LMFDB - Embedded newform 1008.4.k.d.881.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(881,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("1008.881");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.k (of order $2$, degree $1$, not 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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.95072796278784.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 44x^{6} + 8x^{5} + 738x^{4} + 2416x^{3} + 3652x^{2} + 2824x + 946 $ x^8 - 4x^7 - 44x^6 + 8x^5 + 738x^4 + 2416x^3 + 3652x^2 + 2824*x + 946
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$-0.935327 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1008.881
Dual form			1008.4.k.d.881.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.3330 q^{5} +(-16.3875 - 8.62844i) q^{7} +O(q^{10})$	$q+15.3330 q^{5} +(-16.3875 - 8.62844i) q^{7} -20.3470i q^{11} +30.5385i q^{13} -3.45008 q^{17} +147.361i q^{19} -127.191i q^{23} +110.100 q^{25} +216.198i q^{29} +103.541i q^{31} +(-251.269 - 132.300i) q^{35} +236.650 q^{37} +480.305 q^{41} -147.875 q^{43} +455.006 q^{47} +(194.100 + 282.797i) q^{49} +200.783i q^{53} -311.980i q^{55} -160.230 q^{59} -823.092i q^{61} +468.246i q^{65} +407.100 q^{67} +513.872i q^{71} -473.979i q^{73} +(-175.563 + 333.436i) q^{77} +121.300 q^{79} +536.269 q^{83} -52.9000 q^{85} +231.912 q^{89} +(263.500 - 500.450i) q^{91} +2259.49i q^{95} +734.188i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^7	$ 8 q - 16 q^{7} - 40 q^{25} + 512 q^{37} - 32 q^{43} + 632 q^{49} + 2336 q^{67} - 1792 q^{79} - 1344 q^{85} - 2496 q^{91}+O(q^{100}) $ 8 * q - 16 * q^7 - 40 * q^25 + 512 * q^37 - 32 * q^43 + 632 * q^49 + 2336 * q^67 - 1792 * q^79 - 1344 * q^85 - 2496 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	15.3330			1.37142			0.685711	−	0.727874i	$-0.259491\pi$
$5$	15.3330			1.37142			0.685711	+	0.727874i	$0.259491\pi$
$6$		0			0
$7$	−16.3875	−	8.62844i	−0.884842	−	0.465892i
$7$	−16.3875	−	8.62844i	−0.884842	−	0.465892i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	20.3470i		−	0.557713i	−0.960333	−	0.278857i	$-0.910045\pi$
$11$		−	20.3470i		−	0.557713i	0.960333	−	0.278857i	$-0.0899555\pi$
$12$		0			0
$13$			30.5385i			0.651528i	0.945451	+	0.325764i	$0.105621\pi$
$13$			30.5385i			0.651528i	−0.945451	+	0.325764i	$0.894379\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.45008			−0.0492217			−0.0246108	−	0.999697i	$-0.507835\pi$
$17$	−3.45008			−0.0492217			−0.0246108	+	0.999697i	$0.507835\pi$
$18$		0			0
$19$			147.361i			1.77932i	0.456626	+	0.889659i	$0.349058\pi$
$19$			147.361i			1.77932i	−0.456626	+	0.889659i	$0.650942\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	127.191i		−	1.15309i	−0.817065	−	0.576546i	$-0.804400\pi$
$23$		−	127.191i		−	1.15309i	0.817065	−	0.576546i	$-0.195600\pi$
$24$		0			0
$25$	110.100			0.880800
$26$		0			0
$27$		0			0
$28$		0			0
$29$			216.198i			1.38438i	0.721717	+	0.692188i	$0.243353\pi$
$29$			216.198i			1.38438i	−0.721717	+	0.692188i	$0.756647\pi$
$30$		0			0
$31$			103.541i			0.599889i	0.953957	+	0.299945i	$0.0969682\pi$
$31$			103.541i			0.599889i	−0.953957	+	0.299945i	$0.903032\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−251.269	−	132.300i	−1.21349	−	0.638935i
$36$		0			0
$37$	236.650			1.05149			0.525743	−	0.850643i	$-0.323787\pi$
$37$	236.650			1.05149			0.525743	+	0.850643i	$0.323787\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	480.305			1.82954			0.914768	−	0.403979i	$-0.132373\pi$
$41$	480.305			1.82954			0.914768	+	0.403979i	$0.132373\pi$
$42$		0			0
$43$	−147.875			−0.524435			−0.262218	−	0.965009i	$-0.584454\pi$
$43$	−147.875			−0.524435			−0.262218	+	0.965009i	$0.584454\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	455.006			1.41212			0.706059	−	0.708153i	$-0.250472\pi$
$47$	455.006			1.41212			0.706059	+	0.708153i	$0.250472\pi$
$48$		0			0
$49$	194.100	+	282.797i	0.565889	+	0.824481i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			200.783i			0.520371i	0.965559	+	0.260185i	$0.0837837\pi$
$53$			200.783i			0.520371i	−0.965559	+	0.260185i	$0.916216\pi$
$54$		0			0
$55$		−	311.980i		−	0.764861i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−160.230			−0.353562			−0.176781	−	0.984250i	$-0.556568\pi$
$59$	−160.230			−0.353562			−0.176781	+	0.984250i	$0.556568\pi$
$60$		0			0
$61$		−	823.092i		−	1.72764i	−0.503800	−	0.863820i	$-0.668065\pi$
$61$		−	823.092i		−	1.72764i	0.503800	−	0.863820i	$-0.331935\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			468.246i			0.893520i
$66$		0			0
$67$	407.100			0.742316			0.371158	−	0.928570i	$-0.378961\pi$
$67$	407.100			0.742316			0.371158	+	0.928570i	$0.378961\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			513.872i			0.858949i	0.903079	+	0.429475i	$0.141301\pi$
$71$			513.872i			0.858949i	−0.903079	+	0.429475i	$0.858699\pi$
$72$		0			0
$73$		−	473.979i		−	0.759932i	−0.925001	−	0.379966i	$-0.875936\pi$
$73$		−	473.979i		−	0.759932i	0.925001	−	0.379966i	$-0.124064\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−175.563	+	333.436i	−0.259834	+	0.493488i
$78$		0			0
$79$	121.300			0.172751			0.0863753	−	0.996263i	$-0.472472\pi$
$79$	121.300			0.172751			0.0863753	+	0.996263i	$0.472472\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	536.269			0.709195			0.354597	−	0.935019i	$-0.384618\pi$
$83$	536.269			0.709195			0.354597	+	0.935019i	$0.384618\pi$
$84$		0			0
$85$	−52.9000			−0.0675037
$86$		0			0
$87$		0			0
$88$		0			0
$89$	231.912			0.276209			0.138105	−	0.990418i	$-0.455899\pi$
$89$	231.912			0.276209			0.138105	+	0.990418i	$0.455899\pi$
$90$		0			0
$91$	263.500	−	500.450i	0.303542	−	0.576499i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			2259.49i			2.44020i
$96$		0			0
$97$			734.188i			0.768510i	0.923227	+	0.384255i	$0.125542\pi$
$97$			734.188i			0.768510i	−0.923227	+	0.384255i	$0.874458\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−727.550			−0.716771			−0.358386	−	0.933574i	$-0.616673\pi$
$101$	−727.550			−0.716771			−0.358386	+	0.933574i	$0.616673\pi$
$102$		0			0
$103$		−	812.429i		−	0.777195i	−0.921408	−	0.388597i	$-0.872960\pi$
$103$		−	812.429i		−	0.777195i	0.921408	−	0.388597i	$-0.127040\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			884.608i			0.799236i	0.916682	+	0.399618i	$0.130857\pi$
$107$			884.608i			0.799236i	−0.916682	+	0.399618i	$0.869143\pi$
$108$		0			0
$109$	−1601.05			−1.40691			−0.703453	−	0.710742i	$-0.748359\pi$
$109$	−1601.05			−1.40691			−0.703453	+	0.710742i	$0.748359\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	740.553i		−	0.616508i	−0.951304	−	0.308254i	$-0.900255\pi$
$113$		−	740.553i		−	0.616508i	0.951304	−	0.308254i	$-0.0997446\pi$
$114$		0			0
$115$		−	1950.21i		−	1.58138i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	56.5382	+	29.7689i	0.0435534	+	0.0229320i
$120$		0			0
$121$	917.000			0.688956
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−228.462			−0.163474
$126$		0			0
$127$	2428.70			1.69695			0.848473	−	0.529238i	$-0.177522\pi$
$127$	2428.70			1.69695			0.848473	+	0.529238i	$0.177522\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	508.668			0.339256			0.169628	−	0.985508i	$-0.445743\pi$
$131$	508.668			0.339256			0.169628	+	0.985508i	$0.445743\pi$
$132$		0			0
$133$	1271.50	−	2414.88i	0.828970	−	1.57441i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2018.72i		−	1.25891i	−0.777036	−	0.629456i	$-0.783278\pi$
$137$		−	2018.72i		−	1.25891i	0.777036	−	0.629456i	$-0.216722\pi$
$138$		0			0
$139$			3146.55i			1.92005i	0.279920	+	0.960023i	$0.409692\pi$
$139$			3146.55i			1.92005i	−0.279920	+	0.960023i	$0.590308\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	621.367			0.363366
$144$		0			0
$145$			3314.95i			1.89857i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			3056.50i			1.68053i	0.542178	+	0.840264i	$0.317600\pi$
$149$			3056.50i			1.68053i	−0.542178	+	0.840264i	$0.682400\pi$
$150$		0			0
$151$	1801.03			0.970631			0.485316	−	0.874339i	$-0.338705\pi$
$151$	1801.03			0.970631			0.485316	+	0.874339i	$0.338705\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			1587.60i			0.822702i
$156$		0			0
$157$		−	1346.04i		−	0.684239i	−0.939657	−	0.342119i	$-0.888855\pi$
$157$		−	1346.04i		−	0.684239i	0.939657	−	0.342119i	$-0.111145\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1097.46	+	2084.34i	−0.537216	+	1.02030i
$162$		0			0
$163$	746.499			0.358714			0.179357	−	0.983784i	$-0.442598\pi$
$163$	746.499			0.358714			0.179357	+	0.983784i	$0.442598\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2625.39			1.21652			0.608259	−	0.793738i	$-0.291868\pi$
$167$	2625.39			1.21652			0.608259	+	0.793738i	$0.291868\pi$
$168$		0			0
$169$	1264.40			0.575512
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3676.84			1.61587			0.807934	−	0.589274i	$-0.200586\pi$
$173$	3676.84			1.61587			0.807934	+	0.589274i	$0.200586\pi$
$174$		0			0
$175$	−1804.26	−	949.991i	−0.779368	−	0.410358i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			3981.06i			1.66234i	0.556019	+	0.831170i	$0.312328\pi$
$179$			3981.06i			1.66234i	−0.556019	+	0.831170i	$0.687672\pi$
$180$		0			0
$181$			1230.75i			0.505421i	0.967542	+	0.252711i	$0.0813221\pi$
$181$			1230.75i			0.505421i	−0.967542	+	0.252711i	$0.918678\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3628.55			1.44203
$186$		0			0
$187$			70.1988i			0.0274516i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3170.97i		−	1.20127i	−0.799522	−	0.600636i	$-0.794914\pi$
$191$		−	3170.97i		−	1.20127i	0.799522	−	0.600636i	$-0.205086\pi$
$192$		0			0
$193$	−240.201			−0.0895857			−0.0447928	−	0.998996i	$-0.514263\pi$
$193$	−240.201			−0.0895857			−0.0447928	+	0.998996i	$0.514263\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2565.35i			0.927784i	0.885892	+	0.463892i	$0.153547\pi$
$197$			2565.35i			0.927784i	−0.885892	+	0.463892i	$0.846453\pi$
$198$		0			0
$199$		−	1304.74i		−	0.464778i	−0.972623	−	0.232389i	$-0.925346\pi$
$199$		−	1304.74i		−	0.464778i	0.972623	−	0.232389i	$-0.0746543\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1865.45	−	3542.94i	0.644970	−	1.22495i
$204$		0			0
$205$	7364.50			2.50907
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2998.36			0.992349
$210$		0			0
$211$	−1773.22			−0.578549			−0.289274	−	0.957246i	$-0.593414\pi$
$211$	−1773.22			−0.578549			−0.289274	+	0.957246i	$0.593414\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2267.36			−0.719222
$216$		0			0
$217$	893.400	−	1696.78i	0.279484	−	0.530807i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	105.360i		−	0.0320693i
$222$		0			0
$223$			3314.68i			0.995369i	0.867358	+	0.497684i	$0.165816\pi$
$223$			3314.68i			0.995369i	−0.867358	+	0.497684i	$0.834184\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1317.87			−0.385330			−0.192665	−	0.981265i	$-0.561713\pi$
$227$	−1317.87			−0.385330			−0.192665	+	0.981265i	$0.561713\pi$
$228$		0			0
$229$		−	2109.32i		−	0.608679i	−0.952564	−	0.304339i	$-0.901564\pi$
$229$		−	2109.32i		−	0.608679i	0.952564	−	0.304339i	$-0.0984357\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6599.36i			1.85553i	0.373167	+	0.927764i	$0.378272\pi$
$233$			6599.36i			1.85553i	−0.373167	+	0.927764i	$0.621728\pi$
$234$		0			0
$235$	6976.60			1.93661
$236$		0			0
$237$		0			0
$238$		0			0
$239$			300.361i			0.0812919i	0.999174	+	0.0406459i	$0.0129416\pi$
$239$			300.361i			0.0812919i	−0.999174	+	0.0406459i	$0.987058\pi$
$240$		0			0
$241$		−	4573.35i		−	1.22239i	−0.791481	−	0.611194i	$-0.790690\pi$
$241$		−	4573.35i		−	1.22239i	0.791481	−	0.611194i	$-0.209310\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2976.13	+	4336.12i	0.776073	+	1.13071i
$246$		0			0
$247$	−4500.20			−1.15927
$248$		0			0
$249$		0			0
$250$		0			0
$251$	165.982			0.0417399			0.0208699	−	0.999782i	$-0.493356\pi$
$251$	165.982			0.0417399			0.0208699	+	0.999782i	$0.493356\pi$
$252$		0			0
$253$	−2587.95			−0.643095
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6617.33			−1.60614			−0.803069	−	0.595886i	$-0.796801\pi$
$257$	−6617.33			−1.60614			−0.803069	+	0.595886i	$0.796801\pi$
$258$		0			0
$259$	−3878.10	−	2041.92i	−0.930399	−	0.489879i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4568.14i			1.07104i	0.844522	+	0.535520i	$0.179884\pi$
$263$			4568.14i			1.07104i	−0.844522	+	0.535520i	$0.820116\pi$
$264$		0			0
$265$			3078.60i			0.713648i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7679.52			−1.74063			−0.870314	−	0.492498i	$-0.836084\pi$
$269$	−7679.52			−1.74063			−0.870314	+	0.492498i	$0.836084\pi$
$270$		0			0
$271$			2865.38i			0.642286i	0.947031	+	0.321143i	$0.104067\pi$
$271$			2865.38i			0.642286i	−0.947031	+	0.321143i	$0.895933\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	2240.20i		−	0.491234i
$276$		0			0
$277$	168.698			0.0365924			0.0182962	−	0.999833i	$-0.494176\pi$
$277$	168.698			0.0365924			0.0182962	+	0.999833i	$0.494176\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	5412.27i		−	1.14900i	−0.818505	−	0.574499i	$-0.805197\pi$
$281$		−	5412.27i		−	1.14900i	0.818505	−	0.574499i	$-0.194803\pi$
$282$		0			0
$283$		−	3272.95i		−	0.687480i	−0.939065	−	0.343740i	$-0.888306\pi$
$283$		−	3272.95i		−	0.687480i	0.939065	−	0.343740i	$-0.111694\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7870.99	−	4144.28i	−1.61885	−	0.852367i
$288$		0			0
$289$	−4901.10			−0.997577
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7800.65			1.55535			0.777677	−	0.628664i	$-0.216398\pi$
$293$	7800.65			1.55535			0.777677	+	0.628664i	$0.216398\pi$
$294$		0			0
$295$	−2456.80			−0.484883
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3884.22			0.751271
$300$		0			0
$301$	2423.30	+	1275.93i	0.464042	+	0.244330i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12620.4i		−	2.36933i
$306$		0			0
$307$		−	7079.58i		−	1.31613i	−0.752960	−	0.658066i	$-0.771375\pi$
$307$		−	7079.58i		−	1.31613i	0.752960	−	0.658066i	$-0.228625\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5810.43			−1.05942			−0.529710	−	0.848179i	$-0.677699\pi$
$311$	−5810.43			−1.05942			−0.529710	+	0.848179i	$0.677699\pi$
$312$		0			0
$313$		−	4667.86i		−	0.842950i	−0.906840	−	0.421475i	$-0.861513\pi$
$313$		−	4667.86i		−	0.842950i	0.906840	−	0.421475i	$-0.138487\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5099.76i			0.903568i	0.892127	+	0.451784i	$0.149212\pi$
$317$			5099.76i			0.903568i	−0.892127	+	0.451784i	$0.850788\pi$
$318$		0			0
$319$	4398.97			0.772086
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	508.409i		−	0.0875810i
$324$		0			0
$325$			3362.29i			0.573865i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7456.41	−	3926.00i	−1.24950	−	0.657894i
$330$		0			0
$331$	−4072.07			−0.676198			−0.338099	−	0.941111i	$-0.609784\pi$
$331$	−4072.07			−0.676198			−0.338099	+	0.941111i	$0.609784\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6242.05			1.01803
$336$		0			0
$337$	−8456.90			−1.36699			−0.683496	−	0.729954i	$-0.739542\pi$
$337$	−8456.90			−1.36699			−0.683496	+	0.729954i	$0.739542\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2106.75			0.334566
$342$		0			0
$343$	−740.713	−	6309.12i	−0.116603	−	0.993179i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			7869.37i			1.21744i	0.793387	+	0.608718i	$0.208316\pi$
$347$			7869.37i			1.21744i	−0.793387	+	0.608718i	$0.791684\pi$
$348$		0			0
$349$			3417.23i			0.524127i	0.965051	+	0.262063i	$0.0844030\pi$
$349$			3417.23i			0.524127i	−0.965051	+	0.262063i	$0.915597\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7997.28			−1.20581			−0.602907	−	0.797811i	$-0.705991\pi$
$353$	−7997.28			−1.20581			−0.602907	+	0.797811i	$0.705991\pi$
$354$		0			0
$355$			7879.18i			1.17798i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4888.56i			0.718687i	0.933205	+	0.359343i	$0.116999\pi$
$359$			4888.56i			0.718687i	−0.933205	+	0.359343i	$0.883001\pi$
$360$		0			0
$361$	−14856.4			−2.16597
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	7267.50i		−	1.04219i
$366$		0			0
$367$			10494.6i			1.49269i	0.665561	+	0.746344i	$0.268192\pi$
$367$			10494.6i			1.49269i	−0.665561	+	0.746344i	$0.731808\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1732.44	−	3290.33i	0.242437	−	0.460446i
$372$		0			0
$373$	5396.60			0.749130			0.374565	−	0.927201i	$-0.377792\pi$
$373$	5396.60			0.749130			0.374565	+	0.927201i	$0.377792\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6602.36			−0.901960
$378$		0			0
$379$	10159.5			1.37693			0.688466	−	0.725269i	$-0.258285\pi$
$379$	10159.5			1.37693			0.688466	+	0.725269i	$0.258285\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7006.80			0.934806			0.467403	−	0.884044i	$-0.345190\pi$
$383$	7006.80			0.934806			0.467403	+	0.884044i	$0.345190\pi$
$384$		0			0
$385$	−2691.90	+	5112.57i	−0.356343	+	0.676781i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	8693.77i		−	1.13314i	−0.824013	−	0.566570i	$-0.808270\pi$
$389$		−	8693.77i		−	1.13314i	0.824013	−	0.566570i	$-0.191730\pi$
$390$		0			0
$391$			438.819i			0.0567571i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1859.89			0.236914
$396$		0			0
$397$			7508.65i			0.949241i	0.880191	+	0.474620i	$0.157415\pi$
$397$			7508.65i			0.949241i	−0.880191	+	0.474620i	$0.842585\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	2226.04i		−	0.277215i	−0.990347	−	0.138607i	$-0.955737\pi$
$401$		−	2226.04i		−	0.277215i	0.990347	−	0.138607i	$-0.0442626\pi$
$402$		0			0
$403$	−3162.00			−0.390844
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4815.11i		−	0.586428i
$408$		0			0
$409$			4580.56i			0.553776i	0.960902	+	0.276888i	$0.0893031\pi$
$409$			4580.56i			0.553776i	−0.960902	+	0.276888i	$0.910697\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2625.77	+	1382.53i	0.312846	+	0.164722i
$414$		0			0
$415$	8222.60			0.972606
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9723.41			−1.13370			−0.566849	−	0.823821i	$-0.691838\pi$
$419$	−9723.41			−1.13370			−0.566849	+	0.823821i	$0.691838\pi$
$420$		0			0
$421$	6827.70			0.790408			0.395204	−	0.918593i	$-0.370674\pi$
$421$	6827.70			0.790408			0.395204	+	0.918593i	$0.370674\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−379.854			−0.0433544
$426$		0			0
$427$	−7102.00	+	13488.4i	−0.804894	+	1.52869i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	4641.75i		−	0.518759i	−0.965775	−	0.259379i	$-0.916482\pi$
$431$		−	4641.75i		−	0.518759i	0.965775	−	0.259379i	$-0.0835180\pi$
$432$		0			0
$433$		−	3332.86i		−	0.369900i	−0.982748	−	0.184950i	$-0.940788\pi$
$433$		−	3332.86i		−	0.369900i	0.982748	−	0.184950i	$-0.0592124\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18743.0			2.05172
$438$		0			0
$439$		−	15327.7i		−	1.66640i	−0.552970	−	0.833201i	$-0.686506\pi$
$439$		−	15327.7i		−	1.66640i	0.552970	−	0.833201i	$-0.313494\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5633.11i		−	0.604147i	−0.953285	−	0.302073i	$-0.902321\pi$
$443$		−	5633.11i		−	0.604147i	0.953285	−	0.302073i	$-0.0976788\pi$
$444$		0			0
$445$	3555.90			0.378800
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	17327.5i		−	1.82124i	−0.413248	−	0.910619i	$-0.635606\pi$
$449$		−	17327.5i		−	1.82124i	0.413248	−	0.910619i	$-0.364394\pi$
$450$		0			0
$451$		−	9772.76i		−	1.02036i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4040.23	−	7673.38i	0.416284	−	0.790623i
$456$		0			0
$457$	4912.00			0.502787			0.251394	−	0.967885i	$-0.419111\pi$
$457$	4912.00			0.502787			0.251394	+	0.967885i	$0.419111\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6760.33			−0.682993			−0.341497	−	0.939883i	$-0.610934\pi$
$461$	−6760.33			−0.682993			−0.341497	+	0.939883i	$0.610934\pi$
$462$		0			0
$463$	−16397.3			−1.64589			−0.822945	−	0.568121i	$-0.807670\pi$
$463$	−16397.3			−1.64589			−0.822945	+	0.568121i	$0.807670\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12580.3			−1.24657			−0.623284	−	0.781996i	$-0.714202\pi$
$467$	−12580.3			−1.24657			−0.623284	+	0.781996i	$0.714202\pi$
$468$		0			0
$469$	−6671.35	−	3512.64i	−0.656832	−	0.345839i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			3008.81i			0.292485i
$474$		0			0
$475$			16224.5i			1.56722i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12746.3			1.21585			0.607927	−	0.793993i	$-0.292001\pi$
$479$	12746.3			1.21585			0.607927	+	0.793993i	$0.292001\pi$
$480$		0			0
$481$			7226.94i			0.685073i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			11257.3i			1.05395i
$486$		0			0
$487$	−1573.42			−0.146404			−0.0732019	−	0.997317i	$-0.523322\pi$
$487$	−1573.42			−0.146404			−0.0732019	+	0.997317i	$0.523322\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17035.7i		−	1.56581i	−0.622142	−	0.782904i	$-0.713737\pi$
$491$		−	17035.7i		−	1.56581i	0.622142	−	0.782904i	$-0.286263\pi$
$492$		0			0
$493$		−	745.901i		−	0.0681413i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4433.92	−	8421.08i	0.400178	−	0.760034i
$498$		0			0
$499$	−41.2224			−0.00369813			−0.00184907	−	0.999998i	$-0.500589\pi$
$499$	−41.2224			−0.00369813			−0.00184907	+	0.999998i	$0.500589\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6871.06			−0.609076			−0.304538	−	0.952500i	$-0.598502\pi$
$503$	−6871.06			−0.609076			−0.304538	+	0.952500i	$0.598502\pi$
$504$		0			0
$505$	−11155.5			−0.982996
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2205.65			0.192070			0.0960349	−	0.995378i	$-0.469384\pi$
$509$	2205.65			0.192070			0.0960349	+	0.995378i	$0.469384\pi$
$510$		0			0
$511$	−4089.70	+	7767.32i	−0.354046	+	0.672419i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	12457.0i		−	1.06586i
$516$		0			0
$517$		−	9258.01i		−	0.787557i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7724.37			−0.649541			−0.324770	−	0.945793i	$-0.605287\pi$
$521$	−7724.37			−0.649541			−0.324770	+	0.945793i	$0.605287\pi$
$522$		0			0
$523$			2214.30i			0.185133i	0.995706	+	0.0925667i	$0.0295071\pi$
$523$			2214.30i			0.185133i	−0.995706	+	0.0925667i	$0.970493\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	357.226i		−	0.0295276i
$528$		0			0
$529$	−4010.50			−0.329621
$530$		0			0
$531$		0			0
$532$		0			0
$533$			14667.8i			1.19199i
$534$		0			0
$535$			13563.7i			1.09609i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5754.07	−	3949.35i	0.459824	−	0.315604i
$540$		0			0
$541$	13070.3			1.03870			0.519349	−	0.854562i	$-0.326174\pi$
$541$	13070.3			1.03870			0.519349	+	0.854562i	$0.326174\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−24548.8			−1.92946
$546$		0			0
$547$	−19391.3			−1.51574			−0.757872	−	0.652403i	$-0.773761\pi$
$547$	−19391.3			−1.51574			−0.757872	+	0.652403i	$0.773761\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−31859.2			−2.46325
$552$		0			0
$553$	−1987.80	−	1046.63i	−0.152857	−	0.0804832i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			23645.8i			1.79875i	0.437180	+	0.899374i	$0.355977\pi$
$557$			23645.8i			1.79875i	−0.437180	+	0.899374i	$0.644023\pi$
$558$		0			0
$559$		−	4515.88i		−	0.341684i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−23270.4			−1.74198			−0.870988	−	0.491305i	$-0.836520\pi$
$563$	−23270.4			−1.74198			−0.870988	+	0.491305i	$0.836520\pi$
$564$		0			0
$565$		−	11354.9i		−	0.845492i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	13608.8i		−	1.00266i	−0.865257	−	0.501329i	$-0.832845\pi$
$569$		−	13608.8i		−	1.00266i	0.865257	−	0.501329i	$-0.167155\pi$
$570$		0			0
$571$	−21116.3			−1.54762			−0.773809	−	0.633419i	$-0.781651\pi$
$571$	−21116.3			−1.54762			−0.773809	+	0.633419i	$0.781651\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	14003.7i		−	1.01564i
$576$		0			0
$577$			10262.4i			0.740429i	0.928946	+	0.370215i	$0.120716\pi$
$577$			10262.4i			0.740429i	−0.928946	+	0.370215i	$0.879284\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8788.11	−	4627.17i	−0.627525	−	0.330408i
$582$		0			0
$583$	4085.33			0.290218
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13227.4			−0.930072			−0.465036	−	0.885292i	$-0.653959\pi$
$587$	−13227.4			−0.930072			−0.465036	+	0.885292i	$0.653959\pi$
$588$		0			0
$589$	−15258.0			−1.06739
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8258.72			0.571914			0.285957	−	0.958242i	$-0.407689\pi$
$593$	8258.72			0.571914			0.285957	+	0.958242i	$0.407689\pi$
$594$		0			0
$595$	866.899	+	456.445i	0.0597301	+	0.0314494i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			4099.15i			0.279610i	0.990179	+	0.139805i	$0.0446476\pi$
$599$			4099.15i			0.279610i	−0.990179	+	0.139805i	$0.955352\pi$
$600$		0			0
$601$		−	480.297i		−	0.0325985i	−0.999867	−	0.0162993i	$-0.994812\pi$
$601$		−	480.297i		−	0.0325985i	0.999867	−	0.0162993i	$-0.00518844\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14060.3			0.944849
$606$		0			0
$607$		−	12561.5i		−	0.839962i	−0.907533	−	0.419981i	$-0.862037\pi$
$607$		−	12561.5i		−	0.839962i	0.907533	−	0.419981i	$-0.137963\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			13895.2i			0.920033i
$612$		0			0
$613$	30125.4			1.98492			0.992458	−	0.122587i	$-0.0391191\pi$
$613$	30125.4			1.98492			0.992458	+	0.122587i	$0.0391191\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	13044.4i		−	0.851132i	−0.904927	−	0.425566i	$-0.860075\pi$
$617$		−	13044.4i		−	0.851132i	0.904927	−	0.425566i	$-0.139925\pi$
$618$		0			0
$619$		−	9097.71i		−	0.590740i	−0.955383	−	0.295370i	$-0.904557\pi$
$619$		−	9097.71i		−	0.590740i	0.955383	−	0.295370i	$-0.0954428\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3800.46	−	2001.04i	−0.244401	−	0.128684i
$624$		0			0
$625$	−17265.5			−1.10499
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−816.462			−0.0517559
$630$		0			0
$631$	−1769.90			−0.111662			−0.0558310	−	0.998440i	$-0.517781\pi$
$631$	−1769.90			−0.111662			−0.0558310	+	0.998440i	$0.517781\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	37239.2			2.32723
$636$		0			0
$637$	−8636.20	+	5927.52i	−0.537172	+	0.368692i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			9162.76i			0.564598i	0.959326	+	0.282299i	$0.0910970\pi$
$641$			9162.76i			0.564598i	−0.959326	+	0.282299i	$0.908903\pi$
$642$		0			0
$643$		−	24337.4i		−	1.49265i	−0.665581	−	0.746326i	$-0.731816\pi$
$643$		−	24337.4i		−	1.49265i	0.665581	−	0.746326i	$-0.268184\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	30287.2			1.84036			0.920180	−	0.391496i	$-0.128042\pi$
$647$	30287.2			1.84036			0.920180	+	0.391496i	$0.128042\pi$
$648$		0			0
$649$			3260.20i			0.197186i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			16742.6i			1.00335i	0.865057	+	0.501674i	$0.167282\pi$
$653$			16742.6i			1.00335i	−0.865057	+	0.501674i	$0.832718\pi$
$654$		0			0
$655$	7799.40			0.465264
$656$		0			0
$657$		0			0
$658$		0			0
$659$			9709.90i			0.573966i	0.957936	+	0.286983i	$0.0926524\pi$
$659$			9709.90i			0.573966i	−0.957936	+	0.286983i	$0.907348\pi$
$660$		0			0
$661$		−	12089.8i		−	0.711405i	−0.934599	−	0.355703i	$-0.884242\pi$
$661$		−	12089.8i		−	0.711405i	0.934599	−	0.355703i	$-0.115758\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	19495.9	−	37027.4i	1.13687	−	2.15919i
$666$		0			0
$667$	27498.4			1.59631
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16747.4			−0.963529
$672$		0			0
$673$	1800.90			0.103150			0.0515748	−	0.998669i	$-0.483576\pi$
$673$	1800.90			0.103150			0.0515748	+	0.998669i	$0.483576\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−28307.8			−1.60702			−0.803512	−	0.595288i	$-0.797038\pi$
$677$	−28307.8			−1.60702			−0.803512	+	0.595288i	$0.797038\pi$
$678$		0			0
$679$	6334.90	−	12031.5i	0.358043	−	0.680010i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	11836.8i		−	0.663138i	−0.943431	−	0.331569i	$-0.892422\pi$
$683$		−	11836.8i		−	0.663138i	0.943431	−	0.331569i	$-0.107578\pi$
$684$		0			0
$685$		−	30953.0i		−	1.72650i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6131.61			−0.339036
$690$		0			0
$691$			18898.3i			1.04041i	0.854041	+	0.520205i	$0.174145\pi$
$691$			18898.3i			1.04041i	−0.854041	+	0.520205i	$0.825855\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			48245.9i			2.63320i
$696$		0			0
$697$	−1657.09			−0.0900529
$698$		0			0
$699$		0			0
$700$		0			0
$701$			3733.84i			0.201177i	0.994928	+	0.100589i	$0.0320726\pi$
$701$			3733.84i			0.201177i	−0.994928	+	0.100589i	$0.967927\pi$
$702$		0			0
$703$			34873.1i			1.87093i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11922.7	+	6277.62i	0.634229	+	0.333938i
$708$		0			0
$709$	−3344.15			−0.177140			−0.0885699	−	0.996070i	$-0.528230\pi$
$709$	−3344.15			−0.177140			−0.0885699	+	0.996070i	$0.528230\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13169.5			0.691728
$714$		0			0
$715$	9527.40			0.498328
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15255.5			0.791285			0.395642	−	0.918405i	$-0.370522\pi$
$719$	15255.5			0.791285			0.395642	+	0.918405i	$0.370522\pi$
$720$		0			0
$721$	−7010.00	+	13313.7i	−0.362089	+	0.687694i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			23803.4i			1.21936i
$726$		0			0
$727$		−	31743.3i		−	1.61939i	−0.586854	−	0.809693i	$-0.699634\pi$
$727$		−	31743.3i		−	1.61939i	0.586854	−	0.809693i	$-0.300366\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	510.181			0.0258136
$732$		0			0
$733$		−	11147.2i		−	0.561709i	−0.959750	−	0.280854i	$-0.909382\pi$
$733$		−	11147.2i		−	0.561709i	0.959750	−	0.280854i	$-0.0906178\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8283.26i		−	0.414000i
$738$		0			0
$739$	3750.09			0.186670			0.0933352	−	0.995635i	$-0.470247\pi$
$739$	3750.09			0.186670			0.0933352	+	0.995635i	$0.470247\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	12014.8i		−	0.593245i	−0.954995	−	0.296623i	$-0.904140\pi$
$743$		−	12014.8i		−	0.593245i	0.954995	−	0.296623i	$-0.0958603\pi$
$744$		0			0
$745$			46865.3i			2.30471i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7632.79	−	14496.5i	0.372358	−	0.707197i
$750$		0			0
$751$	23857.3			1.15921			0.579603	−	0.814899i	$-0.303207\pi$
$751$	23857.3			1.15921			0.579603	+	0.814899i	$0.303207\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	27615.1			1.33115
$756$		0			0
$757$	18603.5			0.893206			0.446603	−	0.894732i	$-0.352634\pi$
$757$	18603.5			0.893206			0.446603	+	0.894732i	$0.352634\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3964.70			−0.188857			−0.0944286	−	0.995532i	$-0.530102\pi$
$761$	−3964.70			−0.188857			−0.0944286	+	0.995532i	$0.530102\pi$
$762$		0			0
$763$	26237.2	+	13814.6i	1.24489	+	0.655466i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	4893.18i		−	0.230355i
$768$		0			0
$769$			20811.6i			0.975925i	0.872865	+	0.487962i	$0.162260\pi$
$769$			20811.6i			0.975925i	−0.872865	+	0.487962i	$0.837740\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4051.69			0.188524			0.0942621	−	0.995547i	$-0.469951\pi$
$773$	4051.69			0.188524			0.0942621	+	0.995547i	$0.469951\pi$
$774$		0			0
$775$			11399.9i			0.528382i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			70778.4i			3.25533i
$780$		0			0
$781$	10455.7			0.479048
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	20638.7i		−	0.938380i
$786$		0			0
$787$		−	7812.44i		−	0.353854i	−0.984224	−	0.176927i	$-0.943384\pi$
$787$		−	7812.44i		−	0.353854i	0.984224	−	0.176927i	$-0.0566157\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6389.82	+	12135.8i	−0.287226	+	0.545511i
$792$		0			0
$793$	25136.0			1.12561
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20518.5			0.911925			0.455962	−	0.889999i	$-0.349295\pi$
$797$	20518.5			0.911925			0.455962	+	0.889999i	$0.349295\pi$
$798$		0			0
$799$	−1569.81			−0.0695068
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9644.04			−0.423824
$804$		0			0
$805$	−16827.3	+	31959.1i	−0.736751	+	1.39927i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			5958.80i			0.258962i	0.991582	+	0.129481i	$0.0413311\pi$
$809$			5958.80i			0.258962i	−0.991582	+	0.129481i	$0.958669\pi$
$810$		0			0
$811$		−	35305.0i		−	1.52864i	−0.644838	−	0.764319i	$-0.723075\pi$
$811$		−	35305.0i		−	1.52864i	0.644838	−	0.764319i	$-0.276925\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11446.1			0.491948
$816$		0			0
$817$		−	21791.1i		−	0.933137i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			26619.9i			1.13160i	0.824543	+	0.565799i	$0.191432\pi$
$821$			26619.9i			1.13160i	−0.824543	+	0.565799i	$0.808568\pi$
$822$		0			0
$823$	−26093.9			−1.10520			−0.552598	−	0.833448i	$-0.686363\pi$
$823$	−26093.9			−1.10520			−0.552598	+	0.833448i	$0.686363\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	264.904i		−	0.0111386i	−0.999984	−	0.00556930i	$-0.998227\pi$
$827$		−	264.904i		−	0.0111386i	0.999984	−	0.00556930i	$-0.00177277\pi$
$828$		0			0
$829$		−	22454.7i		−	0.940755i	−0.882465	−	0.470377i	$-0.844118\pi$
$829$		−	22454.7i		−	0.940755i	0.882465	−	0.470377i	$-0.155882\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−669.661	−	975.674i	−0.0278540	−	0.0405824i
$834$		0			0
$835$	40255.0			1.66836
$836$		0			0
$837$		0			0
$838$		0			0
$839$	31288.8			1.28750			0.643748	−	0.765238i	$-0.277379\pi$
$839$	31288.8			1.28750			0.643748	+	0.765238i	$0.277379\pi$
$840$		0			0
$841$	−22352.5			−0.916499
$842$		0			0
$843$		0			0
$844$		0			0
$845$	19387.0			0.789270
$846$		0			0
$847$	−15027.3	−	7912.28i	−0.609617	−	0.320979i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	30099.7i		−	1.21246i
$852$		0			0
$853$		−	4559.95i		−	0.183036i	−0.995803	−	0.0915180i	$-0.970828\pi$
$853$		−	4559.95i		−	0.183036i	0.995803	−	0.0915180i	$-0.0291719\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17870.2			−0.712294			−0.356147	−	0.934430i	$-0.615910\pi$
$857$	−17870.2			−0.712294			−0.356147	+	0.934430i	$0.615910\pi$
$858$		0			0
$859$			7666.55i			0.304516i	0.988341	+	0.152258i	$0.0486545\pi$
$859$			7666.55i			0.304516i	−0.988341	+	0.152258i	$0.951346\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			5407.98i			0.213314i	0.994296	+	0.106657i	$0.0340146\pi$
$863$			5407.98i			0.213314i	−0.994296	+	0.106657i	$0.965985\pi$
$864$		0			0
$865$	56376.9			2.21604
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	2468.09i		−	0.0963454i
$870$		0			0
$871$			12432.2i			0.483639i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3743.92	+	1971.27i	0.144649	+	0.0761613i
$876$		0			0
$877$	5694.80			0.219270			0.109635	−	0.993972i	$-0.465032\pi$
$877$	5694.80			0.219270			0.109635	+	0.993972i	$0.465032\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	17761.3			0.679221			0.339611	−	0.940566i	$-0.389705\pi$
$881$	17761.3			0.679221			0.339611	+	0.940566i	$0.389705\pi$
$882$		0			0
$883$	−27753.8			−1.05774			−0.528872	−	0.848701i	$-0.677385\pi$
$883$	−27753.8			−1.05774			−0.528872	+	0.848701i	$0.677385\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22812.0			−0.863532			−0.431766	−	0.901986i	$-0.642109\pi$
$887$	−22812.0			−0.863532			−0.431766	+	0.901986i	$0.642109\pi$
$888$		0			0
$889$	−39800.3	−	20955.9i	−1.50153	−	0.790594i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			67050.4i			2.51260i
$894$		0			0
$895$			61041.5i			2.27977i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22385.4			−0.830473
$900$		0			0
$901$		−	692.718i		−	0.0256135i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			18871.1i			0.693146i
$906$		0			0
$907$	24997.9			0.915151			0.457576	−	0.889171i	$-0.348718\pi$
$907$	24997.9			0.915151			0.457576	+	0.889171i	$0.348718\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	49221.0i		−	1.79008i	−0.445985	−	0.895040i	$-0.647147\pi$
$911$		−	49221.0i		−	1.79008i	0.445985	−	0.895040i	$-0.352853\pi$
$912$		0			0
$913$		−	10911.5i		−	0.395528i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8335.80	−	4389.02i	−0.300188	−	0.158057i
$918$		0			0
$919$	23886.2			0.857380			0.428690	−	0.903452i	$-0.358975\pi$
$919$	23886.2			0.857380			0.428690	+	0.903452i	$0.358975\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−15692.9			−0.559629
$924$		0			0
$925$	26055.1			0.926149
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20799.6			−0.734566			−0.367283	−	0.930109i	$-0.619712\pi$
$929$	−20799.6			−0.734566			−0.367283	+	0.930109i	$0.619712\pi$
$930$		0			0
$931$	−41673.4	+	28602.9i	−1.46701	+	1.00690i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			1076.36i			0.0376477i
$936$		0			0
$937$		−	29654.7i		−	1.03391i	−0.856011	−	0.516957i	$-0.827065\pi$
$937$		−	29654.7i		−	1.03391i	0.856011	−	0.516957i	$-0.172935\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15700.9			0.543927			0.271963	−	0.962308i	$-0.412327\pi$
$941$	15700.9			0.543927			0.271963	+	0.962308i	$0.412327\pi$
$942$		0			0
$943$		−	61090.3i		−	2.10962i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			22277.8i			0.764446i	0.924070	+	0.382223i	$0.124841\pi$
$947$			22277.8i			0.764446i	−0.924070	+	0.382223i	$0.875159\pi$
$948$		0			0
$949$	14474.6			0.495116
$950$		0			0
$951$		0			0
$952$		0			0
$953$			26599.9i			0.904149i	0.891980	+	0.452074i	$0.149316\pi$
$953$			26599.9i			0.904149i	−0.891980	+	0.452074i	$0.850684\pi$
$954$		0			0
$955$		−	48620.3i		−	1.64745i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−17418.4	+	33081.7i	−0.586517	+	1.11394i
$960$		0			0
$961$	19070.2			0.640133
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3682.99			−0.122860
$966$		0			0
$967$	482.698			0.0160523			0.00802613	−	0.999968i	$-0.497445\pi$
$967$	482.698			0.0160523			0.00802613	+	0.999968i	$0.497445\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25071.7			0.828621			0.414310	−	0.910136i	$-0.364023\pi$
$971$	25071.7			0.828621			0.414310	+	0.910136i	$0.364023\pi$
$972$		0			0
$973$	27149.8	−	51564.0i	0.894535	−	1.69894i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			33988.9i			1.11300i	0.830847	+	0.556500i	$0.187856\pi$
$977$			33988.9i			1.11300i	−0.830847	+	0.556500i	$0.812144\pi$
$978$		0			0
$979$		−	4718.71i		−	0.154046i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5527.13			−0.179337			−0.0896685	−	0.995972i	$-0.528581\pi$
$983$	−5527.13			−0.179337			−0.0896685	+	0.995972i	$0.528581\pi$
$984$		0			0
$985$			39334.4i			1.27238i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			18808.3i			0.604722i
$990$		0			0
$991$	−33200.9			−1.06424			−0.532119	−	0.846669i	$-0.678604\pi$
$991$	−33200.9			−1.06424			−0.532119	+	0.846669i	$0.678604\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	20005.6i		−	0.637407i
$996$		0			0
$997$		−	23520.2i		−	0.747134i	−0.927603	−	0.373567i	$-0.878135\pi$
$997$		−	23520.2i		−	0.747134i	0.927603	−	0.373567i	$-0.121865\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.k.d.881.7		8
3.2	odd	2	inner	1008.4.k.d.881.1		8
4.3	odd	2		252.4.f.a.125.8	yes	8
7.6	odd	2	inner	1008.4.k.d.881.2		8
12.11	even	2		252.4.f.a.125.2	yes	8
21.20	even	2	inner	1008.4.k.d.881.8		8
28.3	even	6		1764.4.t.a.1097.8		16
28.11	odd	6		1764.4.t.a.1097.2		16
28.19	even	6		1764.4.t.a.521.7		16
28.23	odd	6		1764.4.t.a.521.1		16
28.27	even	2		252.4.f.a.125.1	✓	8
84.11	even	6		1764.4.t.a.1097.7		16
84.23	even	6		1764.4.t.a.521.8		16
84.47	odd	6		1764.4.t.a.521.2		16
84.59	odd	6		1764.4.t.a.1097.1		16
84.83	odd	2		252.4.f.a.125.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.f.a.125.1	✓	8	28.27	even	2
252.4.f.a.125.2	yes	8	12.11	even	2
252.4.f.a.125.7	yes	8	84.83	odd	2
252.4.f.a.125.8	yes	8	4.3	odd	2
1008.4.k.d.881.1		8	3.2	odd	2	inner
1008.4.k.d.881.2		8	7.6	odd	2	inner
1008.4.k.d.881.7		8	1.1	even	1	trivial
1008.4.k.d.881.8		8	21.20	even	2	inner
1764.4.t.a.521.1		16	28.23	odd	6
1764.4.t.a.521.2		16	84.47	odd	6
1764.4.t.a.521.7		16	28.19	even	6
1764.4.t.a.521.8		16	84.23	even	6
1764.4.t.a.1097.1		16	84.59	odd	6
1764.4.t.a.1097.2		16	28.11	odd	6
1764.4.t.a.1097.7		16	84.11	even	6
1764.4.t.a.1097.8		16	28.3	even	6

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15.3330 q^{5} +(-16.3875 - 8.62844i) q^{7} +O(q^{10})\)	\(q+15.3330 q^{5} +(-16.3875 - 8.62844i) q^{7} -20.3470i q^{11} +30.5385i q^{13} -3.45008 q^{17} +147.361i q^{19} -127.191i q^{23} +110.100 q^{25} +216.198i q^{29} +103.541i q^{31} +(-251.269 - 132.300i) q^{35} +236.650 q^{37} +480.305 q^{41} -147.875 q^{43} +455.006 q^{47} +(194.100 + 282.797i) q^{49} +200.783i q^{53} -311.980i q^{55} -160.230 q^{59} -823.092i q^{61} +468.246i q^{65} +407.100 q^{67} +513.872i q^{71} -473.979i q^{73} +(-175.563 + 333.436i) q^{77} +121.300 q^{79} +536.269 q^{83} -52.9000 q^{85} +231.912 q^{89} +(263.500 - 500.450i) q^{91} +2259.49i q^{95} +734.188i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^7	\( 8 q - 16 q^{7} - 40 q^{25} + 512 q^{37} - 32 q^{43} + 632 q^{49} + 2336 q^{67} - 1792 q^{79} - 1344 q^{85} - 2496 q^{91}+O(q^{100}) \) 8 * q - 16 * q^7 - 40 * q^25 + 512 * q^37 - 32 * q^43 + 632 * q^49 + 2336 * q^67 - 1792 * q^79 - 1344 * q^85 - 2496 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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