0
$825$	−54600.0	−2.30416
$826$	0	0
$827$	− 25899.1i	− 1.08899i	−0.838763	−	0.544497i	$-0.816721\pi$
$827$	− 25899.1i	− 1.08899i	0.838763	−	0.544497i	$-0.183279\pi$
$828$	0	0
$829$	− 25276.5i	− 1.05897i	−0.848318	−	0.529487i	$-0.822384\pi$
$829$	− 25276.5i	− 1.05897i	0.848318	−	0.529487i	$-0.177616\pi$
$830$	0	0
$831$	−30886.4	−1.28934
$832$	0	0
$833$	11830.0	0.492059
$834$	0	0
$835$	36834.2i	1.52659i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10295.5	0.423646	0.211823	−	0.977308i	$-0.432060\pi$
$839$	10295.5	0.423646	0.211823	+	0.977308i	$0.432060\pi$
$840$	0	0
$841$	8709.00	0.357087
$842$	0	0
$843$	7070.85i	0.288889i
$844$	0	0
$845$	33576.8i	1.36695i
$846$	0	0
$847$	14232.6	0.577378
$848$	0	0
$849$	13400.0	0.541681
$850$	0	0
$851$	− 59501.4i	− 2.39681i
$852$	0	0
$853$	21770.4i	0.873860i	0.899495	+	0.436930i	$0.143934\pi$
$853$	21770.4i	0.873860i	−0.899495	+	0.436930i	$0.856066\pi$
$854$	0	0
$855$	19120.2	0.764791
$856$	0	0
$857$	24570.0	0.979341	0.489670	−	0.871908i	$-0.337117\pi$
$857$	24570.0	0.979341	0.489670	+	0.871908i	$0.337117\pi$
$858$	0	0
$859$	− 5837.56i	− 0.231869i	−0.993257	−	0.115934i	$-0.963014\pi$
$859$	− 5837.56i	− 0.231869i	0.993257	−	0.115934i	$-0.0369862\pi$
$860$	0	0
$861$	26045.7i	1.03094i
$862$	0	0
$863$	23713.5	0.935363	0.467681	−	0.883897i	$-0.345089\pi$
$863$	23713.5	0.935363	0.467681	+	0.883897i	$0.345089\pi$
$864$	0	0
$865$	36160.0	1.42136
$866$	0	0
$867$	− 82.2192i	− 0.00322066i
$868$	0	0
$869$	− 30052.8i	− 1.17315i
$870$	0	0
$871$	−3959.80	−0.154044
$872$	0	0
$873$	6370.00	0.246955
$874$	0	0
$875$	28334.0i	1.09470i
$876$	0	0
$877$	43701.7i	1.68267i	0.540514	+	0.841335i	$0.318230\pi$
$877$	43701.7i	1.68267i	−0.540514	+	0.841335i	$0.681770\pi$
$878$	0	0
$879$	27039.8	1.03758
$880$	0	0
$881$	−17038.0	−0.651561	−0.325780	−	0.945446i	$-0.605627\pi$
$881$	−17038.0	−0.651561	−0.325780	+	0.945446i	$0.605627\pi$
$882$	0	0
$883$	− 27492.8i	− 1.04780i	−0.851780	−	0.523900i	$-0.824476\pi$
$883$	− 27492.8i	− 1.04780i	0.851780	−	0.523900i	$-0.175524\pi$
$884$	0	0
$885$	− 9302.04i	− 0.353316i
$886$	0	0
$887$	−43241.0	−1.63686	−0.818428	−	0.574610i	$-0.805154\pi$
$887$	−43241.0	−1.63686	−0.818428	+	0.574610i	$0.805154\pi$
$888$	0	0
$889$	−43008.0	−1.62254
$890$	0	0
$891$	40331.7i	1.51646i
$892$	0	0
$893$	26045.7i	0.976021i
$894$	0	0
$895$	64148.7	2.39582
$896$	0	0
$897$	−17920.0	−0.667036
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 8765.39i	− 0.324104i
$902$	0	0
$903$	19007.0	0.700459
$904$	0	0
$905$	44480.0	1.63377
$906$	0	0
$907$	40154.6i	1.47002i	0.678054	+	0.735012i	$0.262823\pi$
$907$	40154.6i	1.47002i	−0.678054	+	0.735012i	$0.737177\pi$
$908$	0	0
$909$	− 3023.16i	− 0.110310i
$910$	0	0
$911$	11087.4	0.403231	0.201615	−	0.979465i	$-0.435381\pi$
$911$	11087.4	0.403231	0.201615	+	0.979465i	$0.435381\pi$
$912$	0	0
$913$	31640.0	1.14691
$914$	0	0
$915$	26310.2i	0.950587i
$916$	0	0
$917$	3863.93i	0.139147i
$918$	0	0
$919$	1470.78	0.0527928	0.0263964	−	0.999652i	$-0.491597\pi$
$919$	1470.78	0.0527928	0.0263964	+	0.999652i	$0.491597\pi$
$920$	0	0
$921$	−50440.0	−1.80462
$922$	0	0
$923$	2023.86i	0.0721734i
$924$	0	0
$925$	− 73253.6i	− 2.60385i
$926$	0	0
$927$	−2059.09	−0.0729553
$928$	0	0
$929$	32214.0	1.13768	0.568841	−	0.822447i	$-0.307392\pi$
$929$	32214.0	1.13768	0.568841	+	0.822447i	$0.307392\pi$
$930$	0	0
$931$	13895.0i	0.489143i
$932$	0	0
$933$	− 65114.3i	− 2.28483i
$934$	0	0
$935$	55437.2	1.93903
$936$	0	0
$937$	−13650.0	−0.475908	−0.237954	−	0.971276i	$-0.576477\pi$
$937$	−13650.0	−0.475908	−0.237954	+	0.971276i	$0.576477\pi$
$938$	0	0
$939$	13724.3i	0.476970i
$940$	0	0
$941$	− 6529.32i	− 0.226195i	−0.993584	−	0.113098i	$-0.963923\pi$
$941$	− 6529.32i	− 0.226195i	0.993584	−	0.113098i	$-0.0360773\pi$
$942$	0	0
$943$	28827.3	0.995490
$944$	0	0
$945$	35840.0	1.23373
$946$	0	0
$947$	− 11555.0i	− 0.396500i	−0.980151	−	0.198250i	$-0.936474\pi$
$947$	− 11555.0i	− 0.396500i	0.980151	−	0.198250i	$-0.0635258\pi$
$948$	0	0
$949$	− 16278.6i	− 0.556823i
$950$	0	0
$951$	30886.4	1.05317
$952$	0	0
$953$	−10470.0	−0.355883	−0.177942	−	0.984041i	$-0.556944\pi$
$953$	−10470.0	−0.355883	−0.177942	+	0.984041i	$0.556944\pi$
$954$	0	0
$955$	− 40477.2i	− 1.37153i
$956$	0	0
$957$	35061.5i	1.18430i
$958$	0	0
$959$	−43670.9	−1.47050
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	− 22445.8i	− 0.751098i
$964$	0	0
$965$	11269.8i	0.375945i
$966$	0	0
$967$	40706.7	1.35371	0.676856	−	0.736115i	$-0.263342\pi$
$967$	40706.7	1.35371	0.676856	+	0.736115i	$0.263342\pi$
$968$	0	0
$969$	−36400.0	−1.20675
$970$	0	0
$971$	− 49502.3i	− 1.63605i	−0.575183	−	0.818025i	$-0.695069\pi$
$971$	− 49502.3i	− 1.63605i	0.575183	−	0.818025i	$-0.304931\pi$
$972$	0	0
$973$	− 25902.6i	− 0.853443i
$974$	0	0
$975$	−22061.7	−0.724657
$976$	0	0
$977$	−51770.0	−1.69526	−0.847630	−	0.530588i	$-0.821971\pi$
$977$	−51770.0	−1.69526	−0.847630	+	0.530588i	$0.821971\pi$
$978$	0	0
$979$	24172.5i	0.789127i
$980$	0	0
$981$	17906.4i	0.582781i
$982$	0	0
$983$	−22333.3	−0.724639	−0.362320	−	0.932054i	$-0.618015\pi$
$983$	−22333.3	−0.724639	−0.362320	+	0.932054i	$0.618015\pi$
$984$	0	0
$985$	−33600.0	−1.08689
$986$	0	0
$987$	− 45334.4i	− 1.46202i
$988$	0	0
$989$	− 21036.9i	− 0.676376i
$990$	0	0
$991$	52948.2	1.69723	0.848614	−	0.529012i	$-0.177437\pi$
$991$	52948.2	1.69723	0.848614	+	0.529012i	$0.177437\pi$
$992$	0	0
$993$	74200.0	2.37126
$994$	0	0
$995$	70835.0i	2.25691i
$996$	0	0
$997$	17548.7i	0.557444i	0.960372	+	0.278722i	$0.0899108\pi$
$997$	17548.7i	0.557444i	−0.960372	+	0.278722i	$0.910089\pi$
$998$	0	0
$999$	−33262.3	−1.05343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.b.e.65.4	yes	4
3.2	odd	2		1152.4.d.j.577.1		4
4.3	odd	2	inner	128.4.b.e.65.2	yes	4
8.3	odd	2	inner	128.4.b.e.65.3	yes	4
8.5	even	2	inner	128.4.b.e.65.1	✓	4
12.11	even	2		1152.4.d.j.577.2		4
16.3	odd	4		256.4.a.n.1.4		4
16.5	even	4		256.4.a.n.1.3		4
16.11	odd	4		256.4.a.n.1.1		4
16.13	even	4		256.4.a.n.1.2		4
24.5	odd	2		1152.4.d.j.577.3		4
24.11	even	2		1152.4.d.j.577.4		4
48.5	odd	4		2304.4.a.bz.1.4		4
48.11	even	4		2304.4.a.bz.1.3		4
48.29	odd	4		2304.4.a.bz.1.2		4
48.35	even	4		2304.4.a.bz.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.b.e.65.1	✓	4	8.5	even	2	inner
128.4.b.e.65.2	yes	4	4.3	odd	2	inner
128.4.b.e.65.3	yes	4	8.3	odd	2	inner
128.4.b.e.65.4	yes	4	1.1	even	1	trivial
256.4.a.n.1.1		4	16.11	odd	4
256.4.a.n.1.2		4	16.13	even	4
256.4.a.n.1.3		4	16.5	even	4
256.4.a.n.1.4		4	16.3	odd	4
1152.4.d.j.577.1		4	3.2	odd	2
1152.4.d.j.577.2		4	12.11	even	2
1152.4.d.j.577.3		4	24.5	odd	2
1152.4.d.j.577.4		4	24.11	even	2
2304.4.a.bz.1.1		4	48.35	even	4
2304.4.a.bz.1.2		4	48.29	odd	4
2304.4.a.bz.1.3		4	48.11	even	4
2304.4.a.bz.1.4		4	48.5	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.32456i q^{3} +17.8885i q^{5} -22.6274 q^{7} -13.0000 q^{9} +O(q^{10})\)	\(q+6.32456i q^{3} +17.8885i q^{5} -22.6274 q^{7} -13.0000 q^{9} -44.2719i q^{11} -17.8885i q^{13} -113.137 q^{15} +70.0000 q^{17} +82.2192i q^{19} -143.108i q^{21} -158.392 q^{23} -195.000 q^{25} +88.5438i q^{27} +125.220i q^{29} +280.000 q^{33} -404.772i q^{35} +375.659i q^{37} +113.137 q^{39} -182.000 q^{41} +132.816i q^{43} -232.551i q^{45} +316.784 q^{47} +169.000 q^{49} +442.719i q^{51} -125.220i q^{53} +791.960 q^{55} -520.000 q^{57} +82.2192i q^{59} -232.551i q^{61} +294.156 q^{63} +320.000 q^{65} -221.359i q^{67} -1001.76i q^{69} -113.137 q^{71} +910.000 q^{73} -1233.29i q^{75} +1001.76i q^{77} +678.823 q^{79} -911.000 q^{81} +714.675i q^{83} +1252.20i q^{85} -791.960 q^{87} -546.000 q^{89} +404.772i q^{91} -1470.78 q^{95} -490.000 q^{97} +575.535i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 52 q^{9}+O(q^{10}) \) 4 * q - 52 * q^9	\( 4 q - 52 q^{9} + 280 q^{17} - 780 q^{25} + 1120 q^{33} - 728 q^{41} + 676 q^{49} - 2080 q^{57} + 1280 q^{65} + 3640 q^{73} - 3644 q^{81} - 2184 q^{89} - 1960 q^{97}+O(q^{100}) \) 4 * q - 52 * q^9 + 280 * q^17 - 780 * q^25 + 1120 * q^33 - 728 * q^41 + 676 * q^49 - 2080 * q^57 + 1280 * q^65 + 3640 * q^73 - 3644 * q^81 - 2184 * q^89 - 1960 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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