LMFDB - Embedded newform 256.4.a.n.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,4,Mod(1,256)] 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("256.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.1044889615$
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} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.28825$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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
$3$	6.32456	1.21716	0.608581	−	0.793492i	$-0.291739\pi$
$3$	6.32456	1.21716	0.608581	+	0.793492i	$0.291739\pi$
$4$	0	0
$5$	−17.8885	−1.60000	−0.800000	−	0.600000i	$-0.795167\pi$
$5$	−17.8885	−1.60000	−0.800000	+	0.600000i	$0.795167\pi$
$6$	0	0
$7$	22.6274	1.22177	0.610883	−	0.791721i	$-0.290815\pi$
$7$	22.6274	1.22177	0.610883	+	0.791721i	$0.290815\pi$
$8$	0	0
$9$	13.0000	0.481481
$10$	0	0
$11$	44.2719	1.21350	0.606749	−	0.794894i	$-0.292473\pi$
$11$	44.2719	1.21350	0.606749	+	0.794894i	$0.292473\pi$
$12$	0	0
$13$	−17.8885	−0.381645	−0.190823	−	0.981625i	$-0.561116\pi$
$13$	−17.8885	−0.381645	−0.190823	+	0.981625i	$0.561116\pi$
$14$	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
$19$	82.2192	0.992757	0.496378	−	0.868106i	$-0.334663\pi$
$19$	82.2192	0.992757	0.496378	+	0.868106i	$0.334663\pi$
$20$	0	0
$21$	143.108	1.48709
$22$	0	0
$23$	158.392	1.43596	0.717978	−	0.696066i	$-0.245068\pi$
$23$	158.392	1.43596	0.717978	+	0.696066i	$0.245068\pi$
$24$	0	0
$25$	195.000	1.56000
$26$	0	0
$27$	−88.5438	−0.631121
$28$	0	0
$29$	125.220	0.801818	0.400909	−	0.916118i	$-0.368694\pi$
$29$	125.220	0.801818	0.400909	+	0.916118i	$0.368694\pi$
$30$	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
$35$	−404.772	−1.95483
$36$	0	0
$37$	−375.659	−1.66914	−0.834568	−	0.550905i	$-0.814283\pi$
$37$	−375.659	−1.66914	−0.834568	+	0.550905i	$0.814283\pi$
$38$	0	0
$39$	−113.137	−0.464524
$40$	0	0
$41$	182.000	0.693259	0.346630	−	0.938002i	$-0.387326\pi$
$41$	182.000	0.693259	0.346630	+	0.938002i	$0.387326\pi$
$42$	0	0
$43$	−132.816	−0.471028	−0.235514	−	0.971871i	$-0.575677\pi$
$43$	−132.816	−0.471028	−0.235514	+	0.971871i	$0.575677\pi$
$44$	0	0
$45$	−232.551	−0.770370
$46$	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
$51$	442.719	1.21555
$52$	0	0
$53$	125.220	0.324533	0.162267	−	0.986747i	$-0.448120\pi$
$53$	125.220	0.324533	0.162267	+	0.986747i	$0.448120\pi$
$54$	0	0
$55$	−791.960	−1.94160
$56$	0	0
$57$	520.000	1.20835
$58$	0	0
$59$	−82.2192	−0.181424	−0.0907121	−	0.995877i	$-0.528914\pi$
$59$	−82.2192	−0.181424	−0.0907121	+	0.995877i	$0.528914\pi$
$60$	0	0
$61$	−232.551	−0.488117	−0.244058	−	0.969761i	$-0.578479\pi$
$61$	−232.551	−0.488117	−0.244058	+	0.969761i	$0.578479\pi$
$62$	0	0
$63$	294.156	0.588258
$64$	0	0
$65$	320.000	0.610633
$66$	0	0
$67$	−221.359	−0.403632	−0.201816	−	0.979423i	$-0.564684\pi$
$67$	−221.359	−0.403632	−0.201816	+	0.979423i	$0.564684\pi$
$68$	0	0
$69$	1001.76	1.74779
$70$	0	0
$71$	113.137	0.189111	0.0945556	−	0.995520i	$-0.469857\pi$
$71$	113.137	0.189111	0.0945556	+	0.995520i	$0.469857\pi$
$72$	0	0
$73$	−910.000	−1.45901	−0.729503	−	0.683978i	$-0.760249\pi$
$73$	−910.000	−1.45901	−0.729503	+	0.683978i	$0.760249\pi$
$74$	0	0
$75$	1233.29	1.89877
$76$	0	0
$77$	1001.76	1.48261
$78$	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
$83$	714.675	0.945129	0.472565	−	0.881296i	$-0.343328\pi$
$83$	714.675	0.945129	0.472565	+	0.881296i	$0.343328\pi$
$84$	0	0
$85$	−1252.20	−1.59788
$86$	0	0
$87$	791.960	0.975942
$88$	0	0
$89$	546.000	0.650291	0.325145	−	0.945664i	$-0.394587\pi$
$89$	546.000	0.650291	0.325145	+	0.945664i	$0.394587\pi$
$90$	0	0
$91$	−404.772	−0.466281
$92$	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
$99$	575.535	0.584277
$100$	0	0
$101$	−232.551	−0.229106	−0.114553	−	0.993417i	$-0.536544\pi$
$101$	−232.551	−0.229106	−0.114553	+	0.993417i	$0.536544\pi$
$102$	0	0
$103$	−158.392	−0.151523	−0.0757613	−	0.997126i	$-0.524139\pi$
$103$	−158.392	−0.151523	−0.0757613	+	0.997126i	$0.524139\pi$
$104$	0	0
$105$	−2560.00	−2.37934
$106$	0	0
$107$	−1726.60	−1.55997	−0.779986	−	0.625797i	$-0.784774\pi$
$107$	−1726.60	−1.55997	−0.779986	+	0.625797i	$0.784774\pi$
$108$	0	0
$109$	−1377.42	−1.21039	−0.605196	−	0.796077i	$-0.706905\pi$
$109$	−1377.42	−1.21039	−0.605196	+	0.796077i	$0.706905\pi$
$110$	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
$115$	−2833.40	−2.29753
$116$	0	0
$117$	−232.551	−0.183755
$118$	0	0
$119$	1583.92	1.22015
$120$	0	0
$121$	629.000	0.472577
$122$	0	0
$123$	1151.07	0.843808
$124$	0	0
$125$	−1252.20	−0.896000
$126$	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
$131$	−170.763	−0.113890	−0.0569452	−	0.998377i	$-0.518136\pi$
$131$	−170.763	−0.113890	−0.0569452	+	0.998377i	$0.518136\pi$
$132$	0	0
$133$	1860.41	1.21292
$134$	0	0
$135$	1583.92	1.00979
$136$	0	0
$137$	−1930.00	−1.20358	−0.601792	−	0.798653i	$-0.705546\pi$
$137$	−1930.00	−1.20358	−0.601792	+	0.798653i	$0.705546\pi$
$138$	0	0
$139$	−1144.74	−0.698532	−0.349266	−	0.937024i	$-0.613569\pi$
$139$	−1144.74	−0.698532	−0.349266	+	0.937024i	$0.613569\pi$
$140$	0	0
$141$	2003.52	1.19664
$142$	0	0
$143$	−791.960	−0.463126
$144$	0	0
$145$	−2240.00	−1.28291
$146$	0	0
$147$	1068.85	0.599709
$148$	0	0
$149$	1627.86	0.895029	0.447514	−	0.894277i	$-0.352309\pi$
$149$	1627.86	0.895029	0.447514	+	0.894277i	$0.352309\pi$
$150$	0	0
$151$	−2375.88	−1.28044	−0.640219	−	0.768192i	$-0.721157\pi$
$151$	−2375.88	−1.28044	−0.640219	+	0.768192i	$0.721157\pi$
$152$	0	0
$153$	910.000	0.480844
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3488.27	1.77321	0.886605	−	0.462528i	$-0.153057\pi$
$157$	3488.27	1.77321	0.886605	+	0.462528i	$0.153057\pi$
$158$	0	0
$159$	791.960	0.395009
$160$	0	0
$161$	3584.00	1.75440
$162$	0	0
$163$	3497.48	1.68064	0.840318	−	0.542094i	$-0.182368\pi$
$163$	3497.48	1.68064	0.840318	+	0.542094i	$0.182368\pi$
$164$	0	0
$165$	−5008.79	−2.36324
$166$	0	0
$167$	−2059.09	−0.954117	−0.477059	−	0.878872i	$-0.658297\pi$
$167$	−2059.09	−0.954117	−0.477059	+	0.878872i	$0.658297\pi$
$168$	0	0
$169$	−1877.00	−0.854347
$170$	0	0
$171$	1068.85	0.477994
$172$	0	0
$173$	−2021.41	−0.888350	−0.444175	−	0.895940i	$-0.646503\pi$
$173$	−2021.41	−0.888350	−0.444175	+	0.895940i	$0.646503\pi$
$174$	0	0
$175$	4412.35	1.90595
$176$	0	0
$177$	−520.000	−0.220823
$178$	0	0
$179$	−3586.02	−1.49739	−0.748693	−	0.662917i	$-0.769318\pi$
$179$	−3586.02	−1.49739	−0.748693	+	0.662917i	$0.769318\pi$
$180$	0	0
$181$	2486.51	1.02111	0.510554	−	0.859846i	$-0.329440\pi$
$181$	2486.51	1.02111	0.510554	+	0.859846i	$0.329440\pi$
$182$	0	0
$183$	−1470.78	−0.594117
$184$	0	0
$185$	6720.00	2.67062
$186$	0	0
$187$	3099.03	1.21189
$188$	0	0
$189$	−2003.52	−0.771082
$190$	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
$195$	2023.86	0.743238
$196$	0	0
$197$	−1878.30	−0.679305	−0.339653	−	0.940551i	$-0.610310\pi$
$197$	−1878.30	−0.679305	−0.339653	+	0.940551i	$0.610310\pi$
$198$	0	0
$199$	−3959.80	−1.41057	−0.705283	−	0.708926i	$-0.749180\pi$
$199$	−3959.80	−1.41057	−0.705283	+	0.708926i	$0.749180\pi$
$200$	0	0
$201$	−1400.00	−0.491286
$202$	0	0
$203$	2833.40	0.979634
$204$	0	0
$205$	−3255.71	−1.10921
$206$	0	0
$207$	2059.09	0.691386
$208$	0	0
$209$	3640.00	1.20471
$210$	0	0
$211$	2789.13	0.910007	0.455004	−	0.890490i	$-0.349638\pi$
$211$	2789.13	0.910007	0.455004	+	0.890490i	$0.349638\pi$
$212$	0	0
$213$	715.542	0.230179
$214$	0	0
$215$	2375.88	0.753645
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5755.35	−1.77585
$220$	0	0
$221$	−1252.20	−0.381140
$222$	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
$227$	−2219.92	−0.649080	−0.324540	−	0.945872i	$-0.605210\pi$
$227$	−2219.92	−0.649080	−0.324540	+	0.945872i	$0.605210\pi$
$228$	0	0
$229$	4275.36	1.23373	0.616864	−	0.787069i	$-0.288403\pi$
$229$	4275.36	1.23373	0.616864	+	0.787069i	$0.288403\pi$
$230$	0	0
$231$	6335.68	1.80458
$232$	0	0
$233$	5010.00	1.40865	0.704326	−	0.709876i	$-0.251249\pi$
$233$	5010.00	1.40865	0.704326	+	0.709876i	$0.251249\pi$
$234$	0	0
$235$	−5666.80	−1.57303
$236$	0	0
$237$	4293.25	1.17669
$238$	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
$243$	−3370.99	−0.889913
$244$	0	0
$245$	−3023.16	−0.788338
$246$	0	0
$247$	−1470.78	−0.378881
$248$	0	0
$249$	4520.00	1.15037
$250$	0	0
$251$	1233.29	0.310137	0.155069	−	0.987904i	$-0.450440\pi$
$251$	1233.29	0.310137	0.155069	+	0.987904i	$0.450440\pi$
$252$	0	0
$253$	7012.31	1.74253
$254$	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
$259$	−8500.20	−2.03929
$260$	0	0
$261$	1627.86	0.386061
$262$	0	0
$263$	2647.41	0.620708	0.310354	−	0.950621i	$-0.399552\pi$
$263$	2647.41	0.620708	0.310354	+	0.950621i	$0.399552\pi$
$264$	0	0
$265$	−2240.00	−0.519253
$266$	0	0
$267$	3453.21	0.791509
$268$	0	0
$269$	3488.27	0.790644	0.395322	−	0.918543i	$-0.370633\pi$
$269$	3488.27	0.790644	0.395322	+	0.918543i	$0.370633\pi$
$270$	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
$275$	8633.02	1.89306
$276$	0	0
$277$	−4883.57	−1.05930	−0.529649	−	0.848217i	$-0.677676\pi$
$277$	−4883.57	−1.05930	−0.529649	+	0.848217i	$0.677676\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1118.00	−0.237346	−0.118673	−	0.992933i	$-0.537864\pi$
$281$	−1118.00	−0.237346	−0.118673	+	0.992933i	$0.537864\pi$
$282$	0	0
$283$	2118.73	0.445036	0.222518	−	0.974929i	$-0.428572\pi$
$283$	2118.73	0.445036	0.222518	+	0.974929i	$0.428572\pi$
$284$	0	0
$285$	−9302.04	−1.93335
$286$	0	0
$287$	4118.19	0.847000
$288$	0	0
$289$	−13.0000	−0.00264604
$290$	0	0
$291$	−3099.03	−0.624290
$292$	0	0
$293$	4275.36	0.852455	0.426227	−	0.904616i	$-0.359842\pi$
$293$	4275.36	0.852455	0.426227	+	0.904616i	$0.359842\pi$
$294$	0	0
$295$	1470.78	0.290279
$296$	0	0
$297$	−3920.00	−0.765864
$298$	0	0
$299$	−2833.40	−0.548026
$300$	0	0
$301$	−3005.28	−0.575486
$302$	0	0
$303$	−1470.78	−0.278859
$304$	0	0
$305$	4160.00	0.780987
$306$	0	0
$307$	7975.26	1.48265	0.741323	−	0.671148i	$-0.234199\pi$
$307$	7975.26	1.48265	0.741323	+	0.671148i	$0.234199\pi$
$308$	0	0
$309$	−1001.76	−0.184427
$310$	0	0
$311$	10295.5	1.87718	0.938590	−	0.345035i	$-0.112133\pi$
$311$	10295.5	1.87718	0.938590	+	0.345035i	$0.112133\pi$
$312$	0	0
$313$	−2170.00	−0.391871	−0.195936	−	0.980617i	$-0.562774\pi$
$313$	−2170.00	−0.391871	−0.195936	+	0.980617i	$0.562774\pi$
$314$	0	0
$315$	−5262.03	−0.941212
$316$	0	0
$317$	−4883.57	−0.865264	−0.432632	−	0.901571i	$-0.642415\pi$
$317$	−4883.57	−0.865264	−0.432632	+	0.901571i	$0.642415\pi$
$318$	0	0
$319$	5543.72	0.973005
$320$	0	0
$321$	−10920.0	−1.89874
$322$	0	0
$323$	5755.35	0.991443
$324$	0	0
$325$	−3488.27	−0.595367
$326$	0	0
$327$	−8711.56	−1.47324
$328$	0	0
$329$	7168.00	1.20117
$330$	0	0
$331$	11732.1	1.94819	0.974096	−	0.226133i	$-0.0726085\pi$
$331$	11732.1	1.94819	0.974096	+	0.226133i	$0.0726085\pi$
$332$	0	0
$333$	−4883.57	−0.803658
$334$	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
$339$	−5755.35	−0.922087
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3937.17	−0.619788
$344$	0	0
$345$	−17920.0	−2.79646
$346$	0	0
$347$	4117.29	0.636967	0.318483	−	0.947928i	$-0.396826\pi$
$347$	4117.29	0.636967	0.318483	+	0.947928i	$0.396826\pi$
$348$	0	0
$349$	4275.36	0.655745	0.327872	−	0.944722i	$-0.393668\pi$
$349$	4275.36	0.655745	0.327872	+	0.944722i	$0.393668\pi$
$350$	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
$355$	−2023.86	−0.302578
$356$	0	0
$357$	10017.6	1.48512
$358$	0	0
$359$	−10295.5	−1.51358	−0.756789	−	0.653659i	$-0.773233\pi$
$359$	−10295.5	−1.51358	−0.756789	+	0.653659i	$0.773233\pi$
$360$	0	0
$361$	−99.0000	−0.0144336
$362$	0	0
$363$	3978.15	0.575202
$364$	0	0
$365$	16278.6	2.33441
$366$	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
$371$	2833.40	0.396504
$372$	0	0
$373$	−4883.57	−0.677914	−0.338957	−	0.940802i	$-0.610074\pi$
$373$	−4883.57	−0.677914	−0.338957	+	0.940802i	$0.610074\pi$
$374$	0	0
$375$	−7919.60	−1.09058
$376$	0	0
$377$	−2240.00	−0.306010
$378$	0	0
$379$	−3674.57	−0.498021	−0.249010	−	0.968501i	$-0.580105\pi$
$379$	−3674.57	−0.498021	−0.249010	+	0.968501i	$0.580105\pi$
$380$	0	0
$381$	12021.1	1.61643
$382$	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
$387$	−1726.60	−0.226791
$388$	0	0
$389$	−11395.0	−1.48522	−0.742609	−	0.669726i	$-0.766412\pi$
$389$	−11395.0	−1.48522	−0.742609	+	0.669726i	$0.766412\pi$
$390$	0	0
$391$	11087.4	1.43406
$392$	0	0
$393$	−1080.00	−0.138623
$394$	0	0
$395$	−12143.1	−1.54681
$396$	0	0
$397$	−13255.4	−1.67574	−0.837872	−	0.545867i	$-0.816200\pi$
$397$	−13255.4	−1.67574	−0.837872	+	0.545867i	$0.816200\pi$
$398$	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
$405$	16296.5	1.99945
$406$	0	0
$407$	−16631.2	−2.02549
$408$	0	0
$409$	−13594.0	−1.64347	−0.821736	−	0.569868i	$-0.806994\pi$
$409$	−13594.0	−1.64347	−0.821736	+	0.569868i	$0.806994\pi$
$410$	0	0
$411$	−12206.4	−1.46496
$412$	0	0
$413$	−1860.41	−0.221658
$414$	0	0
$415$	−12784.5	−1.51221
$416$	0	0
$417$	−7240.00	−0.850226
$418$	0	0
$419$	−4775.04	−0.556744	−0.278372	−	0.960473i	$-0.589795\pi$
$419$	−4775.04	−0.556744	−0.278372	+	0.960473i	$0.589795\pi$
$420$	0	0
$421$	−7888.85	−0.913252	−0.456626	−	0.889659i	$-0.650942\pi$
$421$	−7888.85	−0.913252	−0.456626	+	0.889659i	$0.650942\pi$
$422$	0	0
$423$	4118.19	0.473365
$424$	0	0
$425$	13650.0	1.55793
$426$	0	0
$427$	−5262.03	−0.596364
$428$	0	0
$429$	−5008.79	−0.563699
$430$	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
$435$	−14167.0	−1.56151
$436$	0	0
$437$	13022.9	1.42556
$438$	0	0
$439$	10295.5	1.11931	0.559654	−	0.828726i	$-0.310934\pi$
$439$	10295.5	1.11931	0.559654	+	0.828726i	$0.310934\pi$
$440$	0	0
$441$	2197.00	0.237231
$442$	0	0
$443$	17753.0	1.90400	0.952000	−	0.306099i	$-0.0990238\pi$
$443$	17753.0	1.90400	0.952000	+	0.306099i	$0.0990238\pi$
$444$	0	0
$445$	−9767.14	−1.04047
$446$	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
$451$	8057.48	0.841268
$452$	0	0
$453$	−15026.4	−1.55850
$454$	0	0
$455$	7240.77	0.746050
$456$	0	0
$457$	−2730.00	−0.279440	−0.139720	−	0.990191i	$-0.544620\pi$
$457$	−2730.00	−0.279440	−0.139720	+	0.990191i	$0.544620\pi$
$458$	0	0
$459$	−6198.06	−0.630285
$460$	0	0
$461$	−10250.1	−1.03557	−0.517784	−	0.855512i	$-0.673243\pi$
$461$	−10250.1	−1.03557	−0.517784	+	0.855512i	$0.673243\pi$
$462$	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
$467$	−13730.6	−1.36055	−0.680275	−	0.732957i	$-0.738140\pi$
$467$	−13730.6	−1.36055	−0.680275	+	0.732957i	$0.738140\pi$
$468$	0	0
$469$	−5008.79	−0.493144
$470$	0	0
$471$	22061.7	2.15828
$472$	0	0
$473$	−5880.00	−0.571591
$474$	0	0
$475$	16032.7	1.54870
$476$	0	0
$477$	1627.86	0.156257
$478$	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
$483$	22667.2	2.13539
$484$	0	0
$485$	8765.39	0.820651
$486$	0	0
$487$	−2059.09	−0.191594	−0.0957972	−	0.995401i	$-0.530540\pi$
$487$	−2059.09	−0.191594	−0.0957972	+	0.995401i	$0.530540\pi$
$488$	0	0
$489$	22120.0	2.04561
$490$	0	0
$491$	5888.16	0.541200	0.270600	−	0.962692i	$-0.412778\pi$
$491$	5888.16	0.541200	0.270600	+	0.962692i	$0.412778\pi$
$492$	0	0
$493$	8765.39	0.800757
$494$	0	0
$495$	−10295.5	−0.934843
$496$	0	0
$497$	2560.00	0.231050
$498$	0	0
$499$	10935.2	0.981012	0.490506	−	0.871438i	$-0.336812\pi$
$499$	10935.2	0.981012	0.490506	+	0.871438i	$0.336812\pi$
$500$	0	0
$501$	−13022.9	−1.16131
$502$	0	0
$503$	−14413.7	−1.27768	−0.638840	−	0.769339i	$-0.720586\pi$
$503$	−14413.7	−1.27768	−0.638840	+	0.769339i	$0.720586\pi$
$504$	0	0
$505$	4160.00	0.366569
$506$	0	0
$507$	−11871.2	−1.03988
$508$	0	0
$509$	12790.3	1.11379	0.556896	−	0.830582i	$-0.311992\pi$
$509$	12790.3	1.11379	0.556896	+	0.830582i	$0.311992\pi$
$510$	0	0
$511$	−20590.9	−1.78256
$512$	0	0
$513$	−7280.00	−0.626549
$514$	0	0
$515$	2833.40	0.242436
$516$	0	0
$517$	14024.6	1.19304
$518$	0	0
$519$	−12784.5	−1.08127
$520$	0	0
$521$	16758.0	1.40918	0.704589	−	0.709616i	$-0.251132\pi$
$521$	16758.0	1.40918	0.704589	+	0.709616i	$0.251132\pi$
$522$	0	0
$523$	−16197.2	−1.35421	−0.677107	−	0.735885i	$-0.736766\pi$
$523$	−16197.2	−1.35421	−0.677107	+	0.735885i	$0.736766\pi$
$524$	0	0
$525$	27906.1	2.31985
$526$	0	0
$527$	0	0
$528$	0	0
$529$	12921.0	1.06197
$530$	0	0
$531$	−1068.85	−0.0873524
$532$	0	0
$533$	−3255.71	−0.264579
$534$	0	0
$535$	30886.4	2.49596
$536$	0	0
$537$	−22680.0	−1.82256
$538$	0	0
$539$	7481.95	0.597904
$540$	0	0
$541$	−375.659	−0.0298537	−0.0149269	−	0.999889i	$-0.504752\pi$
$541$	−375.659	−0.0298537	−0.0149269	+	0.999889i	$0.504752\pi$
$542$	0	0
$543$	15726.1	1.24285
$544$	0	0
$545$	24640.0	1.93663
$546$	0	0
$547$	−1638.06	−0.128041	−0.0640205	−	0.997949i	$-0.520392\pi$
$547$	−1638.06	−0.128041	−0.0640205	+	0.997949i	$0.520392\pi$
$548$	0	0
$549$	−3023.16	−0.235019
$550$	0	0
$551$	10295.5	0.796011
$552$	0	0
$553$	15360.0	1.18115
$554$	0	0
$555$	42501.0	3.25057
$556$	0	0
$557$	−19909.9	−1.51456	−0.757282	−	0.653089i	$-0.773473\pi$
$557$	−19909.9	−1.51456	−0.757282	+	0.653089i	$0.773473\pi$
$558$	0	0
$559$	2375.88	0.179766
$560$	0	0
$561$	19600.0	1.47507
$562$	0	0
$563$	790.569	0.0591803	0.0295902	−	0.999562i	$-0.490580\pi$
$563$	790.569	0.0591803	0.0295902	+	0.999562i	$0.490580\pi$
$564$	0	0
$565$	16278.6	1.21211
$566$	0	0
$567$	−20613.6	−1.52679
$568$	0	0
$569$	20454.0	1.50699	0.753494	−	0.657455i	$-0.228367\pi$
$569$	20454.0	1.50699	0.753494	+	0.657455i	$0.228367\pi$
$570$	0	0
$571$	−5622.53	−0.412076	−0.206038	−	0.978544i	$-0.566057\pi$
$571$	−5622.53	−0.412076	−0.206038	+	0.978544i	$0.566057\pi$
$572$	0	0
$573$	−14310.8	−1.04336
$574$	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
$579$	3984.47	0.285991
$580$	0	0
$581$	16171.2	1.15473
$582$	0	0
$583$	5543.72	0.393820
$584$	0	0
$585$	4160.00	0.294008
$586$	0	0
$587$	−15843.0	−1.11399	−0.556994	−	0.830516i	$-0.688045\pi$
$587$	−15843.0	−1.11399	−0.556994	+	0.830516i	$0.688045\pi$
$588$	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
$595$	−28334.0	−1.95224
$596$	0	0
$597$	−25044.0	−1.71689
$598$	0	0
$599$	−19120.2	−1.30422	−0.652111	−	0.758124i	$-0.726116\pi$
$599$	−19120.2	−1.30422	−0.652111	+	0.758124i	$0.726116\pi$
$600$	0	0
$601$	−4382.00	−0.297413	−0.148707	−	0.988881i	$-0.547511\pi$
$601$	−4382.00	−0.297413	−0.148707	+	0.988881i	$0.547511\pi$
$602$	0	0
$603$	−2877.67	−0.194341
$604$	0	0
$605$	−11251.9	−0.756123
$606$	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
$611$	−5666.80	−0.375212
$612$	0	0
$613$	2128.74	0.140259	0.0701296	−	0.997538i	$-0.477659\pi$
$613$	2128.74	0.140259	0.0701296	+	0.997538i	$0.477659\pi$
$614$	0	0
$615$	−20590.9	−1.35009
$616$	0	0
$617$	−15470.0	−1.00940	−0.504699	−	0.863295i	$-0.668397\pi$
$617$	−15470.0	−1.00940	−0.504699	+	0.863295i	$0.668397\pi$
$618$	0	0
$619$	−19486.0	−1.26528	−0.632639	−	0.774447i	$-0.718028\pi$
$619$	−19486.0	−1.26528	−0.632639	+	0.774447i	$0.718028\pi$
$620$	0	0
$621$	−14024.6	−0.906262
$622$	0	0
$623$	12354.6	0.794503
$624$	0	0
$625$	−1975.00	−0.126400
$626$	0	0
$627$	23021.4	1.46632
$628$	0	0
$629$	−26296.2	−1.66693
$630$	0	0
$631$	−6448.81	−0.406851	−0.203426	−	0.979090i	$-0.565208\pi$
$631$	−6448.81	−0.406851	−0.203426	+	0.979090i	$0.565208\pi$
$632$	0	0
$633$	17640.0	1.10763
$634$	0	0
$635$	−34000.8	−2.12485
$636$	0	0
$637$	−3023.16	−0.188041
$638$	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
$643$	−23293.3	−1.42862	−0.714308	−	0.699832i	$-0.753258\pi$
$643$	−23293.3	−1.42862	−0.714308	+	0.699832i	$0.753258\pi$
$644$	0	0
$645$	15026.4	0.917307
$646$	0	0
$647$	−2059.09	−0.125118	−0.0625590	−	0.998041i	$-0.519926\pi$
$647$	−2059.09	−0.125118	−0.0625590	+	0.998041i	$0.519926\pi$
$648$	0	0
$649$	−3640.00	−0.220158
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14650.7	0.877989	0.438995	−	0.898490i	$-0.355335\pi$
$653$	14650.7	0.877989	0.438995	+	0.898490i	$0.355335\pi$
$654$	0	0
$655$	3054.70	0.182225
$656$	0	0
$657$	−11830.0	−0.702484
$658$	0	0
$659$	10935.2	0.646393	0.323197	−	0.946332i	$-0.395242\pi$
$659$	10935.2	0.646393	0.323197	+	0.946332i	$0.395242\pi$
$660$	0	0
$661$	3774.48	0.222103	0.111052	−	0.993815i	$-0.464578\pi$
$661$	3774.48	0.222103	0.111052	+	0.993815i	$0.464578\pi$
$662$	0	0
$663$	−7919.60	−0.463909
$664$	0	0
$665$	−33280.0	−1.94067
$666$	0	0
$667$	19833.8	1.15138
$668$	0	0
$669$	16028.1	0.926283
$670$	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
$675$	−17266.0	−0.984548
$676$	0	0
$677$	−27566.2	−1.56493	−0.782464	−	0.622695i	$-0.786038\pi$
$677$	−27566.2	−1.56493	−0.782464	+	0.622695i	$0.786038\pi$
$678$	0	0
$679$	−11087.4	−0.626652
$680$	0	0
$681$	−14040.0	−0.790035
$682$	0	0
$683$	2346.41	0.131454	0.0657269	−	0.997838i	$-0.479063\pi$
$683$	2346.41	0.131454	0.0657269	+	0.997838i	$0.479063\pi$
$684$	0	0
$685$	34524.9	1.92573
$686$	0	0
$687$	27039.8	1.50165
$688$	0	0
$689$	−2240.00	−0.123857
$690$	0	0
$691$	10277.4	0.565804	0.282902	−	0.959149i	$-0.408703\pi$
$691$	10277.4	0.565804	0.282902	+	0.959149i	$0.408703\pi$
$692$	0	0
$693$	13022.9	0.713849
$694$	0	0
$695$	20477.8	1.11765
$696$	0	0
$697$	12740.0	0.692341
$698$	0	0
$699$	31686.0	1.71456
$700$	0	0
$701$	626.099	0.0337339	0.0168669	−	0.999858i	$-0.494631\pi$
$701$	626.099	0.0337339	0.0168669	+	0.999858i	$0.494631\pi$
$702$	0	0
$703$	−30886.4	−1.65705
$704$	0	0
$705$	−35840.0	−1.91463
$706$	0	0
$707$	−5262.03	−0.279914
$708$	0	0
$709$	14650.7	0.776050	0.388025	−	0.921649i	$-0.373157\pi$
$709$	14650.7	0.776050	0.388025	+	0.921649i	$0.373157\pi$
$710$	0	0
$711$	8824.69	0.465474
$712$	0	0
$713$	0	0
$714$	0	0
$715$	14167.0	0.741001
$716$	0	0
$717$	−10017.6	−0.521776
$718$	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
$723$	10359.6	0.532889
$724$	0	0
$725$	24417.9	1.25084
$726$	0	0
$727$	−4276.58	−0.218170	−0.109085	−	0.994032i	$-0.534792\pi$
$727$	−4276.58	−0.218170	−0.109085	+	0.994032i	$0.534792\pi$
$728$	0	0
$729$	3277.00	0.166489
$730$	0	0
$731$	−9297.10	−0.470404
$732$	0	0
$733$	9284.15	0.467828	0.233914	−	0.972257i	$-0.424847\pi$
$733$	9284.15	0.467828	0.233914	+	0.972257i	$0.424847\pi$
$734$	0	0
$735$	−19120.2	−0.959535
$736$	0	0
$737$	−9800.00	−0.489807
$738$	0	0
$739$	1726.60	0.0859461	0.0429730	−	0.999076i	$-0.486317\pi$
$739$	1726.60	0.0859461	0.0429730	+	0.999076i	$0.486317\pi$
$740$	0	0
$741$	−9302.04	−0.461159
$742$	0	0
$743$	28352.2	1.39992	0.699959	−	0.714183i	$-0.253201\pi$
$743$	28352.2	1.39992	0.699959	+	0.714183i	$0.253201\pi$
$744$	0	0
$745$	−29120.0	−1.43205
$746$	0	0
$747$	9290.77	0.455062
$748$	0	0
$749$	−39068.6	−1.90592
$750$	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
$755$	42501.0	2.04870
$756$	0	0
$757$	22664.8	1.08820	0.544099	−	0.839021i	$-0.316872\pi$
$757$	22664.8	1.08820	0.544099	+	0.839021i	$0.316872\pi$
$758$	0	0
$759$	44349.7	2.12094
$760$	0	0
$761$	−7098.00	−0.338111	−0.169055	−	0.985607i	$-0.554072\pi$
$761$	−7098.00	−0.338111	−0.169055	+	0.985607i	$0.554072\pi$
$762$	0	0
$763$	−31167.4	−1.47882
$764$	0	0
$765$	−16278.6	−0.769350
$766$	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
$771$	11953.4	0.558355
$772$	0	0
$773$	−6529.32	−0.303808	−0.151904	−	0.988395i	$-0.548540\pi$
$773$	−6529.32	−0.303808	−0.151904	+	0.988395i	$0.548540\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−53760.0	−2.48215
$778$	0	0
$779$	14963.9	0.688238
$780$	0	0
$781$	5008.79	0.229486
$782$	0	0
$783$	−11087.4	−0.506044
$784$	0	0
$785$	−62400.0	−2.83714
$786$	0	0
$787$	2384.36	0.107996	0.0539982	−	0.998541i	$-0.482803\pi$
$787$	2384.36	0.107996	0.0539982	+	0.998541i	$0.482803\pi$
$788$	0	0
$789$	16743.7	0.755502
$790$	0	0
$791$	−20590.9	−0.925575
$792$	0	0
$793$	4160.00	0.186287
$794$	0	0
$795$	−14167.0	−0.632015
$796$	0	0
$797$	29534.0	1.31261	0.656303	−	0.754497i	$-0.272119\pi$
$797$	29534.0	1.31261	0.656303	+	0.754497i	$0.272119\pi$
$798$	0	0
$799$	22174.9	0.981840
$800$	0	0
$801$	7098.00	0.313103
$802$	0	0
$803$	−40287.4	−1.77050
$804$	0	0
$805$	−64112.5	−2.80704
$806$	0	0
$807$	22061.7	0.962342
$808$	0	0
$809$	10934.0	0.475178	0.237589	−	0.971366i	$-0.423643\pi$
$809$	10934.0	0.475178	0.237589	+	0.971366i	$0.423643\pi$
$810$	0	0
$811$	17348.3	0.751146	0.375573	−	0.926793i	$-0.377446\pi$
$811$	17348.3	0.751146	0.375573	+	0.926793i	$0.377446\pi$
$812$	0	0
$813$	−50087.9	−2.16071
$814$	0	0
$815$	−62564.8	−2.68902
$816$	0	0
$817$	−10920.0	−0.467616
$818$	0	0
$819$	−5262.03	−0.224506
$820$	0	0
$821$	−14901.2	−0.633440	−0.316720	−	0.948519i	$-0.602581\pi$
$821$	−14901.2	−0.633440	−0.316720	+	0.948519i	$0.602581\pi$
$822$	0	0
$823$	−17943.5	−0.759991	−0.379995	−	0.924988i	$-0.624074\pi$
$823$	−17943.5	−0.759991	−0.379995	+	0.924988i	$0.624074\pi$
$824$	0	0
$825$	54600.0	2.30416
$826$	0	0
$827$	25899.1	1.08899	0.544497	−	0.838763i	$-0.316721\pi$
$827$	25899.1	1.08899	0.544497	+	0.838763i	$0.316721\pi$
$828$	0	0
$829$	−25276.5	−1.05897	−0.529487	−	0.848318i	$-0.677616\pi$
$829$	−25276.5	−1.05897	−0.529487	+	0.848318i	$0.677616\pi$
$830$	0	0
$831$	−30886.4	−1.28934
$832$	0	0
$833$	11830.0	0.492059
$834$	0	0
$835$	36834.2	1.52659
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10295.5	−0.423646	−0.211823	−	0.977308i	$-0.567940\pi$
$839$	−10295.5	−0.423646	−0.211823	+	0.977308i	$0.567940\pi$
$840$	0	0
$841$	−8709.00	−0.357087
$842$	0	0
$843$	−7070.85	−0.288889
$844$	0	0
$845$	33576.8	1.36695
$846$	0	0
$847$	14232.6	0.577378
$848$	0	0
$849$	13400.0	0.541681
$850$	0	0
$851$	−59501.4	−2.39681
$852$	0	0
$853$	−21770.4	−0.873860	−0.436930	−	0.899495i	$-0.643934\pi$
$853$	−21770.4	−0.873860	−0.436930	+	0.899495i	$0.643934\pi$
$854$	0	0
$855$	−19120.2	−0.764791
$856$	0	0
$857$	−24570.0	−0.979341	−0.489670	−	0.871908i	$-0.662883\pi$
$857$	−24570.0	−0.979341	−0.489670	+	0.871908i	$0.662883\pi$
$858$	0	0
$859$	5837.56	0.231869	0.115934	−	0.993257i	$-0.463014\pi$
$859$	5837.56	0.231869	0.115934	+	0.993257i	$0.463014\pi$
$860$	0	0
$861$	26045.7	1.03094
$862$	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
$867$	−82.2192	−0.00322066
$868$	0	0
$869$	30052.8	1.17315
$870$	0	0
$871$	3959.80	0.154044
$872$	0	0
$873$	−6370.00	−0.246955
$874$	0	0
$875$	−28334.0	−1.09470
$876$	0	0
$877$	43701.7	1.68267	0.841335	−	0.540514i	$-0.181770\pi$
$877$	43701.7	1.68267	0.841335	+	0.540514i	$0.181770\pi$
$878$	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
$883$	−27492.8	−1.04780	−0.523900	−	0.851780i	$-0.675524\pi$
$883$	−27492.8	−1.04780	−0.523900	+	0.851780i	$0.675524\pi$
$884$	0	0
$885$	9302.04	0.353316
$886$	0	0
$887$	43241.0	1.63686	0.818428	−	0.574610i	$-0.194846\pi$
$887$	43241.0	1.63686	0.818428	+	0.574610i	$0.194846\pi$
$888$	0	0
$889$	43008.0	1.62254
$890$	0	0
$891$	−40331.7	−1.51646
$892$	0	0
$893$	26045.7	0.976021
$894$	0	0
$895$	64148.7	2.39582
$896$	0	0
$897$	−17920.0	−0.667036
$898$	0	0
$899$	0	0
$900$	0	0
$901$	8765.39	0.324104
$902$	0	0
$903$	−19007.0	−0.700459
$904$	0	0
$905$	−44480.0	−1.63377
$906$	0	0
$907$	−40154.6	−1.47002	−0.735012	−	0.678054i	$-0.762823\pi$
$907$	−40154.6	−1.47002	−0.735012	+	0.678054i	$0.762823\pi$
$908$	0	0
$909$	−3023.16	−0.110310
$910$	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
$915$	26310.2	0.950587
$916$	0	0
$917$	−3863.93	−0.139147
$918$	0	0
$919$	−1470.78	−0.0527928	−0.0263964	−	0.999652i	$-0.508403\pi$
$919$	−1470.78	−0.0527928	−0.0263964	+	0.999652i	$0.508403\pi$
$920$	0	0
$921$	50440.0	1.80462
$922$	0	0
$923$	−2023.86	−0.0721734
$924$	0	0
$925$	−73253.6	−2.60385
$926$	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
$931$	13895.0	0.489143
$932$	0	0
$933$	65114.3	2.28483
$934$	0	0
$935$	−55437.2	−1.93903
$936$	0	0
$937$	13650.0	0.475908	0.237954	−	0.971276i	$-0.423523\pi$
$937$	13650.0	0.475908	0.237954	+	0.971276i	$0.423523\pi$
$938$	0	0
$939$	−13724.3	−0.476970
$940$	0	0
$941$	−6529.32	−0.226195	−0.113098	−	0.993584i	$-0.536077\pi$
$941$	−6529.32	−0.226195	−0.113098	+	0.993584i	$0.536077\pi$
$942$	0	0
$943$	28827.3	0.995490
$944$	0	0
$945$	35840.0	1.23373
$946$	0	0
$947$	−11555.0	−0.396500	−0.198250	−	0.980151i	$-0.563526\pi$
$947$	−11555.0	−0.396500	−0.198250	+	0.980151i	$0.563526\pi$
$948$	0	0
$949$	16278.6	0.556823
$950$	0	0
$951$	−30886.4	−1.05317
$952$	0	0
$953$	10470.0	0.355883	0.177942	−	0.984041i	$-0.443056\pi$
$953$	10470.0	0.355883	0.177942	+	0.984041i	$0.443056\pi$
$954$	0	0
$955$	40477.2	1.37153
$956$	0	0
$957$	35061.5	1.18430
$958$	0	0
$959$	−43670.9	−1.47050
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	−22445.8	−0.751098
$964$	0	0
$965$	−11269.8	−0.375945
$966$	0	0
$967$	−40706.7	−1.35371	−0.676856	−	0.736115i	$-0.736658\pi$
$967$	−40706.7	−1.35371	−0.676856	+	0.736115i	$0.736658\pi$
$968$	0	0
$969$	36400.0	1.20675
$970$	0	0
$971$	49502.3	1.63605	0.818025	−	0.575183i	$-0.195069\pi$
$971$	49502.3	1.63605	0.818025	+	0.575183i	$0.195069\pi$
$972$	0	0
$973$	−25902.6	−0.853443
$974$	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
$979$	24172.5	0.789127
$980$	0	0
$981$	−17906.4	−0.582781
$982$	0	0
$983$	22333.3	0.724639	0.362320	−	0.932054i	$-0.381985\pi$
$983$	22333.3	0.724639	0.362320	+	0.932054i	$0.381985\pi$
$984$	0	0
$985$	33600.0	1.08689
$986$	0	0
$987$	45334.4	1.46202
$988$	0	0
$989$	−21036.9	−0.676376
$990$	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
$995$	70835.0	2.25691
$996$	0	0
$997$	−17548.7	−0.557444	−0.278722	−	0.960372i	$-0.589911\pi$
$997$	−17548.7	−0.557444	−0.278722	+	0.960372i	$0.589911\pi$
$998$	0	0
$999$	33262.3	1.05343

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

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