LMFDB - Embedded newform 1350.4.c.z.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.c (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:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.z.649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} -24.6969i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} -24.6969i q^{7} +8.00000i q^{8} +44.3939 q^{11} -78.4847i q^{13} -49.3939 q^{14} +16.0000 q^{16} -91.4847i q^{17} +89.4847 q^{19} -88.7878i q^{22} +82.4847i q^{23} -156.969 q^{26} +98.7878i q^{28} +13.4847 q^{29} +268.969 q^{31} -32.0000i q^{32} -182.969 q^{34} -249.091i q^{37} -178.969i q^{38} +298.182 q^{41} +262.879i q^{43} -177.576 q^{44} +164.969 q^{46} +595.393i q^{47} -266.939 q^{49} +313.939i q^{52} -457.485i q^{53} +197.576 q^{56} -26.9694i q^{58} +370.757 q^{59} -449.454 q^{61} -537.939i q^{62} -64.0000 q^{64} -385.909i q^{67} +365.939i q^{68} +775.454 q^{71} +341.818i q^{73} -498.182 q^{74} -357.939 q^{76} -1096.39i q^{77} +371.362 q^{79} -596.363i q^{82} -1187.88i q^{83} +525.757 q^{86} +355.151i q^{88} +197.728 q^{89} -1938.33 q^{91} -329.939i q^{92} +1190.79 q^{94} -860.757i q^{97} +533.878i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4}+O(q^{10}) $ 4 * q - 16 * q^4	$ 4 q - 16 q^{4} + 60 q^{11} - 80 q^{14} + 64 q^{16} + 64 q^{19} - 40 q^{26} - 240 q^{29} + 488 q^{31} - 144 q^{34} + 840 q^{41} - 240 q^{44} + 72 q^{46} + 108 q^{49} + 320 q^{56} + 660 q^{59} - 916 q^{61} - 256 q^{64} + 2220 q^{71} - 1640 q^{74} - 256 q^{76} - 1160 q^{79} + 1280 q^{86} + 1320 q^{89} - 4520 q^{91} + 648 q^{94}+O(q^{100}) $ 4 * q - 16 * q^4 + 60 * q^11 - 80 * q^14 + 64 * q^16 + 64 * q^19 - 40 * q^26 - 240 * q^29 + 488 * q^31 - 144 * q^34 + 840 * q^41 - 240 * q^44 + 72 * q^46 + 108 * q^49 + 320 * q^56 + 660 * q^59 - 916 * q^61 - 256 * q^64 + 2220 * q^71 - 1640 * q^74 - 256 * q^76 - 1160 * q^79 + 1280 * q^86 + 1320 * q^89 - 4520 * q^91 + 648 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 24.6969i	− 1.33351i	−0.745277	−	0.666755i	$-0.767683\pi$
$7$	− 24.6969i	− 1.33351i	0.745277	−	0.666755i	$-0.232317\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	44.3939	1.21684	0.608421	−	0.793615i	$-0.291803\pi$
$11$	44.3939	1.21684	0.608421	+	0.793615i	$0.291803\pi$
$12$	0	0
$13$	− 78.4847i	− 1.67444i	−0.546865	−	0.837221i	$-0.684179\pi$
$13$	− 78.4847i	− 1.67444i	0.546865	−	0.837221i	$-0.315821\pi$
$14$	−49.3939	−0.942933
$15$	0	0
$16$	16.0000	0.250000
$17$	− 91.4847i	− 1.30519i	−0.757705	−	0.652597i	$-0.773680\pi$
$17$	− 91.4847i	− 1.30519i	0.757705	−	0.652597i	$-0.226320\pi$
$18$	0	0
$19$	89.4847	1.08048	0.540242	−	0.841510i	$-0.318333\pi$
$19$	89.4847	1.08048	0.540242	+	0.841510i	$0.318333\pi$
$20$	0	0
$21$	0	0
$22$	− 88.7878i	− 0.860437i
$23$	82.4847i	0.747793i	0.927470	+	0.373897i	$0.121979\pi$
$23$	82.4847i	0.747793i	−0.927470	+	0.373897i	$0.878021\pi$
$24$	0	0
$25$	0	0
$26$	−156.969	−1.18401
$27$	0	0
$28$	98.7878i	0.666755i
$29$	13.4847	0.0863464	0.0431732	−	0.999068i	$-0.486253\pi$
$29$	13.4847	0.0863464	0.0431732	+	0.999068i	$0.486253\pi$
$30$	0	0
$31$	268.969	1.55833	0.779167	−	0.626817i	$-0.215643\pi$
$31$	268.969	1.55833	0.779167	+	0.626817i	$0.215643\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	−182.969	−0.922911
$35$	0	0
$36$	0	0
$37$	− 249.091i	− 1.10676i	−0.832927	−	0.553382i	$-0.813337\pi$
$37$	− 249.091i	− 1.10676i	0.832927	−	0.553382i	$-0.186663\pi$
$38$	− 178.969i	− 0.764018i
$39$	0	0
$40$	0	0
$41$	298.182	1.13581	0.567904	−	0.823095i	$-0.307754\pi$
$41$	298.182	1.13581	0.567904	+	0.823095i	$0.307754\pi$
$42$	0	0
$43$	262.879i	0.932293i	0.884708	+	0.466147i	$0.154358\pi$
$43$	262.879i	0.932293i	−0.884708	+	0.466147i	$0.845642\pi$
$44$	−177.576	−0.608421
$45$	0	0
$46$	164.969	0.528770
$47$	595.393i	1.84781i	0.382624	+	0.923904i	$0.375020\pi$
$47$	595.393i	1.84781i	−0.382624	+	0.923904i	$0.624980\pi$
$48$	0	0
$49$	−266.939	−0.778247
$50$	0	0
$51$	0	0
$52$	313.939i	0.837221i
$53$	− 457.485i	− 1.18567i	−0.805325	−	0.592834i	$-0.798009\pi$
$53$	− 457.485i	− 1.18567i	0.805325	−	0.592834i	$-0.201991\pi$
$54$	0	0
$55$	0	0
$56$	197.576	0.471467
$57$	0	0
$58$	− 26.9694i	− 0.0610561i
$59$	370.757	0.818110	0.409055	−	0.912510i	$-0.365858\pi$
$59$	370.757	0.818110	0.409055	+	0.912510i	$0.365858\pi$
$60$	0	0
$61$	−449.454	−0.943388	−0.471694	−	0.881762i	$-0.656357\pi$
$61$	−449.454	−0.943388	−0.471694	+	0.881762i	$0.656357\pi$
$62$	− 537.939i	− 1.10191i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 385.909i	− 0.703676i	−0.936061	−	0.351838i	$-0.885557\pi$
$67$	− 385.909i	− 0.703676i	0.936061	−	0.351838i	$-0.114443\pi$
$68$	365.939i	0.652597i
$69$	0	0
$70$	0	0
$71$	775.454	1.29619	0.648095	−	0.761560i	$-0.275566\pi$
$71$	775.454	1.29619	0.648095	+	0.761560i	$0.275566\pi$
$72$	0	0
$73$	341.818i	0.548039i	0.961724	+	0.274019i	$0.0883532\pi$
$73$	341.818i	0.548039i	−0.961724	+	0.274019i	$0.911647\pi$
$74$	−498.182	−0.782601
$75$	0	0
$76$	−357.939	−0.540242
$77$	− 1096.39i	− 1.62267i
$78$	0	0
$79$	371.362	0.528880	0.264440	−	0.964402i	$-0.414813\pi$
$79$	371.362	0.528880	0.264440	+	0.964402i	$0.414813\pi$
$80$	0	0
$81$	0	0
$82$	− 596.363i	− 0.803138i
$83$	− 1187.88i	− 1.57092i	−0.618911	−	0.785461i	$-0.712426\pi$
$83$	− 1187.88i	− 1.57092i	0.618911	−	0.785461i	$-0.287574\pi$
$84$	0	0
$85$	0	0
$86$	525.757	0.659231
$87$	0	0
$88$	355.151i	0.430218i
$89$	197.728	0.235495	0.117748	−	0.993044i	$-0.462433\pi$
$89$	197.728	0.235495	0.117748	+	0.993044i	$0.462433\pi$
$90$	0	0
$91$	−1938.33	−2.23288
$92$	− 329.939i	− 0.373897i
$93$	0	0
$94$	1190.79	1.30660
$95$	0	0
$96$	0	0
$97$	− 860.757i	− 0.900996i	−0.892777	−	0.450498i	$-0.851246\pi$
$97$	− 860.757i	− 0.900996i	0.892777	−	0.450498i	$-0.148754\pi$
$98$	533.878i	0.550304i
$99$	0	0
$100$	0	0
$101$	−414.695	−0.408551	−0.204276	−	0.978913i	$-0.565484\pi$
$101$	−414.695	−0.408551	−0.204276	+	0.978913i	$0.565484\pi$
$102$	0	0
$103$	1675.45i	1.60279i	0.598136	+	0.801395i	$0.295908\pi$
$103$	1675.45i	1.60279i	−0.598136	+	0.801395i	$0.704092\pi$
$104$	627.878	0.592004
$105$	0	0
$106$	−914.969	−0.838393
$107$	− 6.21431i	− 0.00561458i	−0.999996	−	0.00280729i	$-0.999106\pi$
$107$	− 6.21431i	− 0.00561458i	0.999996	−	0.00280729i	$-0.000893589\pi$
$108$	0	0
$109$	−1483.94	−1.30400	−0.651998	−	0.758221i	$-0.726069\pi$
$109$	−1483.94	−1.30400	−0.651998	+	0.758221i	$0.726069\pi$
$110$	0	0
$111$	0	0
$112$	− 395.151i	− 0.333377i
$113$	800.847i	0.666702i	0.942803	+	0.333351i	$0.108179\pi$
$113$	800.847i	0.666702i	−0.942803	+	0.333351i	$0.891821\pi$
$114$	0	0
$115$	0	0
$116$	−53.9388	−0.0431732
$117$	0	0
$118$	− 741.514i	− 0.578491i
$119$	−2259.39	−1.74049
$120$	0	0
$121$	639.816	0.480703
$122$	898.908i	0.667076i
$123$	0	0
$124$	−1075.88	−0.779167
$125$	0	0
$126$	0	0
$127$	84.5428i	0.0590706i	0.999564	+	0.0295353i	$0.00940274\pi$
$127$	84.5428i	0.0590706i	−0.999564	+	0.0295353i	$0.990597\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	130.151	0.0868042	0.0434021	−	0.999058i	$-0.486180\pi$
$131$	130.151	0.0868042	0.0434021	+	0.999058i	$0.486180\pi$
$132$	0	0
$133$	− 2210.00i	− 1.44084i
$134$	−771.818	−0.497574
$135$	0	0
$136$	731.878	0.461456
$137$	− 2368.09i	− 1.47678i	−0.674372	−	0.738392i	$-0.735586\pi$
$137$	− 2368.09i	− 1.47678i	0.674372	−	0.738392i	$-0.264414\pi$
$138$	0	0
$139$	−2328.27	−1.42073	−0.710364	−	0.703834i	$-0.751470\pi$
$139$	−2328.27	−1.42073	−0.710364	+	0.703834i	$0.751470\pi$
$140$	0	0
$141$	0	0
$142$	− 1550.91i	− 0.916545i
$143$	− 3484.24i	− 2.03753i
$144$	0	0
$145$	0	0
$146$	683.637	0.387522
$147$	0	0
$148$	996.363i	0.553382i
$149$	633.637	0.348386	0.174193	−	0.984712i	$-0.444268\pi$
$149$	633.637	0.348386	0.174193	+	0.984712i	$0.444268\pi$
$150$	0	0
$151$	−773.730	−0.416988	−0.208494	−	0.978024i	$-0.566856\pi$
$151$	−773.730	−0.416988	−0.208494	+	0.978024i	$0.566856\pi$
$152$	715.878i	0.382009i
$153$	0	0
$154$	−2192.79	−1.14740
$155$	0	0
$156$	0	0
$157$	− 3104.24i	− 1.57800i	−0.614395	−	0.788999i	$-0.710600\pi$
$157$	− 3104.24i	− 1.57800i	0.614395	−	0.788999i	$-0.289400\pi$
$158$	− 742.724i	− 0.373975i
$159$	0	0
$160$	0	0
$161$	2037.12	0.997189
$162$	0	0
$163$	1277.57i	0.613910i	0.951724	+	0.306955i	$0.0993101\pi$
$163$	1277.57i	0.613910i	−0.951724	+	0.306955i	$0.900690\pi$
$164$	−1192.73	−0.567904
$165$	0	0
$166$	−2375.76	−1.11081
$167$	208.423i	0.0965766i	0.998833	+	0.0482883i	$0.0153766\pi$
$167$	208.423i	0.0965766i	−0.998833	+	0.0482883i	$0.984623\pi$
$168$	0	0
$169$	−3962.85	−1.80375
$170$	0	0
$171$	0	0
$172$	− 1051.51i	− 0.466147i
$173$	3409.06i	1.49818i	0.662466	+	0.749092i	$0.269510\pi$
$173$	3409.06i	1.49818i	−0.662466	+	0.749092i	$0.730490\pi$
$174$	0	0
$175$	0	0
$176$	710.302	0.304210
$177$	0	0
$178$	− 395.455i	− 0.166520i
$179$	3422.57	1.42913	0.714567	−	0.699567i	$-0.246624\pi$
$179$	3422.57	1.42913	0.714567	+	0.699567i	$0.246624\pi$
$180$	0	0
$181$	−677.760	−0.278329	−0.139164	−	0.990269i	$-0.544442\pi$
$181$	−677.760	−0.278329	−0.139164	+	0.990269i	$0.544442\pi$
$182$	3876.66i	1.57889i
$183$	0	0
$184$	−659.878	−0.264385
$185$	0	0
$186$	0	0
$187$	− 4061.36i	− 1.58821i
$188$	− 2381.57i	− 0.923904i
$189$	0	0
$190$	0	0
$191$	188.786	0.0715186	0.0357593	−	0.999360i	$-0.488615\pi$
$191$	188.786	0.0715186	0.0357593	+	0.999360i	$0.488615\pi$
$192$	0	0
$193$	886.367i	0.330581i	0.986245	+	0.165290i	$0.0528562\pi$
$193$	886.367i	0.330581i	−0.986245	+	0.165290i	$0.947144\pi$
$194$	−1721.51	−0.637101
$195$	0	0
$196$	1067.76	0.389124
$197$	− 3347.88i	− 1.21079i	−0.795924	−	0.605397i	$-0.793014\pi$
$197$	− 3347.88i	− 1.21079i	0.795924	−	0.605397i	$-0.206986\pi$
$198$	0	0
$199$	933.510	0.332537	0.166268	−	0.986081i	$-0.446828\pi$
$199$	933.510	0.332537	0.166268	+	0.986081i	$0.446828\pi$
$200$	0	0
$201$	0	0
$202$	829.390i	0.288889i
$203$	− 333.031i	− 0.115144i
$204$	0	0
$205$	0	0
$206$	3350.91	1.13334
$207$	0	0
$208$	− 1255.76i	− 0.418610i
$209$	3972.57	1.31478
$210$	0	0
$211$	−2634.60	−0.859590	−0.429795	−	0.902927i	$-0.641414\pi$
$211$	−2634.60	−0.859590	−0.429795	+	0.902927i	$0.641414\pi$
$212$	1829.94i	0.592834i
$213$	0	0
$214$	−12.4286	−0.00397011
$215$	0	0
$216$	0	0
$217$	− 6642.72i	− 2.07805i
$218$	2967.88i	0.922064i
$219$	0	0
$220$	0	0
$221$	−7180.15	−2.18547
$222$	0	0
$223$	− 2112.88i	− 0.634478i	−0.948346	−	0.317239i	$-0.897244\pi$
$223$	− 2112.88i	− 0.634478i	0.948346	−	0.317239i	$-0.102756\pi$
$224$	−790.302	−0.235733
$225$	0	0
$226$	1601.69	0.471430
$227$	167.908i	0.0490945i	0.999699	+	0.0245473i	$0.00781442\pi$
$227$	167.908i	0.0490945i	−0.999699	+	0.0245473i	$0.992186\pi$
$228$	0	0
$229$	4423.75	1.27655	0.638274	−	0.769809i	$-0.279649\pi$
$229$	4423.75	1.27655	0.638274	+	0.769809i	$0.279649\pi$
$230$	0	0
$231$	0	0
$232$	107.878i	0.0305280i
$233$	3591.54i	1.00983i	0.863170	+	0.504913i	$0.168475\pi$
$233$	3591.54i	1.00983i	−0.863170	+	0.504913i	$0.831525\pi$
$234$	0	0
$235$	0	0
$236$	−1483.03	−0.409055
$237$	0	0
$238$	4518.78i	1.23071i
$239$	−356.660	−0.0965290	−0.0482645	−	0.998835i	$-0.515369\pi$
$239$	−356.660	−0.0965290	−0.0482645	+	0.998835i	$0.515369\pi$
$240$	0	0
$241$	−5676.57	−1.51726	−0.758631	−	0.651521i	$-0.774131\pi$
$241$	−5676.57	−1.51726	−0.758631	+	0.651521i	$0.774131\pi$
$242$	− 1279.63i	− 0.339909i
$243$	0	0
$244$	1797.82	0.471694
$245$	0	0
$246$	0	0
$247$	− 7023.18i	− 1.80921i
$248$	2151.76i	0.550954i
$249$	0	0
$250$	0	0
$251$	−5638.32	−1.41788	−0.708940	−	0.705269i	$-0.750826\pi$
$251$	−5638.32	−1.41788	−0.708940	+	0.705269i	$0.750826\pi$
$252$	0	0
$253$	3661.82i	0.909946i
$254$	169.086	0.0417692
$255$	0	0
$256$	256.000	0.0625000
$257$	2821.42i	0.684808i	0.939553	+	0.342404i	$0.111241\pi$
$257$	2821.42i	0.684808i	−0.939553	+	0.342404i	$0.888759\pi$
$258$	0	0
$259$	−6151.78	−1.47588
$260$	0	0
$261$	0	0
$262$	− 260.302i	− 0.0613798i
$263$	− 4272.66i	− 1.00176i	−0.865516	−	0.500881i	$-0.833009\pi$
$263$	− 4272.66i	− 1.00176i	0.865516	−	0.500881i	$-0.166991\pi$
$264$	0	0
$265$	0	0
$266$	−4420.00	−1.01882
$267$	0	0
$268$	1543.64i	0.351838i
$269$	−2582.11	−0.585257	−0.292629	−	0.956226i	$-0.594530\pi$
$269$	−2582.11	−0.585257	−0.292629	+	0.956226i	$0.594530\pi$
$270$	0	0
$271$	−7989.08	−1.79078	−0.895391	−	0.445280i	$-0.853104\pi$
$271$	−7989.08	−1.79078	−0.895391	+	0.445280i	$0.853104\pi$
$272$	− 1463.76i	− 0.326298i
$273$	0	0
$274$	−4736.17	−1.04424
$275$	0	0
$276$	0	0
$277$	8059.68i	1.74823i	0.485720	+	0.874114i	$0.338557\pi$
$277$	8059.68i	1.74823i	−0.485720	+	0.874114i	$0.661443\pi$
$278$	4656.54i	1.00461i
$279$	0	0
$280$	0	0
$281$	2865.29	0.608288	0.304144	−	0.952626i	$-0.401630\pi$
$281$	2865.29	0.608288	0.304144	+	0.952626i	$0.401630\pi$
$282$	0	0
$283$	5634.40i	1.18350i	0.806122	+	0.591749i	$0.201562\pi$
$283$	5634.40i	1.18350i	−0.806122	+	0.591749i	$0.798438\pi$
$284$	−3101.82	−0.648095
$285$	0	0
$286$	−6968.48	−1.44075
$287$	− 7364.17i	− 1.51461i
$288$	0	0
$289$	−3456.45	−0.703531
$290$	0	0
$291$	0	0
$292$	− 1367.27i	− 0.274019i
$293$	− 7475.71i	− 1.49057i	−0.666749	−	0.745283i	$-0.732315\pi$
$293$	− 7475.71i	− 1.49057i	0.666749	−	0.745283i	$-0.267685\pi$
$294$	0	0
$295$	0	0
$296$	1992.73	0.391300
$297$	0	0
$298$	− 1267.27i	− 0.246346i
$299$	6473.79	1.25214
$300$	0	0
$301$	6492.30	1.24322
$302$	1547.46i	0.294855i
$303$	0	0
$304$	1431.76	0.270121
$305$	0	0
$306$	0	0
$307$	6405.44i	1.19081i	0.803427	+	0.595403i	$0.203008\pi$
$307$	6405.44i	1.19081i	−0.803427	+	0.595403i	$0.796992\pi$
$308$	4385.57i	0.811335i
$309$	0	0
$310$	0	0
$311$	−8955.76	−1.63291	−0.816454	−	0.577411i	$-0.804063\pi$
$311$	−8955.76	−1.63291	−0.816454	+	0.577411i	$0.804063\pi$
$312$	0	0
$313$	9563.01i	1.72694i	0.504397	+	0.863472i	$0.331715\pi$
$313$	9563.01i	1.72694i	−0.504397	+	0.863472i	$0.668285\pi$
$314$	−6208.49	−1.11581
$315$	0	0
$316$	−1485.45	−0.264440
$317$	8631.11i	1.52925i	0.644477	+	0.764623i	$0.277075\pi$
$317$	8631.11i	1.52925i	−0.644477	+	0.764623i	$0.722925\pi$
$318$	0	0
$319$	598.638	0.105070
$320$	0	0
$321$	0	0
$322$	− 4074.24i	− 0.705119i
$323$	− 8186.48i	− 1.41024i
$324$	0	0
$325$	0	0
$326$	2555.15	0.434100
$327$	0	0
$328$	2385.45i	0.401569i
$329$	14704.4	2.46407
$330$	0	0
$331$	4159.08	0.690646	0.345323	−	0.938484i	$-0.387769\pi$
$331$	4159.08	0.690646	0.345323	+	0.938484i	$0.387769\pi$
$332$	4751.51i	0.785461i
$333$	0	0
$334$	416.847	0.0682900
$335$	0	0
$336$	0	0
$337$	− 8764.84i	− 1.41677i	−0.705827	−	0.708385i	$-0.749424\pi$
$337$	− 8764.84i	− 1.41677i	0.705827	−	0.708385i	$-0.250576\pi$
$338$	7925.69i	1.27545i
$339$	0	0
$340$	0	0
$341$	11940.6	1.89624
$342$	0	0
$343$	− 1878.48i	− 0.295710i
$344$	−2103.03	−0.329615
$345$	0	0
$346$	6818.11	1.05938
$347$	9002.36i	1.39271i	0.717695	+	0.696357i	$0.245197\pi$
$347$	9002.36i	1.39271i	−0.717695	+	0.696357i	$0.754803\pi$
$348$	0	0
$349$	3012.36	0.462028	0.231014	−	0.972950i	$-0.425796\pi$
$349$	3012.36	0.462028	0.231014	+	0.972950i	$0.425796\pi$
$350$	0	0
$351$	0	0
$352$	− 1420.60i	− 0.215109i
$353$	7740.26i	1.16706i	0.812091	+	0.583530i	$0.198329\pi$
$353$	7740.26i	1.16706i	−0.812091	+	0.583530i	$0.801671\pi$
$354$	0	0
$355$	0	0
$356$	−790.910	−0.117748
$357$	0	0
$358$	− 6845.14i	− 1.01055i
$359$	−3957.57	−0.581818	−0.290909	−	0.956751i	$-0.593958\pi$
$359$	−3957.57	−0.581818	−0.290909	+	0.956751i	$0.593958\pi$
$360$	0	0
$361$	1148.51	0.167446
$362$	1355.52i	0.196808i
$363$	0	0
$364$	7753.33	1.11644
$365$	0	0
$366$	0	0
$367$	− 2802.71i	− 0.398639i	−0.979935	−	0.199319i	$-0.936127\pi$
$367$	− 2802.71i	− 0.398639i	0.979935	−	0.199319i	$-0.0638731\pi$
$368$	1319.76i	0.186948i
$369$	0	0
$370$	0	0
$371$	−11298.5	−1.58110
$372$	0	0
$373$	1660.31i	0.230476i	0.993338	+	0.115238i	$0.0367630\pi$
$373$	1660.31i	0.230476i	−0.993338	+	0.115238i	$0.963237\pi$
$374$	−8122.72	−1.12304
$375$	0	0
$376$	−4763.14	−0.653299
$377$	− 1058.34i	− 0.144582i
$378$	0	0
$379$	−5041.88	−0.683335	−0.341667	−	0.939821i	$-0.610992\pi$
$379$	−5041.88	−0.683335	−0.341667	+	0.939821i	$0.610992\pi$
$380$	0	0
$381$	0	0
$382$	− 377.571i	− 0.0505713i
$383$	11591.9i	1.54653i	0.634085	+	0.773263i	$0.281377\pi$
$383$	11591.9i	1.54653i	−0.634085	+	0.773263i	$0.718623\pi$
$384$	0	0
$385$	0	0
$386$	1772.73	0.233756
$387$	0	0
$388$	3443.03i	0.450498i
$389$	12135.3	1.58171	0.790853	−	0.612006i	$-0.209637\pi$
$389$	12135.3	1.58171	0.790853	+	0.612006i	$0.209637\pi$
$390$	0	0
$391$	7546.09	0.976015
$392$	− 2135.51i	− 0.275152i
$393$	0	0
$394$	−6695.76	−0.856161
$395$	0	0
$396$	0	0
$397$	− 5842.11i	− 0.738557i	−0.929319	−	0.369279i	$-0.879605\pi$
$397$	− 5842.11i	− 0.738557i	0.929319	−	0.369279i	$-0.120395\pi$
$398$	− 1867.02i	− 0.235139i
$399$	0	0
$400$	0	0
$401$	−8535.92	−1.06300	−0.531500	−	0.847058i	$-0.678372\pi$
$401$	−8535.92	−1.06300	−0.531500	+	0.847058i	$0.678372\pi$
$402$	0	0
$403$	− 21110.0i	− 2.60934i
$404$	1658.78	0.204276
$405$	0	0
$406$	−666.061	−0.0814189
$407$	− 11058.1i	− 1.34676i
$408$	0	0
$409$	3211.12	0.388215	0.194107	−	0.980980i	$-0.437819\pi$
$409$	3211.12	0.388215	0.194107	+	0.980980i	$0.437819\pi$
$410$	0	0
$411$	0	0
$412$	− 6701.81i	− 0.801395i
$413$	− 9156.57i	− 1.09096i
$414$	0	0
$415$	0	0
$416$	−2511.51	−0.296002
$417$	0	0
$418$	− 7945.14i	− 0.929688i
$419$	5059.40	0.589899	0.294949	−	0.955513i	$-0.404697\pi$
$419$	5059.40	0.589899	0.294949	+	0.955513i	$0.404697\pi$
$420$	0	0
$421$	−3331.26	−0.385643	−0.192821	−	0.981234i	$-0.561764\pi$
$421$	−3331.26	−0.385643	−0.192821	+	0.981234i	$0.561764\pi$
$422$	5269.20i	0.607822i
$423$	0	0
$424$	3659.88	0.419197
$425$	0	0
$426$	0	0
$427$	11100.1i	1.25802i
$428$	24.8572i	0.00280729i
$429$	0	0
$430$	0	0
$431$	16796.4	1.87715	0.938576	−	0.345074i	$-0.112146\pi$
$431$	16796.4	1.87715	0.938576	+	0.345074i	$0.112146\pi$
$432$	0	0
$433$	− 13623.5i	− 1.51201i	−0.654563	−	0.756007i	$-0.727148\pi$
$433$	− 13623.5i	− 1.51201i	0.654563	−	0.756007i	$-0.272852\pi$
$434$	−13285.4	−1.46940
$435$	0	0
$436$	5935.76	0.651998
$437$	7381.12i	0.807979i
$438$	0	0
$439$	9121.72	0.991699	0.495850	−	0.868408i	$-0.334857\pi$
$439$	9121.72	0.991699	0.495850	+	0.868408i	$0.334857\pi$
$440$	0	0
$441$	0	0
$442$	14360.3i	1.54536i
$443$	− 7659.61i	− 0.821488i	−0.911751	−	0.410744i	$-0.865269\pi$
$443$	− 7659.61i	− 0.821488i	0.911751	−	0.410744i	$-0.134731\pi$
$444$	0	0
$445$	0	0
$446$	−4225.75	−0.448644
$447$	0	0
$448$	1580.60i	0.166689i
$449$	7501.49	0.788457	0.394229	−	0.919012i	$-0.371012\pi$
$449$	7501.49	0.788457	0.394229	+	0.919012i	$0.371012\pi$
$450$	0	0
$451$	13237.4	1.38210
$452$	− 3203.39i	− 0.333351i
$453$	0	0
$454$	335.816	0.0347151
$455$	0	0
$456$	0	0
$457$	14259.5i	1.45959i	0.683665	+	0.729796i	$0.260385\pi$
$457$	14259.5i	1.45959i	−0.683665	+	0.729796i	$0.739615\pi$
$458$	− 8847.50i	− 0.902656i
$459$	0	0
$460$	0	0
$461$	−18059.1	−1.82450	−0.912251	−	0.409633i	$-0.865657\pi$
$461$	−18059.1	−1.82450	−0.912251	+	0.409633i	$0.865657\pi$
$462$	0	0
$463$	9176.80i	0.921127i	0.887627	+	0.460564i	$0.152353\pi$
$463$	9176.80i	0.921127i	−0.887627	+	0.460564i	$0.847647\pi$
$464$	215.755	0.0215866
$465$	0	0
$466$	7183.07	0.714054
$467$	− 15995.7i	− 1.58500i	−0.609874	−	0.792499i	$-0.708780\pi$
$467$	− 15995.7i	− 1.58500i	0.609874	−	0.792499i	$-0.291220\pi$
$468$	0	0
$469$	−9530.78	−0.938359
$470$	0	0
$471$	0	0
$472$	2966.06i	0.289245i
$473$	11670.2i	1.13445i
$474$	0	0
$475$	0	0
$476$	9037.57	0.870244
$477$	0	0
$478$	713.320i	0.0682563i
$479$	−8645.12	−0.824647	−0.412323	−	0.911038i	$-0.635283\pi$
$479$	−8645.12	−0.824647	−0.412323	+	0.911038i	$0.635283\pi$
$480$	0	0
$481$	−19549.8	−1.85321
$482$	11353.1i	1.07287i
$483$	0	0
$484$	−2559.27	−0.240352
$485$	0	0
$486$	0	0
$487$	− 13331.0i	− 1.24042i	−0.784434	−	0.620212i	$-0.787046\pi$
$487$	− 13331.0i	− 1.24042i	0.784434	−	0.620212i	$-0.212954\pi$
$488$	− 3595.63i	− 0.333538i
$489$	0	0
$490$	0	0
$491$	11566.6	1.06313	0.531563	−	0.847019i	$-0.321605\pi$
$491$	11566.6	1.06313	0.531563	+	0.847019i	$0.321605\pi$
$492$	0	0
$493$	− 1233.64i	− 0.112699i
$494$	−14046.4	−1.27930
$495$	0	0
$496$	4303.51	0.389583
$497$	− 19151.3i	− 1.72848i
$498$	0	0
$499$	−1113.62	−0.0999050	−0.0499525	−	0.998752i	$-0.515907\pi$
$499$	−1113.62	−0.0999050	−0.0499525	+	0.998752i	$0.515907\pi$
$500$	0	0
$501$	0	0
$502$	11276.6i	1.00259i
$503$	− 15543.3i	− 1.37782i	−0.724847	−	0.688910i	$-0.758090\pi$
$503$	− 15543.3i	− 1.37782i	0.724847	−	0.688910i	$-0.241910\pi$
$504$	0	0
$505$	0	0
$506$	7323.63	0.643429
$507$	0	0
$508$	− 338.171i	− 0.0295353i
$509$	1544.56	0.134502	0.0672509	−	0.997736i	$-0.478577\pi$
$509$	1544.56	0.134502	0.0672509	+	0.997736i	$0.478577\pi$
$510$	0	0
$511$	8441.87	0.730815
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	5642.85	0.484232
$515$	0	0
$516$	0	0
$517$	26431.8i	2.24849i
$518$	12303.6i	1.04361i
$519$	0	0
$520$	0	0
$521$	2273.20	0.191153	0.0955763	−	0.995422i	$-0.469531\pi$
$521$	2273.20	0.191153	0.0955763	+	0.995422i	$0.469531\pi$
$522$	0	0
$523$	12345.4i	1.03218i	0.856536	+	0.516088i	$0.172612\pi$
$523$	12345.4i	1.03218i	−0.856536	+	0.516088i	$0.827388\pi$
$524$	−520.604	−0.0434021
$525$	0	0
$526$	−8545.32	−0.708353
$527$	− 24606.6i	− 2.03393i
$528$	0	0
$529$	5363.28	0.440805
$530$	0	0
$531$	0	0
$532$	8839.99i	0.720418i
$533$	− 23402.7i	− 1.90184i
$534$	0	0
$535$	0	0
$536$	3087.27	0.248787
$537$	0	0
$538$	5164.22i	0.413839i
$539$	−11850.4	−0.947003
$540$	0	0
$541$	2123.93	0.168789	0.0843947	−	0.996432i	$-0.473104\pi$
$541$	2123.93	0.168789	0.0843947	+	0.996432i	$0.473104\pi$
$542$	15978.2i	1.26627i
$543$	0	0
$544$	−2927.51	−0.230728
$545$	0	0
$546$	0	0
$547$	9720.92i	0.759847i	0.925018	+	0.379924i	$0.124050\pi$
$547$	9720.92i	0.759847i	−0.925018	+	0.379924i	$0.875950\pi$
$548$	9472.35i	0.738392i
$549$	0	0
$550$	0	0
$551$	1206.67	0.0932959
$552$	0	0
$553$	− 9171.51i	− 0.705266i
$554$	16119.4	1.23618
$555$	0	0
$556$	9313.08	0.710364
$557$	− 6289.91i	− 0.478478i	−0.970961	−	0.239239i	$-0.923102\pi$
$557$	− 6289.91i	− 0.478478i	0.970961	−	0.239239i	$-0.0768979\pi$
$558$	0	0
$559$	20631.9	1.56107
$560$	0	0
$561$	0	0
$562$	− 5730.58i	− 0.430125i
$563$	14717.2i	1.10170i	0.834604	+	0.550850i	$0.185696\pi$
$563$	14717.2i	1.10170i	−0.834604	+	0.550850i	$0.814304\pi$
$564$	0	0
$565$	0	0
$566$	11268.8	0.836860
$567$	0	0
$568$	6203.63i	0.458272i
$569$	4164.24	0.306808	0.153404	−	0.988164i	$-0.450976\pi$
$569$	4164.24	0.306808	0.153404	+	0.988164i	$0.450976\pi$
$570$	0	0
$571$	5602.66	0.410620	0.205310	−	0.978697i	$-0.434180\pi$
$571$	5602.66	0.410620	0.205310	+	0.978697i	$0.434180\pi$
$572$	13937.0i	1.01876i
$573$	0	0
$574$	−14728.3	−1.07099
$575$	0	0
$576$	0	0
$577$	14555.0i	1.05014i	0.851058	+	0.525071i	$0.175961\pi$
$577$	14555.0i	1.05014i	−0.851058	+	0.525071i	$0.824039\pi$
$578$	6912.90i	0.497472i
$579$	0	0
$580$	0	0
$581$	−29336.9	−2.09484
$582$	0	0
$583$	− 20309.5i	− 1.44277i
$584$	−2734.55	−0.193761
$585$	0	0
$586$	−14951.4	−1.05399
$587$	24049.4i	1.69101i	0.533964	+	0.845507i	$0.320702\pi$
$587$	24049.4i	1.69101i	−0.533964	+	0.845507i	$0.679298\pi$
$588$	0	0
$589$	24068.6	1.68375
$590$	0	0
$591$	0	0
$592$	− 3985.45i	− 0.276691i
$593$	− 11754.7i	− 0.814008i	−0.913426	−	0.407004i	$-0.866574\pi$
$593$	− 11754.7i	− 0.814008i	0.913426	−	0.407004i	$-0.133426\pi$
$594$	0	0
$595$	0	0
$596$	−2534.55	−0.174193
$597$	0	0
$598$	− 12947.6i	− 0.885394i
$599$	26311.8	1.79478	0.897388	−	0.441243i	$-0.145462\pi$
$599$	26311.8	1.79478	0.897388	+	0.441243i	$0.145462\pi$
$600$	0	0
$601$	12105.5	0.821620	0.410810	−	0.911721i	$-0.365246\pi$
$601$	12105.5	0.821620	0.410810	+	0.911721i	$0.365246\pi$
$602$	− 12984.6i	− 0.879090i
$603$	0	0
$604$	3094.92	0.208494
$605$	0	0
$606$	0	0
$607$	21630.9i	1.44641i	0.690633	+	0.723205i	$0.257332\pi$
$607$	21630.9i	1.44641i	−0.690633	+	0.723205i	$0.742668\pi$
$608$	− 2863.51i	− 0.191004i
$609$	0	0
$610$	0	0
$611$	46729.2	3.09405
$612$	0	0
$613$	− 7459.98i	− 0.491526i	−0.969330	−	0.245763i	$-0.920961\pi$
$613$	− 7459.98i	− 0.491526i	0.969330	−	0.245763i	$-0.0790386\pi$
$614$	12810.9	0.842028
$615$	0	0
$616$	8771.14	0.573700
$617$	8910.37i	0.581390i	0.956816	+	0.290695i	$0.0938866\pi$
$617$	8910.37i	0.581390i	−0.956816	+	0.290695i	$0.906113\pi$
$618$	0	0
$619$	21314.7	1.38402	0.692012	−	0.721886i	$-0.256725\pi$
$619$	21314.7	1.38402	0.692012	+	0.721886i	$0.256725\pi$
$620$	0	0
$621$	0	0
$622$	17911.5i	1.15464i
$623$	− 4883.27i	− 0.314035i
$624$	0	0
$625$	0	0
$626$	19126.0	1.22113
$627$	0	0
$628$	12417.0i	0.788999i
$629$	−22788.0	−1.44454
$630$	0	0
$631$	−9458.91	−0.596757	−0.298378	−	0.954448i	$-0.596446\pi$
$631$	−9458.91	−0.596757	−0.298378	+	0.954448i	$0.596446\pi$
$632$	2970.90i	0.186987i
$633$	0	0
$634$	17262.2	1.08134
$635$	0	0
$636$	0	0
$637$	20950.6i	1.30313i
$638$	− 1197.28i	− 0.0742956i
$639$	0	0
$640$	0	0
$641$	11765.9	0.724999	0.362499	−	0.931984i	$-0.381923\pi$
$641$	11765.9	0.724999	0.362499	+	0.931984i	$0.381923\pi$
$642$	0	0
$643$	13649.1i	0.837118i	0.908190	+	0.418559i	$0.137465\pi$
$643$	13649.1i	0.837118i	−0.908190	+	0.418559i	$0.862535\pi$
$644$	−8148.48	−0.498595
$645$	0	0
$646$	−16373.0	−0.997191
$647$	− 18160.9i	− 1.10352i	−0.834003	−	0.551759i	$-0.813957\pi$
$647$	− 18160.9i	− 1.10352i	0.834003	−	0.551759i	$-0.186043\pi$
$648$	0	0
$649$	16459.3	0.995510
$650$	0	0
$651$	0	0
$652$	− 5110.29i	− 0.306955i
$653$	− 10157.4i	− 0.608713i	−0.952558	−	0.304357i	$-0.901559\pi$
$653$	− 10157.4i	− 0.608713i	0.952558	−	0.304357i	$-0.0984415\pi$
$654$	0	0
$655$	0	0
$656$	4770.91	0.283952
$657$	0	0
$658$	− 29408.8i	− 1.74236i
$659$	7882.10	0.465922	0.232961	−	0.972486i	$-0.425158\pi$
$659$	7882.10	0.465922	0.232961	+	0.972486i	$0.425158\pi$
$660$	0	0
$661$	8860.71	0.521394	0.260697	−	0.965421i	$-0.416048\pi$
$661$	8860.71	0.521394	0.260697	+	0.965421i	$0.416048\pi$
$662$	− 8318.16i	− 0.488360i
$663$	0	0
$664$	9503.02	0.555405
$665$	0	0
$666$	0	0
$667$	1112.28i	0.0645692i
$668$	− 833.694i	− 0.0482883i
$669$	0	0
$670$	0	0
$671$	−19953.0	−1.14795
$672$	0	0
$673$	9099.53i	0.521191i	0.965448	+	0.260595i	$0.0839188\pi$
$673$	9099.53i	0.521191i	−0.965448	+	0.260595i	$0.916081\pi$
$674$	−17529.7	−1.00181
$675$	0	0
$676$	15851.4	0.901877
$677$	2145.34i	0.121790i	0.998144	+	0.0608951i	$0.0193955\pi$
$677$	2145.34i	0.121790i	−0.998144	+	0.0608951i	$0.980604\pi$
$678$	0	0
$679$	−21258.1	−1.20149
$680$	0	0
$681$	0	0
$682$	− 23881.2i	− 1.34085i
$683$	− 11028.3i	− 0.617844i	−0.951087	−	0.308922i	$-0.900032\pi$
$683$	− 11028.3i	− 0.617844i	0.951087	−	0.308922i	$-0.0999682\pi$
$684$	0	0
$685$	0	0
$686$	−3756.96	−0.209098
$687$	0	0
$688$	4206.06i	0.233073i
$689$	−35905.5	−1.98533
$690$	0	0
$691$	16500.0	0.908379	0.454190	−	0.890905i	$-0.349929\pi$
$691$	16500.0	0.908379	0.454190	+	0.890905i	$0.349929\pi$
$692$	− 13636.2i	− 0.749092i
$693$	0	0
$694$	18004.7	0.984798
$695$	0	0
$696$	0	0
$697$	− 27279.1i	− 1.48245i
$698$	− 6024.71i	− 0.326703i
$699$	0	0
$700$	0	0
$701$	−8733.75	−0.470569	−0.235285	−	0.971926i	$-0.575602\pi$
$701$	−8733.75	−0.470569	−0.235285	+	0.971926i	$0.575602\pi$
$702$	0	0
$703$	− 22289.8i	− 1.19584i
$704$	−2841.21	−0.152105
$705$	0	0
$706$	15480.5	0.825237
$707$	10241.7i	0.544807i
$708$	0	0
$709$	−8375.84	−0.443669	−0.221835	−	0.975084i	$-0.571204\pi$
$709$	−8375.84	−0.443669	−0.221835	+	0.975084i	$0.571204\pi$
$710$	0	0
$711$	0	0
$712$	1581.82i	0.0832602i
$713$	22185.9i	1.16531i
$714$	0	0
$715$	0	0
$716$	−13690.3	−0.714567
$717$	0	0
$718$	7915.14i	0.411407i
$719$	4645.44	0.240953	0.120477	−	0.992716i	$-0.461558\pi$
$719$	4645.44	0.240953	0.120477	+	0.992716i	$0.461558\pi$
$720$	0	0
$721$	41378.6	2.13733
$722$	− 2297.02i	− 0.118402i
$723$	0	0
$724$	2711.04	0.139164
$725$	0	0
$726$	0	0
$727$	15759.1i	0.803951i	0.915651	+	0.401975i	$0.131676\pi$
$727$	15759.1i	0.803951i	−0.915651	+	0.401975i	$0.868324\pi$
$728$	− 15506.7i	− 0.789443i
$729$	0	0
$730$	0	0
$731$	24049.4	1.21682
$732$	0	0
$733$	− 18858.2i	− 0.950263i	−0.879915	−	0.475132i	$-0.842400\pi$
$733$	− 18858.2i	− 0.950263i	0.879915	−	0.475132i	$-0.157600\pi$
$734$	−5605.42	−0.281880
$735$	0	0
$736$	2639.51	0.132192
$737$	− 17132.0i	− 0.856263i
$738$	0	0
$739$	−33435.3	−1.66433	−0.832163	−	0.554531i	$-0.812898\pi$
$739$	−33435.3	−1.66433	−0.832163	+	0.554531i	$0.812898\pi$
$740$	0	0
$741$	0	0
$742$	22596.9i	1.11801i
$743$	3313.09i	0.163587i	0.996649	+	0.0817936i	$0.0260648\pi$
$743$	3313.09i	0.163587i	−0.996649	+	0.0817936i	$0.973935\pi$
$744$	0	0
$745$	0	0
$746$	3320.61	0.162971
$747$	0	0
$748$	16245.4i	0.794107i
$749$	−153.474	−0.00748709
$750$	0	0
$751$	17180.9	0.834806	0.417403	−	0.908721i	$-0.362940\pi$
$751$	17180.9	0.834806	0.417403	+	0.908721i	$0.362940\pi$
$752$	9526.29i	0.461952i
$753$	0	0
$754$	−2116.68	−0.102235
$755$	0	0
$756$	0	0
$757$	− 1723.65i	− 0.0827572i	−0.999144	−	0.0413786i	$-0.986825\pi$
$757$	− 1723.65i	− 0.0827572i	0.999144	−	0.0413786i	$-0.0131750\pi$
$758$	10083.8i	0.483191i
$759$	0	0
$760$	0	0
$761$	−33073.9	−1.57547	−0.787733	−	0.616017i	$-0.788745\pi$
$761$	−33073.9	−1.57547	−0.787733	+	0.616017i	$0.788745\pi$
$762$	0	0
$763$	36648.7i	1.73889i
$764$	−755.143	−0.0357593
$765$	0	0
$766$	23183.8	1.09356
$767$	− 29098.8i	− 1.36988i
$768$	0	0
$769$	−15354.5	−0.720021	−0.360011	−	0.932948i	$-0.617227\pi$
$769$	−15354.5	−0.720021	−0.360011	+	0.932948i	$0.617227\pi$
$770$	0	0
$771$	0	0
$772$	− 3545.47i	− 0.165290i
$773$	− 18234.3i	− 0.848438i	−0.905560	−	0.424219i	$-0.860549\pi$
$773$	− 18234.3i	− 0.848438i	0.905560	−	0.424219i	$-0.139451\pi$
$774$	0	0
$775$	0	0
$776$	6886.06	0.318550
$777$	0	0
$778$	− 24270.6i	− 1.11844i
$779$	26682.7	1.22722
$780$	0	0
$781$	34425.4	1.57726
$782$	− 15092.2i	− 0.690147i
$783$	0	0
$784$	−4271.02	−0.194562
$785$	0	0
$786$	0	0
$787$	9327.27i	0.422467i	0.977436	+	0.211233i	$0.0677480\pi$
$787$	9327.27i	0.422467i	−0.977436	+	0.211233i	$0.932252\pi$
$788$	13391.5i	0.605397i
$789$	0	0
$790$	0	0
$791$	19778.5	0.889054
$792$	0	0
$793$	35275.3i	1.57965i
$794$	−11684.2	−0.522239
$795$	0	0
$796$	−3734.04	−0.166268
$797$	22974.9i	1.02110i	0.859849	+	0.510548i	$0.170558\pi$
$797$	22974.9i	1.02110i	−0.859849	+	0.510548i	$0.829442\pi$
$798$	0	0
$799$	54469.3	2.41175
$800$	0	0
$801$	0	0
$802$	17071.8i	0.751655i
$803$	15174.6i	0.666876i
$804$	0	0
$805$	0	0
$806$	−42220.0	−1.84508
$807$	0	0
$808$	− 3317.56i	− 0.144445i
$809$	35297.7	1.53399	0.766996	−	0.641651i	$-0.221750\pi$
$809$	35297.7	1.53399	0.766996	+	0.641651i	$0.221750\pi$
$810$	0	0
$811$	−26680.2	−1.15520	−0.577601	−	0.816319i	$-0.696011\pi$
$811$	−26680.2	−1.15520	−0.577601	+	0.816319i	$0.696011\pi$
$812$	1332.12i	0.0575718i
$813$	0	0
$814$	−22116.2	−0.952301
$815$	0	0
$816$	0	0
$817$	23523.6i	1.00733i
$818$	− 6422.24i	− 0.274509i
$819$	0	0
$820$	0	0
$821$	−5525.16	−0.234871	−0.117436	−	0.993081i	$-0.537467\pi$
$821$	−5525.16	−0.234871	−0.117436	+	0.993081i	$0.537467\pi$
$822$	0	0
$823$	− 26776.2i	− 1.13409i	−0.823686	−	0.567047i	$-0.808086\pi$
$823$	− 26776.2i	− 1.13409i	0.823686	−	0.567047i	$-0.191914\pi$
$824$	−13403.6	−0.566672
$825$	0	0
$826$	−18313.1	−0.771423
$827$	26180.5i	1.10083i	0.834892	+	0.550414i	$0.185530\pi$
$827$	26180.5i	1.10083i	−0.834892	+	0.550414i	$0.814470\pi$
$828$	0	0
$829$	−10283.8	−0.430846	−0.215423	−	0.976521i	$-0.569113\pi$
$829$	−10283.8	−0.430846	−0.215423	+	0.976521i	$0.569113\pi$
$830$	0	0
$831$	0	0
$832$	5023.02i	0.209305i
$833$	24420.8i	1.01576i
$834$	0	0
$835$	0	0
$836$	−15890.3	−0.657389
$837$	0	0
$838$	− 10118.8i	− 0.417122i
$839$	1368.16	0.0562980	0.0281490	−	0.999604i	$-0.491039\pi$
$839$	1368.16	0.0562980	0.0281490	+	0.999604i	$0.491039\pi$
$840$	0	0
$841$	−24207.2	−0.992544
$842$	6662.52i	0.272691i
$843$	0	0
$844$	10538.4	0.429795
$845$	0	0
$846$	0	0
$847$	− 15801.5i	− 0.641023i
$848$	− 7319.76i	− 0.296417i
$849$	0	0
$850$	0	0
$851$	20546.2	0.827631
$852$	0	0
$853$	− 25730.0i	− 1.03280i	−0.856348	−	0.516399i	$-0.827272\pi$
$853$	− 25730.0i	− 1.03280i	0.856348	−	0.516399i	$-0.172728\pi$
$854$	22200.3	0.889553
$855$	0	0
$856$	49.7145	0.00198505
$857$	37389.8i	1.49033i	0.666882	+	0.745163i	$0.267629\pi$
$857$	37389.8i	1.49033i	−0.666882	+	0.745163i	$0.732371\pi$
$858$	0	0
$859$	−6121.40	−0.243143	−0.121571	−	0.992583i	$-0.538793\pi$
$859$	−6121.40	−0.243143	−0.121571	+	0.992583i	$0.538793\pi$
$860$	0	0
$861$	0	0
$862$	− 33592.7i	− 1.32735i
$863$	15542.6i	0.613068i	0.951860	+	0.306534i	$0.0991693\pi$
$863$	15542.6i	0.613068i	−0.951860	+	0.306534i	$0.900831\pi$
$864$	0	0
$865$	0	0
$866$	−27246.9	−1.06916
$867$	0	0
$868$	26570.9i	1.03903i
$869$	16486.2	0.643563
$870$	0	0
$871$	−30288.0	−1.17826
$872$	− 11871.5i	− 0.461032i
$873$	0	0
$874$	14762.2	0.571327
$875$	0	0
$876$	0	0
$877$	− 44484.2i	− 1.71280i	−0.516315	−	0.856399i	$-0.672697\pi$
$877$	− 44484.2i	− 1.71280i	0.516315	−	0.856399i	$-0.327303\pi$
$878$	− 18243.4i	− 0.701237i
$879$	0	0
$880$	0	0
$881$	14946.3	0.571572	0.285786	−	0.958293i	$-0.407745\pi$
$881$	14946.3	0.571572	0.285786	+	0.958293i	$0.407745\pi$
$882$	0	0
$883$	12146.2i	0.462913i	0.972845	+	0.231456i	$0.0743491\pi$
$883$	12146.2i	0.462913i	−0.972845	+	0.231456i	$0.925651\pi$
$884$	28720.6	1.09274
$885$	0	0
$886$	−15319.2	−0.580880
$887$	27998.0i	1.05984i	0.848046	+	0.529922i	$0.177779\pi$
$887$	27998.0i	1.05984i	−0.848046	+	0.529922i	$0.822221\pi$
$888$	0	0
$889$	2087.95	0.0787712
$890$	0	0
$891$	0	0
$892$	8451.51i	0.317239i
$893$	53278.5i	1.99653i
$894$	0	0
$895$	0	0
$896$	3161.21	0.117867
$897$	0	0
$898$	− 15003.0i	− 0.557524i
$899$	3626.97	0.134556
$900$	0	0
$901$	−41852.8	−1.54753
$902$	− 26474.9i	− 0.977292i
$903$	0	0
$904$	−6406.78	−0.235715
$905$	0	0
$906$	0	0
$907$	12778.6i	0.467814i	0.972259	+	0.233907i	$0.0751512\pi$
$907$	12778.6i	0.467814i	−0.972259	+	0.233907i	$0.924849\pi$
$908$	− 671.633i	− 0.0245473i
$909$	0	0
$910$	0	0
$911$	1870.92	0.0680421	0.0340210	−	0.999421i	$-0.489169\pi$
$911$	1870.92	0.0680421	0.0340210	+	0.999421i	$0.489169\pi$
$912$	0	0
$913$	− 52734.5i	− 1.91156i
$914$	28519.1	1.03209
$915$	0	0
$916$	−17695.0	−0.638274
$917$	− 3214.33i	− 0.115754i
$918$	0	0
$919$	16388.0	0.588238	0.294119	−	0.955769i	$-0.404974\pi$
$919$	16388.0	0.588238	0.294119	+	0.955769i	$0.404974\pi$
$920$	0	0
$921$	0	0
$922$	36118.2i	1.29012i
$923$	− 60861.3i	− 2.17039i
$924$	0	0
$925$	0	0
$926$	18353.6	0.651335
$927$	0	0
$928$	− 431.510i	− 0.0152640i
$929$	−43541.4	−1.53773	−0.768864	−	0.639413i	$-0.779178\pi$
$929$	−43541.4	−1.53773	−0.768864	+	0.639413i	$0.779178\pi$
$930$	0	0
$931$	−23886.9	−0.840884
$932$	− 14366.1i	− 0.504913i
$933$	0	0
$934$	−31991.4	−1.12076
$935$	0	0
$936$	0	0
$937$	− 3532.91i	− 0.123175i	−0.998102	−	0.0615875i	$-0.980384\pi$
$937$	− 3532.91i	− 0.123175i	0.998102	−	0.0615875i	$-0.0196163\pi$
$938$	19061.6i	0.663520i
$939$	0	0
$940$	0	0
$941$	−29039.9	−1.00603	−0.503014	−	0.864278i	$-0.667776\pi$
$941$	−29039.9	−1.00603	−0.503014	+	0.864278i	$0.667776\pi$
$942$	0	0
$943$	24595.4i	0.849350i
$944$	5932.11	0.204527
$945$	0	0
$946$	23340.4	0.802179
$947$	5900.88i	0.202484i	0.994862	+	0.101242i	$0.0322817\pi$
$947$	5900.88i	0.202484i	−0.994862	+	0.101242i	$0.967718\pi$
$948$	0	0
$949$	26827.5	0.917658
$950$	0	0
$951$	0	0
$952$	− 18075.1i	− 0.615356i
$953$	− 14406.4i	− 0.489683i	−0.969563	−	0.244842i	$-0.921264\pi$
$953$	− 14406.4i	− 0.489683i	0.969563	−	0.244842i	$-0.0787359\pi$
$954$	0	0
$955$	0	0
$956$	1426.64	0.0482645
$957$	0	0
$958$	17290.2i	0.583113i
$959$	−58484.5	−1.96930
$960$	0	0
$961$	42553.5	1.42840
$962$	39099.6i	1.31042i
$963$	0	0
$964$	22706.3	0.758631
$965$	0	0
$966$	0	0
$967$	27468.1i	0.913459i	0.889606	+	0.456729i	$0.150979\pi$
$967$	27468.1i	0.913459i	−0.889606	+	0.456729i	$0.849021\pi$
$968$	5118.53i	0.169954i
$969$	0	0
$970$	0	0
$971$	44140.1	1.45883	0.729415	−	0.684072i	$-0.239792\pi$
$971$	44140.1	1.45883	0.729415	+	0.684072i	$0.239792\pi$
$972$	0	0
$973$	57501.2i	1.89456i
$974$	−26662.1	−0.877113
$975$	0	0
$976$	−7191.27	−0.235847
$977$	49492.1i	1.62067i	0.585968	+	0.810334i	$0.300715\pi$
$977$	49492.1i	1.62067i	−0.585968	+	0.810334i	$0.699285\pi$
$978$	0	0
$979$	8777.89	0.286560
$980$	0	0
$981$	0	0
$982$	− 23133.3i	− 0.751744i
$983$	2441.95i	0.0792332i	0.999215	+	0.0396166i	$0.0126137\pi$
$983$	2441.95i	0.0792332i	−0.999215	+	0.0396166i	$0.987386\pi$
$984$	0	0
$985$	0	0
$986$	−2467.29	−0.0796900
$987$	0	0
$988$	28092.7i	0.904604i
$989$	−21683.5	−0.697163
$990$	0	0
$991$	−36760.7	−1.17835	−0.589174	−	0.808006i	$-0.700547\pi$
$991$	−36760.7	−1.17835	−0.589174	+	0.808006i	$0.700547\pi$
$992$	− 8607.02i	− 0.275477i
$993$	0	0
$994$	−38302.7	−1.22222
$995$	0	0
$996$	0	0
$997$	− 34634.5i	− 1.10019i	−0.835103	−	0.550093i	$-0.814592\pi$
$997$	− 34634.5i	− 1.10019i	0.835103	−	0.550093i	$-0.185408\pi$
$998$	2227.24i	0.0706435i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.z.649.1		4
3.2	odd	2		1350.4.c.w.649.3		4
5.2	odd	4		1350.4.a.bp.1.2	yes	2
5.3	odd	4		1350.4.a.bc.1.1	✓	2
5.4	even	2	inner	1350.4.c.z.649.4		4
15.2	even	4		1350.4.a.bi.1.2	yes	2
15.8	even	4		1350.4.a.bj.1.1	yes	2
15.14	odd	2		1350.4.c.w.649.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.4.a.bc.1.1	✓	2	5.3	odd	4
1350.4.a.bi.1.2	yes	2	15.2	even	4
1350.4.a.bj.1.1	yes	2	15.8	even	4
1350.4.a.bp.1.2	yes	2	5.2	odd	4
1350.4.c.w.649.2		4	15.14	odd	2
1350.4.c.w.649.3		4	3.2	odd	2
1350.4.c.z.649.1		4	1.1	even	1	trivial
1350.4.c.z.649.4		4	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} -24.6969i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} -24.6969i q^{7} +8.00000i q^{8} +44.3939 q^{11} -78.4847i q^{13} -49.3939 q^{14} +16.0000 q^{16} -91.4847i q^{17} +89.4847 q^{19} -88.7878i q^{22} +82.4847i q^{23} -156.969 q^{26} +98.7878i q^{28} +13.4847 q^{29} +268.969 q^{31} -32.0000i q^{32} -182.969 q^{34} -249.091i q^{37} -178.969i q^{38} +298.182 q^{41} +262.879i q^{43} -177.576 q^{44} +164.969 q^{46} +595.393i q^{47} -266.939 q^{49} +313.939i q^{52} -457.485i q^{53} +197.576 q^{56} -26.9694i q^{58} +370.757 q^{59} -449.454 q^{61} -537.939i q^{62} -64.0000 q^{64} -385.909i q^{67} +365.939i q^{68} +775.454 q^{71} +341.818i q^{73} -498.182 q^{74} -357.939 q^{76} -1096.39i q^{77} +371.362 q^{79} -596.363i q^{82} -1187.88i q^{83} +525.757 q^{86} +355.151i q^{88} +197.728 q^{89} -1938.33 q^{91} -329.939i q^{92} +1190.79 q^{94} -860.757i q^{97} +533.878i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4}+O(q^{10}) \) 4 * q - 16 * q^4	\( 4 q - 16 q^{4} + 60 q^{11} - 80 q^{14} + 64 q^{16} + 64 q^{19} - 40 q^{26} - 240 q^{29} + 488 q^{31} - 144 q^{34} + 840 q^{41} - 240 q^{44} + 72 q^{46} + 108 q^{49} + 320 q^{56} + 660 q^{59} - 916 q^{61} - 256 q^{64} + 2220 q^{71} - 1640 q^{74} - 256 q^{76} - 1160 q^{79} + 1280 q^{86} + 1320 q^{89} - 4520 q^{91} + 648 q^{94}+O(q^{100}) \) 4 * q - 16 * q^4 + 60 * q^11 - 80 * q^14 + 64 * q^16 + 64 * q^19 - 40 * q^26 - 240 * q^29 + 488 * q^31 - 144 * q^34 + 840 * q^41 - 240 * q^44 + 72 * q^46 + 108 * q^49 + 320 * q^56 + 660 * q^59 - 916 * q^61 - 256 * q^64 + 2220 * q^71 - 1640 * q^74 - 256 * q^76 - 1160 * q^79 + 1280 * q^86 + 1320 * q^89 - 4520 * q^91 + 648 * q^94

Introduction

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