LMFDB - Embedded newform 2304.4.a.cd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2304,4,Mod(1,2304)] 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("2304.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.940400653$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.1439868559360000.7
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 84x^{6} + 1807x^{4} - 13356x^{2} + 29241 $ x^8 - 84x^6 + 1807x^4 - 13356*x^2 + 29241
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 1152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.92560$ of defining polynomial
Character	$\chi$	$=$	2304.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-11.8322 q^{5} +20.9762 q^{7} -49.6387 q^{11} -70.1997 q^{13} -118.659 q^{17} -133.866 q^{19} -45.2548 q^{23} +15.0000 q^{25} -130.154 q^{29} +188.786 q^{31} -248.193 q^{35} -210.599 q^{37} +118.659 q^{41} +401.597 q^{43} +316.784 q^{47} +97.0000 q^{49} -556.111 q^{53} +587.333 q^{55} -99.2774 q^{59} +350.999 q^{61} +830.614 q^{65} +267.731 q^{67} -905.097 q^{71} +350.000 q^{73} -1041.23 q^{77} +440.500 q^{79} +1240.97 q^{83} +1403.99 q^{85} -711.955 q^{89} -1472.52 q^{91} +1583.92 q^{95} +770.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 120 q^{25} + 776 q^{49} + 2800 q^{73} + 6160 q^{97}+O(q^{100}) $ 8 * q + 120 * q^25 + 776 * q^49 + 2800 * q^73 + 6160 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−11.8322	−1.05830	−0.529150	−	0.848528i	$-0.677489\pi$
$5$	−11.8322	−1.05830	−0.529150	+	0.848528i	$0.677489\pi$
$6$	0	0
$7$	20.9762	1.13261	0.566304	−	0.824197i	$-0.308373\pi$
$7$	20.9762	1.13261	0.566304	+	0.824197i	$0.308373\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−49.6387	−1.36060	−0.680301	−	0.732933i	$-0.738151\pi$
$11$	−49.6387	−1.36060	−0.680301	+	0.732933i	$0.738151\pi$
$12$	0	0
$13$	−70.1997	−1.49768	−0.748842	−	0.662748i	$-0.769390\pi$
$13$	−70.1997	−1.49768	−0.748842	+	0.662748i	$0.769390\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−118.659	−1.69289	−0.846443	−	0.532479i	$-0.821261\pi$
$17$	−118.659	−1.69289	−0.846443	+	0.532479i	$0.821261\pi$
$18$	0	0
$19$	−133.866	−1.61636	−0.808181	−	0.588934i	$-0.799548\pi$
$19$	−133.866	−1.61636	−0.808181	+	0.588934i	$0.799548\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−45.2548	−0.410273	−0.205137	−	0.978733i	$-0.565764\pi$
$23$	−45.2548	−0.410273	−0.205137	+	0.978733i	$0.565764\pi$
$24$	0	0
$25$	15.0000	0.120000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−130.154	−0.833412	−0.416706	−	0.909041i	$-0.636816\pi$
$29$	−130.154	−0.833412	−0.416706	+	0.909041i	$0.636816\pi$
$30$	0	0
$31$	188.786	1.09377	0.546885	−	0.837207i	$-0.315813\pi$
$31$	188.786	1.09377	0.546885	+	0.837207i	$0.315813\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−248.193	−1.19864
$36$	0	0
$37$	−210.599	−0.935737	−0.467869	−	0.883798i	$-0.654978\pi$
$37$	−210.599	−0.935737	−0.467869	+	0.883798i	$0.654978\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	118.659	0.451987	0.225993	−	0.974129i	$-0.427437\pi$
$41$	118.659	0.451987	0.225993	+	0.974129i	$0.427437\pi$
$42$	0	0
$43$	401.597	1.42425	0.712127	−	0.702050i	$-0.247732\pi$
$43$	401.597	1.42425	0.712127	+	0.702050i	$0.247732\pi$
$44$	0	0
$45$	0	0
$46$	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
$49$	97.0000	0.282799
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−556.111	−1.44128	−0.720640	−	0.693310i	$-0.756152\pi$
$53$	−556.111	−1.44128	−0.720640	+	0.693310i	$0.756152\pi$
$54$	0	0
$55$	587.333	1.43993
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−99.2774	−0.219065	−0.109532	−	0.993983i	$-0.534935\pi$
$59$	−99.2774	−0.219065	−0.109532	+	0.993983i	$0.534935\pi$
$60$	0	0
$61$	350.999	0.736734	0.368367	−	0.929680i	$-0.379917\pi$
$61$	350.999	0.736734	0.368367	+	0.929680i	$0.379917\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	830.614	1.58500
$66$	0	0
$67$	267.731	0.488188	0.244094	−	0.969752i	$-0.421510\pi$
$67$	267.731	0.488188	0.244094	+	0.969752i	$0.421510\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−905.097	−1.51289	−0.756445	−	0.654057i	$-0.773066\pi$
$71$	−905.097	−1.51289	−0.756445	+	0.654057i	$0.773066\pi$
$72$	0	0
$73$	350.000	0.561156	0.280578	−	0.959831i	$-0.409474\pi$
$73$	350.000	0.561156	0.280578	+	0.959831i	$0.409474\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1041.23	−1.54103
$78$	0	0
$79$	440.500	0.627343	0.313671	−	0.949532i	$-0.398441\pi$
$79$	440.500	0.627343	0.313671	+	0.949532i	$0.398441\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1240.97	1.64113	0.820565	−	0.571553i	$-0.193659\pi$
$83$	1240.97	1.64113	0.820565	+	0.571553i	$0.193659\pi$
$84$	0	0
$85$	1403.99	1.79158
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−711.955	−0.847945	−0.423972	−	0.905675i	$-0.639365\pi$
$89$	−711.955	−0.847945	−0.423972	+	0.905675i	$0.639365\pi$
$90$	0	0
$91$	−1472.52	−1.69629
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1583.92	1.71060
$96$	0	0
$97$	770.000	0.805996	0.402998	−	0.915201i	$-0.367968\pi$
$97$	770.000	0.805996	0.402998	+	0.915201i	$0.367968\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−366.797	−0.361363	−0.180681	−	0.983542i	$-0.557830\pi$
$101$	−366.797	−0.361363	−0.180681	+	0.983542i	$0.557830\pi$
$102$	0	0
$103$	−1279.55	−1.22405	−0.612027	−	0.790837i	$-0.709645\pi$
$103$	−1279.55	−1.22405	−0.612027	+	0.790837i	$0.709645\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1489.16	1.34544	0.672722	−	0.739895i	$-0.265125\pi$
$107$	1489.16	1.34544	0.672722	+	0.739895i	$0.265125\pi$
$108$	0	0
$109$	−350.999	−0.308436	−0.154218	−	0.988037i	$-0.549286\pi$
$109$	−350.999	−0.308436	−0.154218	+	0.988037i	$0.549286\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1661.23	−1.38297	−0.691483	−	0.722392i	$-0.743042\pi$
$113$	−1661.23	−1.38297	−0.691483	+	0.722392i	$0.743042\pi$
$114$	0	0
$115$	535.462	0.434192
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2489.02	−1.91738
$120$	0	0
$121$	1133.00	0.851240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1301.54	0.931304
$126$	0	0
$127$	−2202.50	−1.53890	−0.769449	−	0.638708i	$-0.779469\pi$
$127$	−2202.50	−1.53890	−0.769449	+	0.638708i	$0.779469\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1786.99	−1.19183	−0.595917	−	0.803046i	$-0.703211\pi$
$131$	−1786.99	−1.19183	−0.595917	+	0.803046i	$0.703211\pi$
$132$	0	0
$133$	−2807.99	−1.83070
$134$	0	0
$135$	0	0
$136$	0	0
$137$	830.614	0.517987	0.258993	−	0.965879i	$-0.416609\pi$
$137$	830.614	0.517987	0.258993	+	0.965879i	$0.416609\pi$
$138$	0	0
$139$	−267.731	−0.163372	−0.0816858	−	0.996658i	$-0.526030\pi$
$139$	−267.731	−0.163372	−0.0816858	+	0.996658i	$0.526030\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3484.62	2.03775
$144$	0	0
$145$	1540.00	0.882000
$146$	0	0
$147$	0	0
$148$	0	0
$149$	650.769	0.357806	0.178903	−	0.983867i	$-0.442745\pi$
$149$	650.769	0.357806	0.178903	+	0.983867i	$0.442745\pi$
$150$	0	0
$151$	440.500	0.237400	0.118700	−	0.992930i	$-0.462127\pi$
$151$	440.500	0.237400	0.118700	+	0.992930i	$0.462127\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2233.74	−1.15754
$156$	0	0
$157$	−210.599	−0.107055	−0.0535275	−	0.998566i	$-0.517046\pi$
$157$	−210.599	−0.107055	−0.0535275	+	0.998566i	$0.517046\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−949.273	−0.464678
$162$	0	0
$163$	−133.866	−0.0643262	−0.0321631	−	0.999483i	$-0.510240\pi$
$163$	−133.866	−0.0643262	−0.0321631	+	0.999483i	$0.510240\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3484.62	−1.61466	−0.807330	−	0.590100i	$-0.799088\pi$
$167$	−3484.62	−1.61466	−0.807330	+	0.590100i	$0.799088\pi$
$168$	0	0
$169$	2731.00	1.24306
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−911.076	−0.400392	−0.200196	−	0.979756i	$-0.564158\pi$
$173$	−911.076	−0.400392	−0.200196	+	0.979756i	$0.564158\pi$
$174$	0	0
$175$	314.643	0.135913
$176$	0	0
$177$	0	0
$178$	0	0
$179$	893.497	0.373090	0.186545	−	0.982446i	$-0.440271\pi$
$179$	893.497	0.373090	0.186545	+	0.982446i	$0.440271\pi$
$180$	0	0
$181$	−350.999	−0.144141	−0.0720705	−	0.997400i	$-0.522961\pi$
$181$	−350.999	−0.144141	−0.0720705	+	0.997400i	$0.522961\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2491.84	0.990291
$186$	0	0
$187$	5890.09	2.30335
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3846.66	1.45725	0.728625	−	0.684913i	$-0.240160\pi$
$191$	3846.66	1.45725	0.728625	+	0.684913i	$0.240160\pi$
$192$	0	0
$193$	4230.00	1.57763	0.788814	−	0.614632i	$-0.210696\pi$
$193$	4230.00	1.57763	0.788814	+	0.614632i	$0.210696\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4082.10	1.47633	0.738166	−	0.674620i	$-0.235692\pi$
$197$	4082.10	1.47633	0.738166	+	0.674620i	$0.235692\pi$
$198$	0	0
$199$	−4803.54	−1.71113	−0.855563	−	0.517698i	$-0.826789\pi$
$199$	−4803.54	−1.71113	−0.855563	+	0.517698i	$0.826789\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2730.13	−0.943928
$204$	0	0
$205$	−1403.99	−0.478338
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6644.91	2.19923
$210$	0	0
$211$	−2409.58	−0.786172	−0.393086	−	0.919502i	$-0.628593\pi$
$211$	−2409.58	−0.786172	−0.393086	+	0.919502i	$0.628593\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4751.76	−1.50729
$216$	0	0
$217$	3960.00	1.23881
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8329.84	2.53541
$222$	0	0
$223$	−104.881	−0.0314948	−0.0157474	−	0.999876i	$-0.505013\pi$
$223$	−104.881	−0.0314948	−0.0157474	+	0.999876i	$0.505013\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2233.74	−0.653122	−0.326561	−	0.945176i	$-0.605890\pi$
$227$	−2233.74	−0.653122	−0.326561	+	0.945176i	$0.605890\pi$
$228$	0	0
$229$	6668.97	1.92445	0.962223	−	0.272263i	$-0.0877719\pi$
$229$	6668.97	1.92445	0.962223	+	0.272263i	$0.0877719\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4983.69	−1.40125	−0.700627	−	0.713528i	$-0.747096\pi$
$233$	−4983.69	−1.40125	−0.700627	+	0.713528i	$0.747096\pi$
$234$	0	0
$235$	−3748.24	−1.04046
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2489.02	0.673645	0.336822	−	0.941568i	$-0.390648\pi$
$239$	2489.02	0.673645	0.336822	+	0.941568i	$0.390648\pi$
$240$	0	0
$241$	3402.00	0.909303	0.454652	−	0.890669i	$-0.349764\pi$
$241$	3402.00	0.909303	0.454652	+	0.890669i	$0.349764\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1147.72	−0.299286
$246$	0	0
$247$	9397.33	2.42080
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5509.90	1.38558	0.692792	−	0.721138i	$-0.256380\pi$
$251$	5509.90	1.38558	0.692792	+	0.721138i	$0.256380\pi$
$252$	0	0
$253$	2246.39	0.558219
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4271.73	−1.03682	−0.518411	−	0.855132i	$-0.673476\pi$
$257$	−4271.73	−1.03682	−0.518411	+	0.855132i	$0.673476\pi$
$258$	0	0
$259$	−4417.56	−1.05982
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1629.17	−0.381974	−0.190987	−	0.981593i	$-0.561169\pi$
$263$	−1629.17	−0.381974	−0.190987	+	0.981593i	$0.561169\pi$
$264$	0	0
$265$	6580.00	1.52531
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5998.90	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$269$	−5998.90	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$270$	0	0
$271$	3419.12	0.766408	0.383204	−	0.923664i	$-0.374821\pi$
$271$	3419.12	0.766408	0.383204	+	0.923664i	$0.374821\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−744.580	−0.163272
$276$	0	0
$277$	−4001.38	−0.867942	−0.433971	−	0.900927i	$-0.642888\pi$
$277$	−4001.38	−0.867942	−0.433971	+	0.900927i	$0.642888\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3322.46	−0.705342	−0.352671	−	0.935747i	$-0.614727\pi$
$281$	−3322.46	−0.705342	−0.352671	+	0.935747i	$0.614727\pi$
$282$	0	0
$283$	−4283.70	−0.899786	−0.449893	−	0.893082i	$-0.648538\pi$
$283$	−4283.70	−0.899786	−0.449893	+	0.893082i	$0.648538\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2489.02	0.511923
$288$	0	0
$289$	9167.00	1.86587
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3892.78	0.776173	0.388086	−	0.921623i	$-0.373136\pi$
$293$	3892.78	0.776173	0.388086	+	0.921623i	$0.373136\pi$
$294$	0	0
$295$	1174.67	0.231836
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3176.88	0.614460
$300$	0	0
$301$	8423.97	1.61312
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4153.07	−0.779686
$306$	0	0
$307$	9906.05	1.84159	0.920795	−	0.390046i	$-0.127541\pi$
$307$	9906.05	1.84159	0.920795	+	0.390046i	$0.127541\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3167.84	0.577594	0.288797	−	0.957390i	$-0.406745\pi$
$311$	3167.84	0.577594	0.288797	+	0.957390i	$0.406745\pi$
$312$	0	0
$313$	5390.00	0.973357	0.486679	−	0.873581i	$-0.338208\pi$
$313$	5390.00	0.973357	0.486679	+	0.873581i	$0.338208\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−35.4965	−0.00628921	−0.00314461	−	0.999995i	$-0.501001\pi$
$317$	−35.4965	−0.00628921	−0.00314461	+	0.999995i	$0.501001\pi$
$318$	0	0
$319$	6460.66	1.13394
$320$	0	0
$321$	0	0
$322$	0	0
$323$	15884.4	2.73632
$324$	0	0
$325$	−1053.00	−0.179722
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6644.91	1.11351
$330$	0	0
$331$	2677.31	0.444587	0.222294	−	0.974980i	$-0.428646\pi$
$331$	2677.31	0.444587	0.222294	+	0.974980i	$0.428646\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3167.84	−0.516649
$336$	0	0
$337$	4310.00	0.696679	0.348339	−	0.937369i	$-0.386746\pi$
$337$	4310.00	0.696679	0.348339	+	0.937369i	$0.386746\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9371.07	−1.48819
$342$	0	0
$343$	−5160.14	−0.812307
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3722.90	−0.575953	−0.287977	−	0.957637i	$-0.592983\pi$
$347$	−3722.90	−0.575953	−0.287977	+	0.957637i	$0.592983\pi$
$348$	0	0
$349$	−12285.0	−1.88424	−0.942118	−	0.335282i	$-0.891168\pi$
$349$	−12285.0	−1.88424	−0.942118	+	0.335282i	$0.891168\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−711.955	−0.107347	−0.0536736	−	0.998559i	$-0.517093\pi$
$353$	−711.955	−0.107347	−0.0536736	+	0.998559i	$0.517093\pi$
$354$	0	0
$355$	10709.2	1.60109
$356$	0	0
$357$	0	0
$358$	0	0
$359$	905.097	0.133062	0.0665309	−	0.997784i	$-0.478807\pi$
$359$	905.097	0.133062	0.0665309	+	0.997784i	$0.478807\pi$
$360$	0	0
$361$	11061.0	1.61263
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4141.26	−0.593872
$366$	0	0
$367$	−2831.78	−0.402774	−0.201387	−	0.979512i	$-0.564545\pi$
$367$	−2831.78	−0.402774	−0.201387	+	0.979512i	$0.564545\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11665.1	−1.63240
$372$	0	0
$373$	−7792.17	−1.08167	−0.540835	−	0.841128i	$-0.681892\pi$
$373$	−7792.17	−1.08167	−0.540835	+	0.841128i	$0.681892\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9136.76	1.24819
$378$	0	0
$379$	−4953.03	−0.671293	−0.335646	−	0.941988i	$-0.608955\pi$
$379$	−4953.03	−0.671293	−0.335646	+	0.941988i	$0.608955\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11721.0	−1.56375	−0.781874	−	0.623437i	$-0.785736\pi$
$383$	−11721.0	−1.56375	−0.781874	+	0.623437i	$0.785736\pi$
$384$	0	0
$385$	12320.0	1.63087
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7111.13	−0.926860	−0.463430	−	0.886134i	$-0.653381\pi$
$389$	−7111.13	−0.926860	−0.463430	+	0.886134i	$0.653381\pi$
$390$	0	0
$391$	5369.90	0.694546
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5212.06	−0.663917
$396$	0	0
$397$	−10319.4	−1.30457	−0.652284	−	0.757974i	$-0.726189\pi$
$397$	−10319.4	−1.30457	−0.652284	+	0.757974i	$0.726189\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2491.84	0.310316	0.155158	−	0.987890i	$-0.450411\pi$
$401$	2491.84	0.310316	0.155158	+	0.987890i	$0.450411\pi$
$402$	0	0
$403$	−13252.7	−1.63812
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10453.9	1.27317
$408$	0	0
$409$	5306.00	0.641479	0.320739	−	0.947167i	$-0.396069\pi$
$409$	5306.00	0.641479	0.320739	+	0.947167i	$0.396069\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2082.46	−0.248114
$414$	0	0
$415$	−14683.3	−1.73681
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9580.27	−1.11701	−0.558504	−	0.829502i	$-0.688625\pi$
$419$	−9580.27	−1.11701	−0.558504	+	0.829502i	$0.688625\pi$
$420$	0	0
$421$	9476.96	1.09710	0.548550	−	0.836118i	$-0.315180\pi$
$421$	9476.96	1.09710	0.548550	+	0.836118i	$0.315180\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1779.89	−0.203146
$426$	0	0
$427$	7362.61	0.834430
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5656.85	−0.632207	−0.316103	−	0.948725i	$-0.602375\pi$
$431$	−5656.85	−0.632207	−0.316103	+	0.948725i	$0.602375\pi$
$432$	0	0
$433$	2030.00	0.225302	0.112651	−	0.993635i	$-0.464066\pi$
$433$	2030.00	0.225302	0.112651	+	0.993635i	$0.464066\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6058.07	0.663150
$438$	0	0
$439$	−7824.11	−0.850625	−0.425313	−	0.905046i	$-0.639836\pi$
$439$	−7824.11	−0.850625	−0.425313	+	0.905046i	$0.639836\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4219.29	0.452516	0.226258	−	0.974067i	$-0.427351\pi$
$443$	4219.29	0.452516	0.226258	+	0.974067i	$0.427351\pi$
$444$	0	0
$445$	8423.97	0.897380
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17442.9	1.83337	0.916683	−	0.399615i	$-0.130856\pi$
$449$	17442.9	1.83337	0.916683	+	0.399615i	$0.130856\pi$
$450$	0	0
$451$	−5890.09	−0.614974
$452$	0	0
$453$	0	0
$454$	0	0
$455$	17423.1	1.79518
$456$	0	0
$457$	7530.00	0.770763	0.385381	−	0.922757i	$-0.374070\pi$
$457$	7530.00	0.770763	0.385381	+	0.922757i	$0.374070\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17002.8	−1.71779	−0.858894	−	0.512154i	$-0.828848\pi$
$461$	−17002.8	−1.71779	−0.858894	+	0.512154i	$0.828848\pi$
$462$	0	0
$463$	3670.83	0.368462	0.184231	−	0.982883i	$-0.441021\pi$
$463$	3670.83	0.368462	0.184231	+	0.982883i	$0.441021\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7197.61	−0.713203	−0.356601	−	0.934257i	$-0.616065\pi$
$467$	−7197.61	−0.713203	−0.356601	+	0.934257i	$0.616065\pi$
$468$	0	0
$469$	5615.98	0.552925
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−19934.7	−1.93784
$474$	0	0
$475$	−2007.98	−0.193963
$476$	0	0
$477$	0	0
$478$	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
$481$	14784.0	1.40144
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9110.76	−0.852986
$486$	0	0
$487$	−7194.83	−0.669464	−0.334732	−	0.942313i	$-0.608646\pi$
$487$	−7194.83	−0.669464	−0.334732	+	0.942313i	$0.608646\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6155.20	−0.565744	−0.282872	−	0.959158i	$-0.591287\pi$
$491$	−6155.20	−0.565744	−0.282872	+	0.959158i	$0.591287\pi$
$492$	0	0
$493$	15443.9	1.41087
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18985.5	−1.71351
$498$	0	0
$499$	−11780.2	−1.05682	−0.528410	−	0.848989i	$-0.677212\pi$
$499$	−11780.2	−1.05682	−0.528410	+	0.848989i	$0.677212\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6969.24	0.617780	0.308890	−	0.951098i	$-0.400043\pi$
$503$	6969.24	0.617780	0.308890	+	0.951098i	$0.400043\pi$
$504$	0	0
$505$	4340.00	0.382431
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9524.89	0.829437	0.414718	−	0.909950i	$-0.363880\pi$
$509$	9524.89	0.829437	0.414718	+	0.909950i	$0.363880\pi$
$510$	0	0
$511$	7341.66	0.635569
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15139.8	1.29542
$516$	0	0
$517$	−15724.7	−1.33767
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3203.80	−0.269407	−0.134703	−	0.990886i	$-0.543008\pi$
$521$	−3203.80	−0.269407	−0.134703	+	0.990886i	$0.543008\pi$
$522$	0	0
$523$	9504.46	0.794648	0.397324	−	0.917678i	$-0.369939\pi$
$523$	9504.46	0.794648	0.397324	+	0.917678i	$0.369939\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22401.1	−1.85163
$528$	0	0
$529$	−10119.0	−0.831676
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8329.84	−0.676933
$534$	0	0
$535$	−17620.0	−1.42389
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4814.95	−0.384777
$540$	0	0
$541$	13688.9	1.08786	0.543931	−	0.839130i	$-0.316935\pi$
$541$	13688.9	1.08786	0.543931	+	0.839130i	$0.316935\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4153.07	0.326418
$546$	0	0
$547$	−3078.91	−0.240667	−0.120333	−	0.992734i	$-0.538396\pi$
$547$	−3078.91	−0.240667	−0.120333	+	0.992734i	$0.538396\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17423.1	1.34710
$552$	0	0
$553$	9240.00	0.710533
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21333.4	1.62284	0.811422	−	0.584460i	$-0.198694\pi$
$557$	21333.4	1.62284	0.811422	+	0.584460i	$0.198694\pi$
$558$	0	0
$559$	−28192.0	−2.13308
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17621.7	1.31913	0.659563	−	0.751650i	$-0.270741\pi$
$563$	17621.7	1.31913	0.659563	+	0.751650i	$0.270741\pi$
$564$	0	0
$565$	19655.9	1.46359
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2491.84	0.183591	0.0917957	−	0.995778i	$-0.470739\pi$
$569$	2491.84	0.183591	0.0917957	+	0.995778i	$0.470739\pi$
$570$	0	0
$571$	7764.21	0.569040	0.284520	−	0.958670i	$-0.408166\pi$
$571$	7764.21	0.569040	0.284520	+	0.958670i	$0.408166\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−678.823	−0.0492328
$576$	0	0
$577$	20090.0	1.44949	0.724747	−	0.689015i	$-0.241957\pi$
$577$	20090.0	1.44949	0.724747	+	0.689015i	$0.241957\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	26030.8	1.85876
$582$	0	0
$583$	27604.6	1.96101
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5956.64	0.418836	0.209418	−	0.977826i	$-0.432843\pi$
$587$	5956.64	0.418836	0.209418	+	0.977826i	$0.432843\pi$
$588$	0	0
$589$	−25271.9	−1.76793
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4034.41	−0.279382	−0.139691	−	0.990195i	$-0.544611\pi$
$593$	−4034.41	−0.279382	−0.139691	+	0.990195i	$0.544611\pi$
$594$	0	0
$595$	29450.4	2.02916
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16518.0	1.12672	0.563362	−	0.826210i	$-0.309508\pi$
$599$	16518.0	1.12672	0.563362	+	0.826210i	$0.309508\pi$
$600$	0	0
$601$	−11438.0	−0.776316	−0.388158	−	0.921593i	$-0.626888\pi$
$601$	−11438.0	−0.776316	−0.388158	+	0.921593i	$0.626888\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13405.8	−0.900867
$606$	0	0
$607$	15753.1	1.05338	0.526688	−	0.850059i	$-0.323434\pi$
$607$	15753.1	1.05338	0.526688	+	0.850059i	$0.323434\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22238.1	−1.47244
$612$	0	0
$613$	−5545.78	−0.365403	−0.182701	−	0.983168i	$-0.558484\pi$
$613$	−5545.78	−0.365403	−0.182701	+	0.983168i	$0.558484\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16612.3	1.08393	0.541965	−	0.840401i	$-0.317680\pi$
$617$	16612.3	1.08393	0.541965	+	0.840401i	$0.317680\pi$
$618$	0	0
$619$	−17670.3	−1.14738	−0.573690	−	0.819073i	$-0.694488\pi$
$619$	−17670.3	−1.14738	−0.573690	+	0.819073i	$0.694488\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14934.1	−0.960388
$624$	0	0
$625$	−17275.0	−1.10560
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24989.5	1.58410
$630$	0	0
$631$	5726.50	0.361281	0.180640	−	0.983549i	$-0.442183\pi$
$631$	5726.50	0.361281	0.180640	+	0.983549i	$0.442183\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	26060.3	1.62862
$636$	0	0
$637$	−6809.37	−0.423543
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19104.1	−1.17717	−0.588586	−	0.808434i	$-0.700315\pi$
$641$	−19104.1	−1.17717	−0.588586	+	0.808434i	$0.700315\pi$
$642$	0	0
$643$	10307.7	0.632184	0.316092	−	0.948729i	$-0.397629\pi$
$643$	10307.7	0.632184	0.316092	+	0.948729i	$0.397629\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9186.73	−0.558219	−0.279109	−	0.960259i	$-0.590039\pi$
$647$	−9186.73	−0.558219	−0.279109	+	0.960259i	$0.590039\pi$
$648$	0	0
$649$	4928.00	0.298060
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26728.8	1.60181	0.800904	−	0.598793i	$-0.204353\pi$
$653$	26728.8	1.60181	0.800904	+	0.598793i	$0.204353\pi$
$654$	0	0
$655$	21144.0	1.26132
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19458.4	1.15021	0.575107	−	0.818079i	$-0.304961\pi$
$659$	19458.4	1.15021	0.575107	+	0.818079i	$0.304961\pi$
$660$	0	0
$661$	5966.98	0.351117	0.175559	−	0.984469i	$-0.443827\pi$
$661$	5966.98	0.351117	0.175559	+	0.984469i	$0.443827\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	33224.6	1.93743
$666$	0	0
$667$	5890.09	0.341927
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17423.1	−1.00240
$672$	0	0
$673$	2270.00	0.130018	0.0650090	−	0.997885i	$-0.479292\pi$
$673$	2270.00	0.130018	0.0650090	+	0.997885i	$0.479292\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16103.6	−0.914196	−0.457098	−	0.889416i	$-0.651111\pi$
$677$	−16103.6	−0.914196	−0.457098	+	0.889416i	$0.651111\pi$
$678$	0	0
$679$	16151.7	0.912877
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20600.1	1.15408	0.577042	−	0.816714i	$-0.304207\pi$
$683$	20600.1	1.15408	0.577042	+	0.816714i	$0.304207\pi$
$684$	0	0
$685$	−9827.96	−0.548185
$686$	0	0
$687$	0	0
$688$	0	0
$689$	39038.9	2.15858
$690$	0	0
$691$	7362.61	0.405335	0.202668	−	0.979248i	$-0.435039\pi$
$691$	7362.61	0.405335	0.202668	+	0.979248i	$0.435039\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3167.84	0.172896
$696$	0	0
$697$	−14080.0	−0.765162
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13855.5	−0.746524	−0.373262	−	0.927726i	$-0.621761\pi$
$701$	−13855.5	−0.746524	−0.373262	+	0.927726i	$0.621761\pi$
$702$	0	0
$703$	28192.0	1.51249
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7694.00	−0.409282
$708$	0	0
$709$	−28430.9	−1.50599	−0.752993	−	0.658028i	$-0.771391\pi$
$709$	−28430.9	−1.50599	−0.752993	+	0.658028i	$0.771391\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8543.46	−0.448745
$714$	0	0
$715$	−41230.6	−2.15656
$716$	0	0
$717$	0	0
$718$	0	0
$719$	28510.5	1.47881	0.739405	−	0.673261i	$-0.235107\pi$
$719$	28510.5	1.47881	0.739405	+	0.673261i	$0.235107\pi$
$720$	0	0
$721$	−26840.0	−1.38637
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1952.31	−0.100009
$726$	0	0
$727$	−18689.8	−0.953460	−0.476730	−	0.879050i	$-0.658178\pi$
$727$	−18689.8	−0.953460	−0.476730	+	0.879050i	$0.658178\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−47653.1	−2.41110
$732$	0	0
$733$	−14110.1	−0.711010	−0.355505	−	0.934674i	$-0.615691\pi$
$733$	−14110.1	−0.711010	−0.355505	+	0.934674i	$0.615691\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13289.8	−0.664229
$738$	0	0
$739$	−32930.9	−1.63922	−0.819610	−	0.572921i	$-0.805810\pi$
$739$	−32930.9	−1.63922	−0.819610	+	0.572921i	$0.805810\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17151.6	0.846878	0.423439	−	0.905925i	$-0.360823\pi$
$743$	17151.6	0.846878	0.423439	+	0.905925i	$0.360823\pi$
$744$	0	0
$745$	−7700.00	−0.378666
$746$	0	0
$747$	0	0
$748$	0	0
$749$	31236.9	1.52386
$750$	0	0
$751$	−12480.8	−0.606434	−0.303217	−	0.952922i	$-0.598061\pi$
$751$	−12480.8	−0.606434	−0.303217	+	0.952922i	$0.598061\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5212.06	−0.251240
$756$	0	0
$757$	−12144.6	−0.583093	−0.291546	−	0.956557i	$-0.594170\pi$
$757$	−12144.6	−0.583093	−0.291546	+	0.956557i	$0.594170\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−36665.7	−1.74656	−0.873279	−	0.487221i	$-0.838011\pi$
$761$	−36665.7	−1.74656	−0.873279	+	0.487221i	$0.838011\pi$
$762$	0	0
$763$	−7362.61	−0.349337
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6969.24	0.328090
$768$	0	0
$769$	16646.0	0.780585	0.390293	−	0.920691i	$-0.372374\pi$
$769$	16646.0	0.780585	0.390293	+	0.920691i	$0.372374\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7395.10	0.344092	0.172046	−	0.985089i	$-0.444962\pi$
$773$	7395.10	0.344092	0.172046	+	0.985089i	$0.444962\pi$
$774$	0	0
$775$	2831.78	0.131252
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−15884.4	−0.730574
$780$	0	0
$781$	44927.8	2.05844
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2491.84	0.113296
$786$	0	0
$787$	−11110.8	−0.503251	−0.251626	−	0.967825i	$-0.580965\pi$
$787$	−11110.8	−0.503251	−0.251626	+	0.967825i	$0.580965\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−34846.2	−1.56636
$792$	0	0
$793$	−24640.0	−1.10339
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2614.91	−0.116217	−0.0581084	−	0.998310i	$-0.518507\pi$
$797$	−2614.91	−0.116217	−0.0581084	+	0.998310i	$0.518507\pi$
$798$	0	0
$799$	−37589.3	−1.66435
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17373.5	−0.763511
$804$	0	0
$805$	11232.0	0.491769
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39038.9	−1.69658	−0.848290	−	0.529532i	$-0.822368\pi$
$809$	−39038.9	−1.69658	−0.848290	+	0.529532i	$0.822368\pi$
$810$	0	0
$811$	29584.3	1.28094	0.640472	−	0.767982i	$-0.278739\pi$
$811$	29584.3	1.28094	0.640472	+	0.767982i	$0.278739\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1583.92	0.0680764
$816$	0	0
$817$	−53760.0	−2.30211
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21877.7	0.930007	0.465003	−	0.885309i	$-0.346053\pi$
$821$	21877.7	0.930007	0.465003	+	0.885309i	$0.346053\pi$
$822$	0	0
$823$	35093.1	1.48635	0.743177	−	0.669094i	$-0.233318\pi$
$823$	35093.1	1.48635	0.743177	+	0.669094i	$0.233318\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31272.4	−1.31493	−0.657465	−	0.753485i	$-0.728371\pi$
$827$	−31272.4	−1.31493	−0.657465	+	0.753485i	$0.728371\pi$
$828$	0	0
$829$	22112.9	0.926433	0.463217	−	0.886245i	$-0.346695\pi$
$829$	22112.9	0.926433	0.463217	+	0.886245i	$0.346695\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11509.9	−0.478746
$834$	0	0
$835$	41230.6	1.70880
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31678.4	−1.30353	−0.651764	−	0.758422i	$-0.725971\pi$
$839$	−31678.4	−1.30353	−0.651764	+	0.758422i	$0.725971\pi$
$840$	0	0
$841$	−7449.00	−0.305425
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−32313.6	−1.31553
$846$	0	0
$847$	23766.0	0.964120
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9530.63	0.383908
$852$	0	0
$853$	16918.1	0.679092	0.339546	−	0.940589i	$-0.389727\pi$
$853$	16918.1	0.679092	0.339546	+	0.940589i	$0.389727\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18154.9	0.723638	0.361819	−	0.932248i	$-0.382156\pi$
$857$	18154.9	0.723638	0.361819	+	0.932248i	$0.382156\pi$
$858$	0	0
$859$	48325.5	1.91949	0.959746	−	0.280868i	$-0.0906224\pi$
$859$	48325.5	1.91949	0.959746	+	0.280868i	$0.0906224\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41227.2	1.62617	0.813087	−	0.582142i	$-0.197785\pi$
$863$	41227.2	1.62617	0.813087	+	0.582142i	$0.197785\pi$
$864$	0	0
$865$	10780.0	0.423735
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21865.8	−0.853564
$870$	0	0
$871$	−18794.7	−0.731151
$872$	0	0
$873$	0	0
$874$	0	0
$875$	27301.3	1.05480
$876$	0	0
$877$	−24359.3	−0.937919	−0.468960	−	0.883220i	$-0.655371\pi$
$877$	−24359.3	−0.937919	−0.468960	+	0.883220i	$0.655371\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9018.10	0.344867	0.172433	−	0.985021i	$-0.444837\pi$
$881$	9018.10	0.344867	0.172433	+	0.985021i	$0.444837\pi$
$882$	0	0
$883$	−44577.2	−1.69892	−0.849459	−	0.527655i	$-0.823071\pi$
$883$	−44577.2	−1.69892	−0.849459	+	0.527655i	$0.823071\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1900.70	0.0719497	0.0359748	−	0.999353i	$-0.488546\pi$
$887$	1900.70	0.0719497	0.0359748	+	0.999353i	$0.488546\pi$
$888$	0	0
$889$	−46200.0	−1.74297
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−42406.5	−1.58911
$894$	0	0
$895$	−10572.0	−0.394841
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−24571.2	−0.911562
$900$	0	0
$901$	65987.7	2.43992
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4153.07	0.152544
$906$	0	0
$907$	33868.0	1.23988	0.619938	−	0.784650i	$-0.287158\pi$
$907$	33868.0	1.23988	0.619938	+	0.784650i	$0.287158\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	15160.4	0.551356	0.275678	−	0.961250i	$-0.411098\pi$
$911$	15160.4	0.551356	0.275678	+	0.961250i	$0.411098\pi$
$912$	0	0
$913$	−61600.0	−2.23293
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−37484.3	−1.34988
$918$	0	0
$919$	−44784.1	−1.60750	−0.803750	−	0.594967i	$-0.797165\pi$
$919$	−44784.1	−1.60750	−0.803750	+	0.594967i	$0.797165\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	63537.5	2.26583
$924$	0	0
$925$	−3158.99	−0.112288
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16730.9	0.590877	0.295438	−	0.955362i	$-0.404534\pi$
$929$	16730.9	0.590877	0.295438	+	0.955362i	$0.404534\pi$
$930$	0	0
$931$	−12985.0	−0.457105
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−69692.4	−2.43763
$936$	0	0
$937$	6090.00	0.212328	0.106164	−	0.994349i	$-0.466143\pi$
$937$	6090.00	0.212328	0.106164	+	0.994349i	$0.466143\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41684.7	1.44408	0.722042	−	0.691850i	$-0.243204\pi$
$941$	41684.7	1.44408	0.722042	+	0.691850i	$0.243204\pi$
$942$	0	0
$943$	−5369.90	−0.185438
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4467.48	−0.153298	−0.0766492	−	0.997058i	$-0.524422\pi$
$947$	−4467.48	−0.153298	−0.0766492	+	0.997058i	$0.524422\pi$
$948$	0	0
$949$	−24569.9	−0.840435
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12459.2	0.423498	0.211749	−	0.977324i	$-0.432084\pi$
$953$	12459.2	0.423498	0.211749	+	0.977324i	$0.432084\pi$
$954$	0	0
$955$	−45514.3	−1.54221
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17423.1	0.586675
$960$	0	0
$961$	5849.00	0.196334
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−50050.0	−1.66960
$966$	0	0
$967$	−45665.1	−1.51861	−0.759303	−	0.650737i	$-0.774460\pi$
$967$	−45665.1	−1.51861	−0.759303	+	0.650737i	$0.774460\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24869.0	−0.821919	−0.410960	−	0.911654i	$-0.634806\pi$
$971$	−24869.0	−0.821919	−0.410960	+	0.911654i	$0.634806\pi$
$972$	0	0
$973$	−5615.98	−0.185036
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40700.1	−1.33277	−0.666383	−	0.745610i	$-0.732158\pi$
$977$	−40700.1	−1.33277	−0.666383	+	0.745610i	$0.732158\pi$
$978$	0	0
$979$	35340.5	1.15372
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13304.9	−0.431700	−0.215850	−	0.976427i	$-0.569252\pi$
$983$	−13304.9	−0.431700	−0.215850	+	0.976427i	$0.569252\pi$
$984$	0	0
$985$	−48300.0	−1.56240
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18174.2	−0.584334
$990$	0	0
$991$	−17179.5	−0.550681	−0.275340	−	0.961347i	$-0.588791\pi$
$991$	−17179.5	−0.550681	−0.275340	+	0.961347i	$0.588791\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	56836.3	1.81089
$996$	0	0
$997$	−17058.5	−0.541875	−0.270937	−	0.962597i	$-0.587334\pi$
$997$	−17058.5	−0.541875	−0.270937	+	0.962597i	$0.587334\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.a.cd.1.3		8
3.2	odd	2	inner	2304.4.a.cd.1.7		8
4.3	odd	2	inner	2304.4.a.cd.1.1		8
8.3	odd	2	inner	2304.4.a.cd.1.6		8
8.5	even	2	inner	2304.4.a.cd.1.8		8
12.11	even	2	inner	2304.4.a.cd.1.5		8
16.3	odd	4		1152.4.d.q.577.7	yes	8
16.5	even	4		1152.4.d.q.577.2	yes	8
16.11	odd	4		1152.4.d.q.577.4	yes	8
16.13	even	4		1152.4.d.q.577.5	yes	8
24.5	odd	2	inner	2304.4.a.cd.1.4		8
24.11	even	2	inner	2304.4.a.cd.1.2		8
48.5	odd	4		1152.4.d.q.577.6	yes	8
48.11	even	4		1152.4.d.q.577.8	yes	8
48.29	odd	4		1152.4.d.q.577.1	✓	8
48.35	even	4		1152.4.d.q.577.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.d.q.577.1	✓	8	48.29	odd	4
1152.4.d.q.577.2	yes	8	16.5	even	4
1152.4.d.q.577.3	yes	8	48.35	even	4
1152.4.d.q.577.4	yes	8	16.11	odd	4
1152.4.d.q.577.5	yes	8	16.13	even	4
1152.4.d.q.577.6	yes	8	48.5	odd	4
1152.4.d.q.577.7	yes	8	16.3	odd	4
1152.4.d.q.577.8	yes	8	48.11	even	4
2304.4.a.cd.1.1		8	4.3	odd	2	inner
2304.4.a.cd.1.2		8	24.11	even	2	inner
2304.4.a.cd.1.3		8	1.1	even	1	trivial
2304.4.a.cd.1.4		8	24.5	odd	2	inner
2304.4.a.cd.1.5		8	12.11	even	2	inner
2304.4.a.cd.1.6		8	8.3	odd	2	inner
2304.4.a.cd.1.7		8	3.2	odd	2	inner
2304.4.a.cd.1.8		8	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)

\(f(q)\)	\(=\)	\(q-11.8322 q^{5} +20.9762 q^{7} -49.6387 q^{11} -70.1997 q^{13} -118.659 q^{17} -133.866 q^{19} -45.2548 q^{23} +15.0000 q^{25} -130.154 q^{29} +188.786 q^{31} -248.193 q^{35} -210.599 q^{37} +118.659 q^{41} +401.597 q^{43} +316.784 q^{47} +97.0000 q^{49} -556.111 q^{53} +587.333 q^{55} -99.2774 q^{59} +350.999 q^{61} +830.614 q^{65} +267.731 q^{67} -905.097 q^{71} +350.000 q^{73} -1041.23 q^{77} +440.500 q^{79} +1240.97 q^{83} +1403.99 q^{85} -711.955 q^{89} -1472.52 q^{91} +1583.92 q^{95} +770.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 120 q^{25} + 776 q^{49} + 2800 q^{73} + 6160 q^{97}+O(q^{100}) \) 8 * q + 120 * q^25 + 776 * q^49 + 2800 * q^73 + 6160 * q^97

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