LMFDB - Embedded newform 720.5.e.b.271.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,5,Mod(271,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("720.271");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	720.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$74.4263734204$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.4
Root			$0.809017 + 1.40126i$ of defining polynomial
Character	$\chi$	$=$	720.271
Dual form			720.5.e.b.271.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.1803 q^{5} +36.5889i q^{7} +O(q^{10})$	$q+11.1803 q^{5} +36.5889i q^{7} -162.858i q^{11} +52.7477 q^{13} -214.328 q^{17} +369.785i q^{19} -664.500i q^{23} +125.000 q^{25} -300.845 q^{29} -310.463i q^{31} +409.076i q^{35} -2268.22 q^{37} -442.073 q^{41} -1508.36i q^{43} +3602.90i q^{47} +1062.25 q^{49} +1752.87 q^{53} -1820.81i q^{55} -4023.17i q^{59} -3091.66 q^{61} +589.737 q^{65} -8303.66i q^{67} -2518.07i q^{71} +8403.94 q^{73} +5958.81 q^{77} +9383.40i q^{79} -10882.0i q^{83} -2396.26 q^{85} -13536.8 q^{89} +1929.98i q^{91} +4134.32i q^{95} -10089.5 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 272 q^{13} + 216 q^{17} + 500 q^{25} - 3672 q^{29} - 3760 q^{37} + 432 q^{41} + 4732 q^{49} - 3024 q^{53} + 5504 q^{61} + 5400 q^{65} + 22024 q^{73} + 18576 q^{77} - 12000 q^{85} - 8424 q^{89} - 35528 q^{97}+O(q^{100}) $ 4 * q - 272 * q^13 + 216 * q^17 + 500 * q^25 - 3672 * q^29 - 3760 * q^37 + 432 * q^41 + 4732 * q^49 - 3024 * q^53 + 5504 * q^61 + 5400 * q^65 + 22024 * q^73 + 18576 * q^77 - 12000 * q^85 - 8424 * q^89 - 35528 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	11.1803	0.447214
$5$	11.1803	0.447214
$6$	0	0
$7$	36.5889i	0.746712i	0.927688	+	0.373356i	$0.121793\pi$
$7$	36.5889i	0.746712i	−0.927688	+	0.373356i	$0.878207\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 162.858i	− 1.34594i	−0.739671	−	0.672968i	$-0.765019\pi$
$11$	− 162.858i	− 1.34594i	0.739671	−	0.672968i	$-0.234981\pi$
$12$	0	0
$13$	52.7477	0.312116	0.156058	−	0.987748i	$-0.450121\pi$
$13$	52.7477	0.312116	0.156058	+	0.987748i	$0.450121\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−214.328	−0.741620	−0.370810	−	0.928709i	$-0.620920\pi$
$17$	−214.328	−0.741620	−0.370810	+	0.928709i	$0.620920\pi$
$18$	0	0
$19$	369.785i	1.02433i	0.858886	+	0.512167i	$0.171157\pi$
$19$	369.785i	1.02433i	−0.858886	+	0.512167i	$0.828843\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 664.500i	− 1.25614i	−0.778155	−	0.628072i	$-0.783844\pi$
$23$	− 664.500i	− 1.25614i	0.778155	−	0.628072i	$-0.216156\pi$
$24$	0	0
$25$	125.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−300.845	−0.357723	−0.178862	−	0.983874i	$-0.557241\pi$
$29$	−300.845	−0.357723	−0.178862	+	0.983874i	$0.557241\pi$
$30$	0	0
$31$	− 310.463i	− 0.323063i	−0.986868	−	0.161531i	$-0.948357\pi$
$31$	− 310.463i	− 0.323063i	0.986868	−	0.161531i	$-0.0516433\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	409.076i	0.333940i
$36$	0	0
$37$	−2268.22	−1.65685	−0.828424	−	0.560102i	$-0.810762\pi$
$37$	−2268.22	−1.65685	−0.828424	+	0.560102i	$0.810762\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−442.073	−0.262982	−0.131491	−	0.991317i	$-0.541976\pi$
$41$	−442.073	−0.262982	−0.131491	+	0.991317i	$0.541976\pi$
$42$	0	0
$43$	− 1508.36i	− 0.815769i	−0.913033	−	0.407885i	$-0.866267\pi$
$43$	− 1508.36i	− 0.815769i	0.913033	−	0.407885i	$-0.133733\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3602.90i	1.63101i	0.578750	+	0.815505i	$0.303541\pi$
$47$	3602.90i	1.63101i	−0.578750	+	0.815505i	$0.696459\pi$
$48$	0	0
$49$	1062.25	0.442421
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1752.87	0.624019	0.312009	−	0.950079i	$-0.398998\pi$
$53$	1752.87	0.624019	0.312009	+	0.950079i	$0.398998\pi$
$54$	0	0
$55$	− 1820.81i	− 0.601921i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4023.17i	− 1.15575i	−0.816125	−	0.577876i	$-0.803882\pi$
$59$	− 4023.17i	− 1.15575i	0.816125	−	0.577876i	$-0.196118\pi$
$60$	0	0
$61$	−3091.66	−0.830869	−0.415435	−	0.909623i	$-0.636370\pi$
$61$	−3091.66	−0.830869	−0.415435	+	0.909623i	$0.636370\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	589.737	0.139583
$66$	0	0
$67$	− 8303.66i	− 1.84978i	−0.380235	−	0.924890i	$-0.624157\pi$
$67$	− 8303.66i	− 1.84978i	0.380235	−	0.924890i	$-0.375843\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 2518.07i	− 0.499517i	−0.968308	−	0.249759i	$-0.919649\pi$
$71$	− 2518.07i	− 0.499517i	0.968308	−	0.249759i	$-0.0803513\pi$
$72$	0	0
$73$	8403.94	1.57702	0.788510	−	0.615021i	$-0.210853\pi$
$73$	8403.94	1.57702	0.788510	+	0.615021i	$0.210853\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5958.81	1.00503
$78$	0	0
$79$	9383.40i	1.50351i	0.659443	+	0.751754i	$0.270792\pi$
$79$	9383.40i	1.50351i	−0.659443	+	0.751754i	$0.729208\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 10882.0i	− 1.57961i	−0.613355	−	0.789807i	$-0.710181\pi$
$83$	− 10882.0i	− 1.57961i	0.613355	−	0.789807i	$-0.289819\pi$
$84$	0	0
$85$	−2396.26	−0.331663
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13536.8	−1.70897	−0.854487	−	0.519473i	$-0.826128\pi$
$89$	−13536.8	−1.70897	−0.854487	+	0.519473i	$0.826128\pi$
$90$	0	0
$91$	1929.98i	0.233061i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4134.32i	0.458096i
$96$	0	0
$97$	−10089.5	−1.07232	−0.536161	−	0.844116i	$-0.680126\pi$
$97$	−10089.5	−1.07232	−0.536161	+	0.844116i	$0.680126\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7115.63	−0.697542	−0.348771	−	0.937208i	$-0.613401\pi$
$101$	−7115.63	−0.697542	−0.348771	+	0.937208i	$0.613401\pi$
$102$	0	0
$103$	− 4118.03i	− 0.388164i	−0.980985	−	0.194082i	$-0.937827\pi$
$103$	− 4118.03i	− 0.388164i	0.980985	−	0.194082i	$-0.0621728\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9841.44i	0.859590i	0.902927	+	0.429795i	$0.141414\pi$
$107$	9841.44i	0.859590i	−0.902927	+	0.429795i	$0.858586\pi$
$108$	0	0
$109$	−5405.53	−0.454973	−0.227486	−	0.973781i	$-0.573051\pi$
$109$	−5405.53	−0.454973	−0.227486	+	0.973781i	$0.573051\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2199.20	−0.172230	−0.0861150	−	0.996285i	$-0.527445\pi$
$113$	−2199.20	−0.172230	−0.0861150	+	0.996285i	$0.527445\pi$
$114$	0	0
$115$	− 7429.34i	− 0.561765i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 7842.03i	− 0.553777i
$120$	0	0
$121$	−11881.8	−0.811546
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1397.54	0.0894427
$126$	0	0
$127$	− 29699.6i	− 1.84138i	−0.390297	−	0.920689i	$-0.627628\pi$
$127$	− 29699.6i	− 1.84138i	0.390297	−	0.920689i	$-0.372372\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	405.420i	0.0236245i	0.999930	+	0.0118123i	$0.00376004\pi$
$131$	405.420i	0.0236245i	−0.999930	+	0.0118123i	$0.996240\pi$
$132$	0	0
$133$	−13530.0	−0.764883
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10732.5	0.571823	0.285911	−	0.958256i	$-0.407704\pi$
$137$	10732.5	0.571823	0.285911	+	0.958256i	$0.407704\pi$
$138$	0	0
$139$	− 25829.6i	− 1.33687i	−0.743771	−	0.668434i	$-0.766965\pi$
$139$	− 25829.6i	− 1.33687i	0.743771	−	0.668434i	$-0.233035\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8590.40i	− 0.420089i
$144$	0	0
$145$	−3363.55	−0.159979
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−22952.6	−1.03386	−0.516928	−	0.856029i	$-0.672924\pi$
$149$	−22952.6	−1.03386	−0.516928	+	0.856029i	$0.672924\pi$
$150$	0	0
$151$	− 30515.3i	− 1.33833i	−0.743113	−	0.669166i	$-0.766651\pi$
$151$	− 30515.3i	− 1.33833i	0.743113	−	0.669166i	$-0.233349\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3471.09i	− 0.144478i
$156$	0	0
$157$	−4554.13	−0.184759	−0.0923796	−	0.995724i	$-0.529447\pi$
$157$	−4554.13	−0.184759	−0.0923796	+	0.995724i	$0.529447\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24313.3	0.937978
$162$	0	0
$163$	− 16938.1i	− 0.637513i	−0.947837	−	0.318756i	$-0.896735\pi$
$163$	− 16938.1i	− 0.637513i	0.947837	−	0.318756i	$-0.103265\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 34072.3i	− 1.22171i	−0.791742	−	0.610856i	$-0.790825\pi$
$167$	− 34072.3i	− 1.22171i	0.791742	−	0.610856i	$-0.209175\pi$
$168$	0	0
$169$	−25778.7	−0.902583
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1867.89	0.0624107	0.0312054	−	0.999513i	$-0.490065\pi$
$173$	1867.89	0.0624107	0.0312054	+	0.999513i	$0.490065\pi$
$174$	0	0
$175$	4573.61i	0.149342i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9737.81i	− 0.303917i	−0.988387	−	0.151959i	$-0.951442\pi$
$179$	− 9737.81i	− 0.303917i	0.988387	−	0.151959i	$-0.0485581\pi$
$180$	0	0
$181$	64023.4	1.95426	0.977128	−	0.212652i	$-0.0682100\pi$
$181$	64023.4	1.95426	0.977128	+	0.212652i	$0.0682100\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−25359.5	−0.740965
$186$	0	0
$187$	34905.1i	0.998174i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 26107.5i	− 0.715645i	−0.933790	−	0.357823i	$-0.883519\pi$
$191$	− 26107.5i	− 0.715645i	0.933790	−	0.357823i	$-0.116481\pi$
$192$	0	0
$193$	−32913.1	−0.883598	−0.441799	−	0.897114i	$-0.645660\pi$
$193$	−32913.1	−0.883598	−0.441799	+	0.897114i	$0.645660\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−68426.0	−1.76315	−0.881574	−	0.472047i	$-0.843515\pi$
$197$	−68426.0	−1.76315	−0.881574	+	0.472047i	$0.843515\pi$
$198$	0	0
$199$	− 22073.0i	− 0.557386i	−0.960380	−	0.278693i	$-0.910099\pi$
$199$	− 22073.0i	− 0.557386i	0.960380	−	0.278693i	$-0.0899012\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 11007.6i	− 0.267116i
$204$	0	0
$205$	−4942.52	−0.117609
$206$	0	0
$207$	0	0
$208$	0	0
$209$	60222.5	1.37869
$210$	0	0
$211$	− 28488.9i	− 0.639898i	−0.947435	−	0.319949i	$-0.896334\pi$
$211$	− 28488.9i	− 0.639898i	0.947435	−	0.319949i	$-0.103666\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 16863.9i	− 0.364823i
$216$	0	0
$217$	11359.5	0.241235
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11305.3	−0.231472
$222$	0	0
$223$	− 37138.6i	− 0.746820i	−0.927666	−	0.373410i	$-0.878188\pi$
$223$	− 37138.6i	− 0.746820i	0.927666	−	0.373410i	$-0.121812\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 48406.9i	− 0.939410i	−0.882823	−	0.469705i	$-0.844360\pi$
$227$	− 48406.9i	− 0.939410i	0.882823	−	0.469705i	$-0.155640\pi$
$228$	0	0
$229$	7933.55	0.151285	0.0756427	−	0.997135i	$-0.475899\pi$
$229$	7933.55	0.151285	0.0756427	+	0.997135i	$0.475899\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−74672.1	−1.37545	−0.687727	−	0.725969i	$-0.741392\pi$
$233$	−74672.1	−1.37545	−0.687727	+	0.725969i	$0.741392\pi$
$234$	0	0
$235$	40281.6i	0.729410i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	95027.6i	1.66362i	0.555061	+	0.831809i	$0.312695\pi$
$239$	95027.6i	1.66362i	−0.555061	+	0.831809i	$0.687305\pi$
$240$	0	0
$241$	−102393.	−1.76293	−0.881464	−	0.472252i	$-0.843441\pi$
$241$	−102393.	−1.76293	−0.881464	+	0.472252i	$0.843441\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11876.3	0.197857
$246$	0	0
$247$	19505.3i	0.319712i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	77450.3i	1.22935i	0.788780	+	0.614675i	$0.210713\pi$
$251$	77450.3i	1.22935i	−0.788780	+	0.614675i	$0.789287\pi$
$252$	0	0
$253$	−108219.	−1.69069
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30750.0	0.465563	0.232782	−	0.972529i	$-0.425217\pi$
$257$	30750.0	0.465563	0.232782	+	0.972529i	$0.425217\pi$
$258$	0	0
$259$	− 82991.8i	− 1.23719i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	32614.5i	0.471519i	0.971811	+	0.235759i	$0.0757577\pi$
$263$	32614.5i	0.471519i	−0.971811	+	0.235759i	$0.924242\pi$
$264$	0	0
$265$	19597.7	0.279070
$266$	0	0
$267$	0	0
$268$	0	0
$269$	101613.	1.40425	0.702124	−	0.712054i	$-0.252235\pi$
$269$	101613.	1.40425	0.702124	+	0.712054i	$0.252235\pi$
$270$	0	0
$271$	18100.7i	0.246467i	0.992378	+	0.123233i	$0.0393263\pi$
$271$	18100.7i	0.246467i	−0.992378	+	0.123233i	$0.960674\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 20357.3i	− 0.269187i
$276$	0	0
$277$	93653.6	1.22058	0.610288	−	0.792179i	$-0.291054\pi$
$277$	93653.6	1.22058	0.610288	+	0.792179i	$0.291054\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3668.36	−0.0464579	−0.0232289	−	0.999730i	$-0.507395\pi$
$281$	−3668.36	−0.0464579	−0.0232289	+	0.999730i	$0.507395\pi$
$282$	0	0
$283$	− 66383.8i	− 0.828876i	−0.910078	−	0.414438i	$-0.863978\pi$
$283$	− 66383.8i	− 0.828876i	0.910078	−	0.414438i	$-0.136022\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 16175.0i	− 0.196372i
$288$	0	0
$289$	−37584.4	−0.450000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	96944.2	1.12924	0.564620	−	0.825351i	$-0.309023\pi$
$293$	96944.2	1.12924	0.564620	+	0.825351i	$0.309023\pi$
$294$	0	0
$295$	− 44980.4i	− 0.516868i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 35050.8i	− 0.392063i
$300$	0	0
$301$	55189.1	0.609145
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−34565.9	−0.371576
$306$	0	0
$307$	117094.i	1.24239i	0.783656	+	0.621195i	$0.213353\pi$
$307$	117094.i	1.24239i	−0.783656	+	0.621195i	$0.786647\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	47400.0i	0.490069i	0.969514	+	0.245035i	$0.0787993\pi$
$311$	47400.0i	0.490069i	−0.969514	+	0.245035i	$0.921201\pi$
$312$	0	0
$313$	168773.	1.72272	0.861361	−	0.507994i	$-0.169613\pi$
$313$	168773.	1.72272	0.861361	+	0.507994i	$0.169613\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	54885.4	0.546184	0.273092	−	0.961988i	$-0.411954\pi$
$317$	54885.4	0.546184	0.273092	+	0.961988i	$0.411954\pi$
$318$	0	0
$319$	48995.2i	0.481473i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 79255.3i	− 0.759667i
$324$	0	0
$325$	6593.46	0.0624233
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−131826.	−1.21789
$330$	0	0
$331$	132847.i	1.21254i	0.795260	+	0.606268i	$0.207334\pi$
$331$	132847.i	1.21254i	−0.795260	+	0.606268i	$0.792666\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 92837.8i	− 0.827247i
$336$	0	0
$337$	−87602.0	−0.771355	−0.385678	−	0.922634i	$-0.626032\pi$
$337$	−87602.0	−0.771355	−0.385678	+	0.922634i	$0.626032\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−50561.6	−0.434822
$342$	0	0
$343$	126717.i	1.07707i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	84305.9i	0.700163i	0.936719	+	0.350081i	$0.113846\pi$
$347$	84305.9i	0.700163i	−0.936719	+	0.350081i	$0.886154\pi$
$348$	0	0
$349$	92865.1	0.762433	0.381216	−	0.924486i	$-0.375505\pi$
$349$	92865.1	0.762433	0.381216	+	0.924486i	$0.375505\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−233027.	−1.87007	−0.935033	−	0.354561i	$-0.884630\pi$
$353$	−233027.	−1.87007	−0.935033	+	0.354561i	$0.884630\pi$
$354$	0	0
$355$	− 28152.8i	− 0.223391i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 91097.6i	− 0.706835i	−0.935466	−	0.353417i	$-0.885020\pi$
$359$	− 91097.6i	− 0.706835i	0.935466	−	0.353417i	$-0.114980\pi$
$360$	0	0
$361$	−6419.78	−0.0492613
$362$	0	0
$363$	0	0
$364$	0	0
$365$	93959.0	0.705265
$366$	0	0
$367$	− 33168.2i	− 0.246258i	−0.992391	−	0.123129i	$-0.960707\pi$
$367$	− 33168.2i	− 0.246258i	0.992391	−	0.123129i	$-0.0392929\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	64135.5i	0.465962i
$372$	0	0
$373$	25147.9	0.180752	0.0903762	−	0.995908i	$-0.471193\pi$
$373$	25147.9	0.180752	0.0903762	+	0.995908i	$0.471193\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15868.9	−0.111651
$378$	0	0
$379$	160183.i	1.11517i	0.830121	+	0.557583i	$0.188271\pi$
$379$	160183.i	1.11517i	−0.830121	+	0.557583i	$0.811729\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 98909.6i	− 0.674281i	−0.941454	−	0.337141i	$-0.890540\pi$
$383$	− 98909.6i	− 0.674281i	0.941454	−	0.337141i	$-0.109460\pi$
$384$	0	0
$385$	66621.5	0.449462
$386$	0	0
$387$	0	0
$388$	0	0
$389$	44918.2	0.296840	0.148420	−	0.988924i	$-0.452581\pi$
$389$	44918.2	0.296840	0.148420	+	0.988924i	$0.452581\pi$
$390$	0	0
$391$	142421.i	0.931581i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	104910.i	0.672390i
$396$	0	0
$397$	−14849.1	−0.0942147	−0.0471073	−	0.998890i	$-0.515000\pi$
$397$	−14849.1	−0.0942147	−0.0471073	+	0.998890i	$0.515000\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	213981.	1.33072	0.665361	−	0.746522i	$-0.268278\pi$
$401$	213981.	1.33072	0.665361	+	0.746522i	$0.268278\pi$
$402$	0	0
$403$	− 16376.2i	− 0.100833i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	369399.i	2.23001i
$408$	0	0
$409$	43520.4	0.260163	0.130082	−	0.991503i	$-0.458476\pi$
$409$	43520.4	0.260163	0.130082	+	0.991503i	$0.458476\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	147203.	0.863014
$414$	0	0
$415$	− 121664.i	− 0.706425i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	28184.6i	0.160540i	0.996773	+	0.0802701i	$0.0255783\pi$
$419$	28184.6i	0.160540i	−0.996773	+	0.0802701i	$0.974422\pi$
$420$	0	0
$421$	211952.	1.19584	0.597921	−	0.801555i	$-0.295994\pi$
$421$	211952.	1.19584	0.597921	+	0.801555i	$0.295994\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−26791.0	−0.148324
$426$	0	0
$427$	− 113121.i	− 0.620420i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	212480.i	1.14384i	0.820310	+	0.571919i	$0.193801\pi$
$431$	212480.i	1.14384i	−0.820310	+	0.571919i	$0.806199\pi$
$432$	0	0
$433$	26538.3	0.141546	0.0707730	−	0.997492i	$-0.477453\pi$
$433$	26538.3	0.141546	0.0707730	+	0.997492i	$0.477453\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	245722.	1.28671
$438$	0	0
$439$	276500.i	1.43472i	0.696703	+	0.717359i	$0.254649\pi$
$439$	276500.i	1.43472i	−0.696703	+	0.717359i	$0.745351\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 163600.i	− 0.833637i	−0.908990	−	0.416818i	$-0.863145\pi$
$443$	− 163600.i	− 0.833637i	0.908990	−	0.416818i	$-0.136855\pi$
$444$	0	0
$445$	−151346.	−0.764276
$446$	0	0
$447$	0	0
$448$	0	0
$449$	121506.	0.602704	0.301352	−	0.953513i	$-0.402562\pi$
$449$	121506.	0.602704	0.301352	+	0.953513i	$0.402562\pi$
$450$	0	0
$451$	71995.2i	0.353957i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	21577.8i	0.104228i
$456$	0	0
$457$	−270788.	−1.29657	−0.648286	−	0.761397i	$-0.724514\pi$
$457$	−270788.	−1.29657	−0.648286	+	0.761397i	$0.724514\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−76883.8	−0.361770	−0.180885	−	0.983504i	$-0.557896\pi$
$461$	−76883.8	−0.361770	−0.180885	+	0.983504i	$0.557896\pi$
$462$	0	0
$463$	4544.05i	0.0211973i	0.999944	+	0.0105987i	$0.00337372\pi$
$463$	4544.05i	0.0211973i	−0.999944	+	0.0105987i	$0.996626\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	238931.i	1.09557i	0.836620	+	0.547783i	$0.184528\pi$
$467$	238931.i	1.09557i	−0.836620	+	0.547783i	$0.815472\pi$
$468$	0	0
$469$	303822.	1.38125
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−245649.	−1.09797
$474$	0	0
$475$	46223.1i	0.204867i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 15941.5i	− 0.0694797i	−0.999396	−	0.0347398i	$-0.988940\pi$
$479$	− 15941.5i	− 0.0694797i	0.999396	−	0.0347398i	$-0.0110603\pi$
$480$	0	0
$481$	−119644.	−0.517129
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−112804.	−0.479557
$486$	0	0
$487$	87667.0i	0.369640i	0.982772	+	0.184820i	$0.0591701\pi$
$487$	87667.0i	0.369640i	−0.982772	+	0.184820i	$0.940830\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	317165.i	1.31560i	0.753194	+	0.657798i	$0.228512\pi$
$491$	317165.i	1.31560i	−0.753194	+	0.657798i	$0.771488\pi$
$492$	0	0
$493$	64479.6	0.265295
$494$	0	0
$495$	0	0
$496$	0	0
$497$	92133.3	0.372996
$498$	0	0
$499$	242643.i	0.974468i	0.873272	+	0.487234i	$0.161994\pi$
$499$	242643.i	0.974468i	−0.873272	+	0.487234i	$0.838006\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 381517.i	− 1.50792i	−0.656922	−	0.753959i	$-0.728142\pi$
$503$	− 381517.i	− 1.50792i	0.656922	−	0.753959i	$-0.271858\pi$
$504$	0	0
$505$	−79555.1	−0.311950
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−232841.	−0.898720	−0.449360	−	0.893351i	$-0.648348\pi$
$509$	−232841.	−0.898720	−0.449360	+	0.893351i	$0.648348\pi$
$510$	0	0
$511$	307491.i	1.17758i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 46041.0i	− 0.173592i
$516$	0	0
$517$	586762.	2.19524
$518$	0	0
$519$	0	0
$520$	0	0
$521$	353752.	1.30324	0.651619	−	0.758546i	$-0.274090\pi$
$521$	353752.	1.30324	0.651619	+	0.758546i	$0.274090\pi$
$522$	0	0
$523$	95102.2i	0.347686i	0.984773	+	0.173843i	$0.0556185\pi$
$523$	95102.2i	0.347686i	−0.984773	+	0.173843i	$0.944381\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	66541.0i	0.239590i
$528$	0	0
$529$	−161720.	−0.577898
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−23318.3	−0.0820810
$534$	0	0
$535$	110031.i	0.384420i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 172997.i	− 0.595470i
$540$	0	0
$541$	513368.	1.75402	0.877009	−	0.480473i	$-0.159535\pi$
$541$	513368.	1.75402	0.877009	+	0.480473i	$0.159535\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−60435.7	−0.203470
$546$	0	0
$547$	− 36282.1i	− 0.121260i	−0.998160	−	0.0606301i	$-0.980689\pi$
$547$	− 36282.1i	− 0.121260i	0.998160	−	0.0606301i	$-0.0193110\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 111248.i	− 0.366428i
$552$	0	0
$553$	−343328.	−1.12269
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−187444.	−0.604174	−0.302087	−	0.953280i	$-0.597683\pi$
$557$	−187444.	−0.604174	−0.302087	+	0.953280i	$0.597683\pi$
$558$	0	0
$559$	− 79562.3i	− 0.254615i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 43663.9i	− 0.137754i	−0.997625	−	0.0688772i	$-0.978058\pi$
$563$	− 43663.9i	− 0.137754i	0.997625	−	0.0688772i	$-0.0219417\pi$
$564$	0	0
$565$	−24587.9	−0.0770236
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−281148.	−0.868380	−0.434190	−	0.900821i	$-0.642965\pi$
$569$	−281148.	−0.868380	−0.434190	+	0.900821i	$0.642965\pi$
$570$	0	0
$571$	47208.6i	0.144793i	0.997376	+	0.0723967i	$0.0230648\pi$
$571$	47208.6i	0.144793i	−0.997376	+	0.0723967i	$0.976935\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 83062.5i	− 0.251229i
$576$	0	0
$577$	54896.3	0.164889	0.0824445	−	0.996596i	$-0.473727\pi$
$577$	54896.3	0.164889	0.0824445	+	0.996596i	$0.473727\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	398159.	1.17952
$582$	0	0
$583$	− 285469.i	− 0.839890i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 388030.i	− 1.12613i	−0.826412	−	0.563066i	$-0.809622\pi$
$587$	− 388030.i	− 1.12613i	0.826412	−	0.563066i	$-0.190378\pi$
$588$	0	0
$589$	114805.	0.330924
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−105032.	−0.298683	−0.149342	−	0.988786i	$-0.547715\pi$
$593$	−105032.	−0.298683	−0.149342	+	0.988786i	$0.547715\pi$
$594$	0	0
$595$	− 87676.6i	− 0.247656i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	700009.i	1.95097i	0.220075	+	0.975483i	$0.429370\pi$
$599$	700009.i	1.95097i	−0.220075	+	0.975483i	$0.570630\pi$
$600$	0	0
$601$	−62068.8	−0.171840	−0.0859201	−	0.996302i	$-0.527383\pi$
$601$	−62068.8	−0.171840	−0.0859201	+	0.996302i	$0.527383\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−132843.	−0.362934
$606$	0	0
$607$	291592.i	0.791405i	0.918379	+	0.395703i	$0.129499\pi$
$607$	291592.i	0.791405i	−0.918379	+	0.395703i	$0.870501\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	190045.i	0.509065i
$612$	0	0
$613$	−319676.	−0.850724	−0.425362	−	0.905023i	$-0.639853\pi$
$613$	−319676.	−0.850724	−0.425362	+	0.905023i	$0.639853\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	566436.	1.48792	0.743962	−	0.668222i	$-0.232944\pi$
$617$	566436.	1.48792	0.743962	+	0.668222i	$0.232944\pi$
$618$	0	0
$619$	− 507691.i	− 1.32501i	−0.749059	−	0.662503i	$-0.769494\pi$
$619$	− 507691.i	− 1.32501i	0.749059	−	0.662503i	$-0.230506\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 495296.i	− 1.27611i
$624$	0	0
$625$	15625.0	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	486144.	1.22875
$630$	0	0
$631$	− 637776.i	− 1.60181i	−0.598795	−	0.800903i	$-0.704353\pi$
$631$	− 637776.i	− 1.60181i	0.598795	−	0.800903i	$-0.295647\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 332051.i	− 0.823489i
$636$	0	0
$637$	56031.3	0.138087
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−150086.	−0.365277	−0.182639	−	0.983180i	$-0.558464\pi$
$641$	−150086.	−0.365277	−0.182639	+	0.983180i	$0.558464\pi$
$642$	0	0
$643$	− 405951.i	− 0.981864i	−0.871198	−	0.490932i	$-0.836656\pi$
$643$	− 405951.i	− 0.981864i	0.871198	−	0.490932i	$-0.163344\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 22789.6i	− 0.0544412i	−0.999629	−	0.0272206i	$-0.991334\pi$
$647$	− 22789.6i	− 0.0544412i	0.999629	−	0.0272206i	$-0.00866566\pi$
$648$	0	0
$649$	−655207.	−1.55557
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−420289.	−0.985648	−0.492824	−	0.870129i	$-0.664035\pi$
$653$	−420289.	−0.985648	−0.492824	+	0.870129i	$0.664035\pi$
$654$	0	0
$655$	4532.74i	0.0105652i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	566312.i	1.30402i	0.758210	+	0.652011i	$0.226074\pi$
$659$	566312.i	1.30402i	−0.758210	+	0.652011i	$0.773926\pi$
$660$	0	0
$661$	−133072.	−0.304568	−0.152284	−	0.988337i	$-0.548663\pi$
$661$	−133072.	−0.304568	−0.152284	+	0.988337i	$0.548663\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−151270.	−0.342066
$666$	0	0
$667$	199912.i	0.449352i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	503503.i	1.11830i
$672$	0	0
$673$	−579942.	−1.28043	−0.640213	−	0.768197i	$-0.721154\pi$
$673$	−579942.	−1.28043	−0.640213	+	0.768197i	$0.721154\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−288127.	−0.628646	−0.314323	−	0.949316i	$-0.601778\pi$
$677$	−288127.	−0.628646	−0.314323	+	0.949316i	$0.601778\pi$
$678$	0	0
$679$	− 369163.i	− 0.800716i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 356621.i	− 0.764479i	−0.924063	−	0.382240i	$-0.875153\pi$
$683$	− 356621.i	− 0.764479i	0.924063	−	0.382240i	$-0.124847\pi$
$684$	0	0
$685$	119993.	0.255727
$686$	0	0
$687$	0	0
$688$	0	0
$689$	92459.7	0.194766
$690$	0	0
$691$	379750.i	0.795319i	0.917533	+	0.397660i	$0.130178\pi$
$691$	379750.i	0.795319i	−0.917533	+	0.397660i	$0.869822\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 288784.i	− 0.597866i
$696$	0	0
$697$	94748.6	0.195033
$698$	0	0
$699$	0	0
$700$	0	0
$701$	592953.	1.20666	0.603329	−	0.797493i	$-0.293841\pi$
$701$	592953.	1.20666	0.603329	+	0.797493i	$0.293841\pi$
$702$	0	0
$703$	− 838755.i	− 1.69717i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 260353.i	− 0.520863i
$708$	0	0
$709$	−461296.	−0.917671	−0.458835	−	0.888521i	$-0.651733\pi$
$709$	−461296.	−0.917671	−0.458835	+	0.888521i	$0.651733\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−206303.	−0.405813
$714$	0	0
$715$	− 96043.6i	− 0.187869i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 320044.i	− 0.619087i	−0.950885	−	0.309544i	$-0.899824\pi$
$719$	− 320044.i	− 0.619087i	0.950885	−	0.309544i	$-0.100176\pi$
$720$	0	0
$721$	150674.	0.289847
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−37605.7	−0.0715446
$726$	0	0
$727$	680321.i	1.28720i	0.765363	+	0.643599i	$0.222560\pi$
$727$	680321.i	1.28720i	−0.765363	+	0.643599i	$0.777440\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	323283.i	0.604991i
$732$	0	0
$733$	714211.	1.32929	0.664643	−	0.747161i	$-0.268584\pi$
$733$	714211.	1.32929	0.664643	+	0.747161i	$0.268584\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.35232e6	−2.48969
$738$	0	0
$739$	− 147574.i	− 0.270223i	−0.990830	−	0.135111i	$-0.956861\pi$
$739$	− 147574.i	− 0.270223i	0.990830	−	0.135111i	$-0.0431392\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26402.2i	0.0478258i	0.999714	+	0.0239129i	$0.00761244\pi$
$743$	26402.2i	0.0478258i	−0.999714	+	0.0239129i	$0.992388\pi$
$744$	0	0
$745$	−256618.	−0.462354
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−360088.	−0.641866
$750$	0	0
$751$	432471.i	0.766791i	0.923584	+	0.383395i	$0.125245\pi$
$751$	432471.i	0.766791i	−0.923584	+	0.383395i	$0.874755\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 341172.i	− 0.598520i
$756$	0	0
$757$	−308072.	−0.537601	−0.268801	−	0.963196i	$-0.586627\pi$
$757$	−308072.	−0.537601	−0.268801	+	0.963196i	$0.586627\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−596768.	−1.03047	−0.515236	−	0.857048i	$-0.672296\pi$
$761$	−596768.	−1.03047	−0.515236	+	0.857048i	$0.672296\pi$
$762$	0	0
$763$	− 197783.i	− 0.339734i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 212213.i	− 0.360729i
$768$	0	0
$769$	69759.1	0.117964	0.0589818	−	0.998259i	$-0.481215\pi$
$769$	69759.1	0.117964	0.0589818	+	0.998259i	$0.481215\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	55125.2	0.0922552	0.0461276	−	0.998936i	$-0.485312\pi$
$773$	55125.2	0.0922552	0.0461276	+	0.998936i	$0.485312\pi$
$774$	0	0
$775$	− 38807.9i	− 0.0646126i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 163472.i	− 0.269382i
$780$	0	0
$781$	−410088.	−0.672319
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−50916.7	−0.0826269
$786$	0	0
$787$	− 259759.i	− 0.419393i	−0.977767	−	0.209696i	$-0.932752\pi$
$787$	− 259759.i	− 0.419393i	0.977767	−	0.209696i	$-0.0672475\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 80466.5i	− 0.128606i
$792$	0	0
$793$	−163078.	−0.259328
$794$	0	0
$795$	0	0
$796$	0	0
$797$	896807.	1.41183	0.705915	−	0.708296i	$-0.250536\pi$
$797$	896807.	1.41183	0.705915	+	0.708296i	$0.250536\pi$
$798$	0	0
$799$	− 772203.i	− 1.20959i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.36865e6i	− 2.12257i
$804$	0	0
$805$	271831.	0.419477
$806$	0	0
$807$	0	0
$808$	0	0
$809$	262986.	0.401824	0.200912	−	0.979609i	$-0.435610\pi$
$809$	262986.	0.401824	0.200912	+	0.979609i	$0.435610\pi$
$810$	0	0
$811$	226369.i	0.344172i	0.985082	+	0.172086i	$0.0550507\pi$
$811$	226369.i	0.344172i	−0.985082	+	0.172086i	$0.944949\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 189373.i	− 0.285104i
$816$	0	0
$817$	557768.	0.835621
$818$	0	0
$819$	0	0
$820$	0	0
$821$	659303.	0.978135	0.489067	−	0.872246i	$-0.337337\pi$
$821$	659303.	0.978135	0.489067	+	0.872246i	$0.337337\pi$
$822$	0	0
$823$	− 588601.i	− 0.869003i	−0.900671	−	0.434502i	$-0.856924\pi$
$823$	− 588601.i	− 0.869003i	0.900671	−	0.434502i	$-0.143076\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	913221.i	1.33526i	0.744494	+	0.667629i	$0.232691\pi$
$827$	913221.i	1.33526i	−0.744494	+	0.667629i	$0.767309\pi$
$828$	0	0
$829$	353959.	0.515043	0.257522	−	0.966273i	$-0.417094\pi$
$829$	353959.	0.515043	0.257522	+	0.966273i	$0.417094\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−227671.	−0.328108
$834$	0	0
$835$	− 380940.i	− 0.546366i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 885043.i	− 1.25730i	−0.777686	−	0.628652i	$-0.783607\pi$
$839$	− 885043.i	− 1.25730i	0.777686	−	0.628652i	$-0.216393\pi$
$840$	0	0
$841$	−616773.	−0.872034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−288214.	−0.403648
$846$	0	0
$847$	− 434744.i	− 0.605991i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.50724e6i	2.08124i
$852$	0	0
$853$	102468.	0.140828	0.0704141	−	0.997518i	$-0.477568\pi$
$853$	102468.	0.140828	0.0704141	+	0.997518i	$0.477568\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	404804.	0.551167	0.275583	−	0.961277i	$-0.411129\pi$
$857$	404804.	0.551167	0.275583	+	0.961277i	$0.411129\pi$
$858$	0	0
$859$	124674.i	0.168962i	0.996425	+	0.0844808i	$0.0269232\pi$
$859$	124674.i	0.168962i	−0.996425	+	0.0844808i	$0.973077\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 920667.i	− 1.23618i	−0.786108	−	0.618089i	$-0.787907\pi$
$863$	− 920667.i	− 1.23618i	0.786108	−	0.618089i	$-0.212093\pi$
$864$	0	0
$865$	20883.7	0.0279109
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.52816e6	2.02363
$870$	0	0
$871$	− 437999.i	− 0.577347i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	51134.5i	0.0667880i
$876$	0	0
$877$	−699098.	−0.908947	−0.454474	−	0.890760i	$-0.650173\pi$
$877$	−699098.	−0.908947	−0.454474	+	0.890760i	$0.650173\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−524442.	−0.675687	−0.337843	−	0.941202i	$-0.609697\pi$
$881$	−524442.	−0.675687	−0.337843	+	0.941202i	$0.609697\pi$
$882$	0	0
$883$	− 1.24169e6i	− 1.59254i	−0.604942	−	0.796270i	$-0.706804\pi$
$883$	− 1.24169e6i	− 1.59254i	0.604942	−	0.796270i	$-0.293196\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.06805e6i	1.35751i	0.734363	+	0.678757i	$0.237481\pi$
$887$	1.06805e6i	1.35751i	−0.734363	+	0.678757i	$0.762519\pi$
$888$	0	0
$889$	1.08668e6	1.37498
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.33230e6	−1.67070
$894$	0	0
$895$	− 108872.i	− 0.135916i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	93401.4i	0.115567i
$900$	0	0
$901$	−375689.	−0.462785
$902$	0	0
$903$	0	0
$904$	0	0
$905$	715803.	0.873970
$906$	0	0
$907$	− 1.09531e6i	− 1.33144i	−0.746200	−	0.665722i	$-0.768124\pi$
$907$	− 1.09531e6i	− 1.33144i	0.746200	−	0.665722i	$-0.231876\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	174846.i	0.210678i	0.994436	+	0.105339i	$0.0335928\pi$
$911$	174846.i	0.210678i	−0.994436	+	0.105339i	$0.966407\pi$
$912$	0	0
$913$	−1.77222e6	−2.12606
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14833.9	−0.0176407
$918$	0	0
$919$	299047.i	0.354086i	0.984203	+	0.177043i	$0.0566531\pi$
$919$	299047.i	0.354086i	−0.984203	+	0.177043i	$0.943347\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 132822.i	− 0.155908i
$924$	0	0
$925$	−283528.	−0.331370
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.07473e6	−1.24528	−0.622639	−	0.782509i	$-0.713940\pi$
$929$	−1.07473e6	−1.24528	−0.622639	+	0.782509i	$0.713940\pi$
$930$	0	0
$931$	392805.i	0.453187i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	390251.i	0.446397i
$936$	0	0
$937$	−1.38751e6	−1.58036	−0.790180	−	0.612875i	$-0.790013\pi$
$937$	−1.38751e6	−1.58036	−0.790180	+	0.612875i	$0.790013\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	590849.	0.667264	0.333632	−	0.942703i	$-0.391726\pi$
$941$	590849.	0.667264	0.333632	+	0.942703i	$0.391726\pi$
$942$	0	0
$943$	293757.i	0.330343i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 232949.i	− 0.259753i	−0.991530	−	0.129877i	$-0.958542\pi$
$947$	− 232949.i	− 0.259753i	0.991530	−	0.129877i	$-0.0414581\pi$
$948$	0	0
$949$	443288.	0.492214
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−740127.	−0.814930	−0.407465	−	0.913221i	$-0.633587\pi$
$953$	−740127.	−0.814930	−0.407465	+	0.913221i	$0.633587\pi$
$954$	0	0
$955$	− 291890.i	− 0.320046i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	392692.i	0.426987i
$960$	0	0
$961$	827133.	0.895630
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−367980.	−0.395157
$966$	0	0
$967$	− 974550.i	− 1.04220i	−0.853496	−	0.521100i	$-0.825522\pi$
$967$	− 974550.i	− 1.04220i	0.853496	−	0.521100i	$-0.174478\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 519990.i	− 0.551514i	−0.961227	−	0.275757i	$-0.911071\pi$
$971$	− 519990.i	− 0.551514i	0.961227	−	0.275757i	$-0.0889285\pi$
$972$	0	0
$973$	945078.	0.998256
$974$	0	0
$975$	0	0
$976$	0	0
$977$	561487.	0.588235	0.294117	−	0.955769i	$-0.404974\pi$
$977$	561487.	0.588235	0.294117	+	0.955769i	$0.404974\pi$
$978$	0	0
$979$	2.20458e6i	2.30017i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.52223e6i	− 1.57533i	−0.616101	−	0.787667i	$-0.711289\pi$
$983$	− 1.52223e6i	− 1.57533i	0.616101	−	0.787667i	$-0.288711\pi$
$984$	0	0
$985$	−765026.	−0.788503
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.00230e6	−1.02472
$990$	0	0
$991$	1.40959e6i	1.43531i	0.696400	+	0.717654i	$0.254784\pi$
$991$	1.40959e6i	1.43531i	−0.696400	+	0.717654i	$0.745216\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 246784.i	− 0.249271i
$996$	0	0
$997$	753052.	0.757591	0.378795	−	0.925480i	$-0.376338\pi$
$997$	753052.	0.757591	0.378795	+	0.925480i	$0.376338\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.5.e.b.271.4		4
3.2	odd	2		80.5.b.b.31.2	✓	4
4.3	odd	2	inner	720.5.e.b.271.3		4
12.11	even	2		80.5.b.b.31.3	yes	4
15.2	even	4		400.5.h.c.399.4		8
15.8	even	4		400.5.h.c.399.6		8
15.14	odd	2		400.5.b.h.351.3		4
24.5	odd	2		320.5.b.b.191.3		4
24.11	even	2		320.5.b.b.191.2		4
60.23	odd	4		400.5.h.c.399.3		8
60.47	odd	4		400.5.h.c.399.5		8
60.59	even	2		400.5.b.h.351.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.5.b.b.31.2	✓	4	3.2	odd	2
80.5.b.b.31.3	yes	4	12.11	even	2
320.5.b.b.191.2		4	24.11	even	2
320.5.b.b.191.3		4	24.5	odd	2
400.5.b.h.351.2		4	60.59	even	2
400.5.b.h.351.3		4	15.14	odd	2
400.5.h.c.399.3		8	60.23	odd	4
400.5.h.c.399.4		8	15.2	even	4
400.5.h.c.399.5		8	60.47	odd	4
400.5.h.c.399.6		8	15.8	even	4
720.5.e.b.271.3		4	4.3	odd	2	inner
720.5.e.b.271.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	720.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.1803 q^{5} +36.5889i q^{7} +O(q^{10})\)	\(q+11.1803 q^{5} +36.5889i q^{7} -162.858i q^{11} +52.7477 q^{13} -214.328 q^{17} +369.785i q^{19} -664.500i q^{23} +125.000 q^{25} -300.845 q^{29} -310.463i q^{31} +409.076i q^{35} -2268.22 q^{37} -442.073 q^{41} -1508.36i q^{43} +3602.90i q^{47} +1062.25 q^{49} +1752.87 q^{53} -1820.81i q^{55} -4023.17i q^{59} -3091.66 q^{61} +589.737 q^{65} -8303.66i q^{67} -2518.07i q^{71} +8403.94 q^{73} +5958.81 q^{77} +9383.40i q^{79} -10882.0i q^{83} -2396.26 q^{85} -13536.8 q^{89} +1929.98i q^{91} +4134.32i q^{95} -10089.5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 272 q^{13} + 216 q^{17} + 500 q^{25} - 3672 q^{29} - 3760 q^{37} + 432 q^{41} + 4732 q^{49} - 3024 q^{53} + 5504 q^{61} + 5400 q^{65} + 22024 q^{73} + 18576 q^{77} - 12000 q^{85} - 8424 q^{89} - 35528 q^{97}+O(q^{100}) \) 4 * q - 272 * q^13 + 216 * q^17 + 500 * q^25 - 3672 * q^29 - 3760 * q^37 + 432 * q^41 + 4732 * q^49 - 3024 * q^53 + 5504 * q^61 + 5400 * q^65 + 22024 * q^73 + 18576 * q^77 - 12000 * q^85 - 8424 * q^89 - 35528 * q^97

Introduction

L-functions

Modular forms

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Database

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