LMFDB - Embedded newform 405.5.c.b.161.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,5,Mod(161,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	405.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,-256,0,0,52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$41.8648350490$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.19
Character	$\chi$	$=$	405.161
Dual form			405.5.c.b.161.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.23097i q^{2} +14.4847 q^{4} -11.1803i q^{5} +42.2629 q^{7} +37.5258i q^{8} +13.7627 q^{10} -27.6338i q^{11} +236.614 q^{13} +52.0244i q^{14} +185.562 q^{16} -304.483i q^{17} -315.542 q^{19} -161.944i q^{20} +34.0165 q^{22} +840.845i q^{23} -125.000 q^{25} +291.266i q^{26} +612.165 q^{28} -776.278i q^{29} +1666.24 q^{31} +828.835i q^{32} +374.810 q^{34} -472.513i q^{35} -209.474 q^{37} -388.424i q^{38} +419.552 q^{40} -2103.41i q^{41} -1611.02 q^{43} -400.267i q^{44} -1035.06 q^{46} +370.631i q^{47} -614.850 q^{49} -153.872i q^{50} +3427.29 q^{52} -2527.06i q^{53} -308.955 q^{55} +1585.95i q^{56} +955.577 q^{58} +1376.56i q^{59} +800.400 q^{61} +2051.09i q^{62} +1948.72 q^{64} -2645.43i q^{65} +2385.40 q^{67} -4410.35i q^{68} +581.651 q^{70} +1978.96i q^{71} +10301.8 q^{73} -257.857i q^{74} -4570.54 q^{76} -1167.88i q^{77} +5831.54 q^{79} -2074.65i q^{80} +2589.23 q^{82} +170.358i q^{83} -3404.22 q^{85} -1983.12i q^{86} +1036.98 q^{88} +11188.1i q^{89} +9999.99 q^{91} +12179.4i q^{92} -456.237 q^{94} +3527.87i q^{95} -1032.89 q^{97} -756.864i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 256 q^{4} + 52 q^{7} - 20 q^{13} + 2048 q^{16} + 508 q^{19} - 1344 q^{22} - 4000 q^{25} - 1664 q^{28} + 2944 q^{31} + 1188 q^{34} + 2068 q^{37} - 3300 q^{40} - 1136 q^{43} + 5724 q^{46} + 3348 q^{49}+ \cdots - 46532 q^{97}+O(q^{100}) $ 32 * q - 256 * q^4 + 52 * q^7 - 20 * q^13 + 2048 * q^16 + 508 * q^19 - 1344 * q^22 - 4000 * q^25 - 1664 * q^28 + 2944 * q^31 + 1188 * q^34 + 2068 * q^37 - 3300 * q^40 - 1136 * q^43 + 5724 * q^46 + 3348 * q^49 - 3068 * q^52 + 16380 * q^58 + 8956 * q^61 - 37216 * q^64 - 25676 * q^67 - 4800 * q^70 + 29236 * q^73 - 12380 * q^76 + 7024 * q^79 + 34692 * q^82 + 8700 * q^85 + 32256 * q^88 - 48904 * q^91 - 10236 * q^94 - 46532 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.23097i	0.307743i	0.988091	+	0.153872i	$0.0491742\pi$
$2$	1.23097i	0.307743i	−0.988091	+	0.153872i	$0.950826\pi$
$3$	0	0
$3$	0	0
$4$	14.4847	0.905294
$5$	− 11.1803i	− 0.447214i
$5$	− 11.1803i	− 0.447214i
$6$	0	0
$7$	42.2629	0.862507	0.431254	−	0.902231i	$-0.358071\pi$
$7$	42.2629	0.862507	0.431254	+	0.902231i	$0.358071\pi$
$8$	37.5258i	0.586341i
$9$	0	0
$10$	13.7627	0.137627
$11$	− 27.6338i	− 0.228378i	−0.993459	−	0.114189i	$-0.963573\pi$
$11$	− 27.6338i	− 0.228378i	0.993459	−	0.114189i	$-0.0364270\pi$
$12$	0	0
$13$	236.614	1.40008	0.700042	−	0.714102i	$-0.253165\pi$
$13$	236.614	1.40008	0.700042	+	0.714102i	$0.253165\pi$
$14$	52.0244i	0.265431i
$15$	0	0
$16$	185.562	0.724852
$17$	− 304.483i	− 1.05357i	−0.849997	−	0.526787i	$-0.823397\pi$
$17$	− 304.483i	− 1.05357i	0.849997	−	0.526787i	$-0.176603\pi$
$18$	0	0
$19$	−315.542	−0.874079	−0.437039	−	0.899442i	$-0.643973\pi$
$19$	−315.542	−0.874079	−0.437039	+	0.899442i	$0.643973\pi$
$20$	− 161.944i	− 0.404860i
$21$	0	0
$22$	34.0165	0.0702819
$23$	840.845i	1.58950i	0.606937	+	0.794750i	$0.292398\pi$
$23$	840.845i	1.58950i	−0.606937	+	0.794750i	$0.707602\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	291.266i	0.430866i
$27$	0	0
$28$	612.165	0.780823
$29$	− 776.278i	− 0.923042i	−0.887129	−	0.461521i	$-0.847304\pi$
$29$	− 776.278i	− 0.923042i	0.887129	−	0.461521i	$-0.152696\pi$
$30$	0	0
$31$	1666.24	1.73386	0.866928	−	0.498433i	$-0.166091\pi$
$31$	1666.24	1.73386	0.866928	+	0.498433i	$0.166091\pi$
$32$	828.835i	0.809409i
$33$	0	0
$34$	374.810	0.324230
$35$	− 472.513i	− 0.385725i
$36$	0	0
$37$	−209.474	−0.153013	−0.0765063	−	0.997069i	$-0.524377\pi$
$37$	−209.474	−0.153013	−0.0765063	+	0.997069i	$0.524377\pi$
$38$	− 388.424i	− 0.268992i
$39$	0	0
$40$	419.552	0.262220
$41$	− 2103.41i	− 1.25128i	−0.780111	−	0.625641i	$-0.784837\pi$
$41$	− 2103.41i	− 1.25128i	0.780111	−	0.625641i	$-0.215163\pi$
$42$	0	0
$43$	−1611.02	−0.871293	−0.435647	−	0.900118i	$-0.643480\pi$
$43$	−1611.02	−0.871293	−0.435647	+	0.900118i	$0.643480\pi$
$44$	− 400.267i	− 0.206750i
$45$	0	0
$46$	−1035.06	−0.489158
$47$	370.631i	0.167782i	0.996475	+	0.0838912i	$0.0267348\pi$
$47$	370.631i	0.167782i	−0.996475	+	0.0838912i	$0.973265\pi$
$48$	0	0
$49$	−614.850	−0.256081
$50$	− 153.872i	− 0.0615486i
$51$	0	0
$52$	3427.29	1.26749
$53$	− 2527.06i	− 0.899631i	−0.893121	−	0.449816i	$-0.851490\pi$
$53$	− 2527.06i	− 0.899631i	0.893121	−	0.449816i	$-0.148510\pi$
$54$	0	0
$55$	−308.955	−0.102134
$56$	1585.95i	0.505724i
$57$	0	0
$58$	955.577	0.284060
$59$	1376.56i	0.395449i	0.980258	+	0.197724i	$0.0633551\pi$
$59$	1376.56i	0.395449i	−0.980258	+	0.197724i	$0.936645\pi$
$60$	0	0
$61$	800.400	0.215104	0.107552	−	0.994199i	$-0.465699\pi$
$61$	800.400	0.215104	0.107552	+	0.994199i	$0.465699\pi$
$62$	2051.09i	0.533582i
$63$	0	0
$64$	1948.72	0.475761
$65$	− 2645.43i	− 0.626137i
$66$	0	0
$67$	2385.40	0.531387	0.265693	−	0.964058i	$-0.414399\pi$
$67$	2385.40	0.531387	0.265693	+	0.964058i	$0.414399\pi$
$68$	− 4410.35i	− 0.953794i
$69$	0	0
$70$	581.651	0.118704
$71$	1978.96i	0.392573i	0.980547	+	0.196287i	$0.0628883\pi$
$71$	1978.96i	0.392573i	−0.980547	+	0.196287i	$0.937112\pi$
$72$	0	0
$73$	10301.8	1.93315	0.966577	−	0.256376i	$-0.0825285\pi$
$73$	10301.8	1.93315	0.966577	+	0.256376i	$0.0825285\pi$
$74$	− 257.857i	− 0.0470886i
$75$	0	0
$76$	−4570.54	−0.791298
$77$	− 1167.88i	− 0.196978i
$78$	0	0
$79$	5831.54	0.934392	0.467196	−	0.884154i	$-0.345264\pi$
$79$	5831.54	0.934392	0.467196	+	0.884154i	$0.345264\pi$
$80$	− 2074.65i	− 0.324164i
$81$	0	0
$82$	2589.23	0.385074
$83$	170.358i	0.0247290i	0.999924	+	0.0123645i	$0.00393585\pi$
$83$	170.358i	0.0247290i	−0.999924	+	0.0123645i	$0.996064\pi$
$84$	0	0
$85$	−3404.22	−0.471173
$86$	− 1983.12i	− 0.268134i
$87$	0	0
$88$	1036.98	0.133908
$89$	11188.1i	1.41246i	0.707982	+	0.706230i	$0.249606\pi$
$89$	11188.1i	1.41246i	−0.707982	+	0.706230i	$0.750394\pi$
$90$	0	0
$91$	9999.99	1.20758
$92$	12179.4i	1.43896i
$93$	0	0
$94$	−456.237	−0.0516339
$95$	3527.87i	0.390900i
$96$	0	0
$97$	−1032.89	−0.109777	−0.0548884	−	0.998492i	$-0.517480\pi$
$97$	−1032.89	−0.109777	−0.0548884	+	0.998492i	$0.517480\pi$
$98$	− 756.864i	− 0.0788072i
$99$	0	0
$100$	−1810.59	−0.181059
$101$	4381.07i	0.429475i	0.976672	+	0.214737i	$0.0688895\pi$
$101$	4381.07i	0.429475i	−0.976672	+	0.214737i	$0.931110\pi$
$102$	0	0
$103$	11711.1	1.10388	0.551942	−	0.833882i	$-0.313887\pi$
$103$	11711.1	1.10388	0.551942	+	0.833882i	$0.313887\pi$
$104$	8879.15i	0.820927i
$105$	0	0
$106$	3110.75	0.276855
$107$	− 1773.37i	− 0.154893i	−0.996997	−	0.0774464i	$-0.975323\pi$
$107$	− 1773.37i	− 0.154893i	0.996997	−	0.0774464i	$-0.0246767\pi$
$108$	0	0
$109$	−7552.17	−0.635651	−0.317826	−	0.948149i	$-0.602953\pi$
$109$	−7552.17	−0.635651	−0.317826	+	0.948149i	$0.602953\pi$
$110$	− 380.315i	− 0.0314310i
$111$	0	0
$112$	7842.38	0.625190
$113$	− 25119.2i	− 1.96721i	−0.180349	−	0.983603i	$-0.557723\pi$
$113$	− 25119.2i	− 1.96721i	0.180349	−	0.983603i	$-0.442277\pi$
$114$	0	0
$115$	9400.93	0.710846
$116$	− 11244.2i	− 0.835625i
$117$	0	0
$118$	−1694.50	−0.121697
$119$	− 12868.3i	− 0.908715i
$120$	0	0
$121$	13877.4	0.947843
$122$	985.271i	0.0661966i
$123$	0	0
$124$	24134.9	1.56965
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	11630.0	0.721063	0.360531	−	0.932747i	$-0.382595\pi$
$127$	11630.0	0.721063	0.360531	+	0.932747i	$0.382595\pi$
$128$	15660.2i	0.955822i
$129$	0	0
$130$	3256.45	0.192689
$131$	26389.9i	1.53778i	0.639380	+	0.768891i	$0.279191\pi$
$131$	26389.9i	1.53778i	−0.639380	+	0.768891i	$0.720809\pi$
$132$	0	0
$133$	−13335.7	−0.753899
$134$	2936.36i	0.163531i
$135$	0	0
$136$	11426.0	0.617754
$137$	− 32851.0i	− 1.75028i	−0.483869	−	0.875141i	$-0.660769\pi$
$137$	− 32851.0i	− 1.75028i	0.483869	−	0.875141i	$-0.339231\pi$
$138$	0	0
$139$	11678.3	0.604438	0.302219	−	0.953239i	$-0.402273\pi$
$139$	11678.3	0.604438	0.302219	+	0.953239i	$0.402273\pi$
$140$	− 6844.21i	− 0.349195i
$141$	0	0
$142$	−2436.05	−0.120812
$143$	− 6538.55i	− 0.319749i
$144$	0	0
$145$	−8679.06	−0.412797
$146$	12681.2i	0.594915i
$147$	0	0
$148$	−3034.17	−0.138521
$149$	38427.6i	1.73089i	0.501000	+	0.865447i	$0.332966\pi$
$149$	38427.6i	1.73089i	−0.501000	+	0.865447i	$0.667034\pi$
$150$	0	0
$151$	−41728.6	−1.83012	−0.915060	−	0.403317i	$-0.867857\pi$
$151$	−41728.6	−1.83012	−0.915060	+	0.403317i	$0.867857\pi$
$152$	− 11841.0i	− 0.512508i
$153$	0	0
$154$	1437.63	0.0606187
$155$	− 18629.1i	− 0.775404i
$156$	0	0
$157$	849.964	0.0344827	0.0172413	−	0.999851i	$-0.494512\pi$
$157$	849.964	0.0344827	0.0172413	+	0.999851i	$0.494512\pi$
$158$	7178.47i	0.287553i
$159$	0	0
$160$	9266.66	0.361979
$161$	35536.5i	1.37095i
$162$	0	0
$163$	−13884.8	−0.522594	−0.261297	−	0.965259i	$-0.584150\pi$
$163$	−13884.8	−0.522594	−0.261297	+	0.965259i	$0.584150\pi$
$164$	− 30467.2i	− 1.13278i
$165$	0	0
$166$	−209.706	−0.00761019
$167$	− 4568.71i	− 0.163818i	−0.996640	−	0.0819088i	$-0.973898\pi$
$167$	− 4568.71i	− 0.163818i	0.996640	−	0.0819088i	$-0.0261016\pi$
$168$	0	0
$169$	27425.3	0.960236
$170$	− 4190.50i	− 0.145000i
$171$	0	0
$172$	−23335.2	−0.788777
$173$	28421.5i	0.949631i	0.880085	+	0.474815i	$0.157485\pi$
$173$	28421.5i	0.949631i	−0.880085	+	0.474815i	$0.842515\pi$
$174$	0	0
$175$	−5282.86	−0.172501
$176$	− 5127.78i	− 0.165541i
$177$	0	0
$178$	−13772.2	−0.434675
$179$	− 28162.8i	− 0.878963i	−0.898252	−	0.439481i	$-0.855162\pi$
$179$	− 28162.8i	− 0.878963i	0.898252	−	0.439481i	$-0.144838\pi$
$180$	0	0
$181$	−44153.7	−1.34775	−0.673876	−	0.738844i	$-0.735372\pi$
$181$	−44153.7	−1.34775	−0.673876	+	0.738844i	$0.735372\pi$
$182$	12309.7i	0.371625i
$183$	0	0
$184$	−31553.4	−0.931989
$185$	2341.99i	0.0684293i
$186$	0	0
$187$	−8414.02	−0.240614
$188$	5368.49i	0.151892i
$189$	0	0
$190$	−4342.71	−0.120297
$191$	29592.7i	0.811182i	0.914055	+	0.405591i	$0.132934\pi$
$191$	29592.7i	0.811182i	−0.914055	+	0.405591i	$0.867066\pi$
$192$	0	0
$193$	−6759.19	−0.181460	−0.0907298	−	0.995876i	$-0.528920\pi$
$193$	−6759.19	−0.181460	−0.0907298	+	0.995876i	$0.528920\pi$
$194$	− 1271.46i	− 0.0337831i
$195$	0	0
$196$	−8905.93	−0.231829
$197$	− 21665.5i	− 0.558259i	−0.960253	−	0.279130i	$-0.909954\pi$
$197$	− 21665.5i	− 0.558259i	0.960253	−	0.279130i	$-0.0900459\pi$
$198$	0	0
$199$	−55987.8	−1.41380	−0.706899	−	0.707314i	$-0.749907\pi$
$199$	−55987.8	−1.41380	−0.706899	+	0.707314i	$0.749907\pi$
$200$	− 4690.73i	− 0.117268i
$201$	0	0
$202$	−5392.98	−0.132168
$203$	− 32807.7i	− 0.796131i
$204$	0	0
$205$	−23516.8	−0.559590
$206$	14416.1i	0.339713i
$207$	0	0
$208$	43906.6	1.01485
$209$	8719.64i	0.199621i
$210$	0	0
$211$	−44330.1	−0.995711	−0.497856	−	0.867260i	$-0.665879\pi$
$211$	−44330.1	−0.995711	−0.497856	+	0.867260i	$0.665879\pi$
$212$	− 36603.8i	− 0.814431i
$213$	0	0
$214$	2182.97	0.0476672
$215$	18011.8i	0.389654i
$216$	0	0
$217$	70419.9	1.49546
$218$	− 9296.52i	− 0.195617i
$219$	0	0
$220$	−4475.13	−0.0924613
$221$	− 72045.0i	− 1.47509i
$222$	0	0
$223$	−64428.3	−1.29559	−0.647794	−	0.761816i	$-0.724308\pi$
$223$	−64428.3	−1.29559	−0.647794	+	0.761816i	$0.724308\pi$
$224$	35028.9i	0.698122i
$225$	0	0
$226$	30921.1	0.605394
$227$	− 15789.0i	− 0.306409i	−0.988195	−	0.153205i	$-0.951041\pi$
$227$	− 15789.0i	− 0.306409i	0.988195	−	0.153205i	$-0.0489593\pi$
$228$	0	0
$229$	−36599.8	−0.697924	−0.348962	−	0.937137i	$-0.613466\pi$
$229$	−36599.8	−0.697924	−0.348962	+	0.937137i	$0.613466\pi$
$230$	11572.3i	0.218758i
$231$	0	0
$232$	29130.5	0.541218
$233$	− 5152.64i	− 0.0949114i	−0.998873	−	0.0474557i	$-0.984889\pi$
$233$	− 5152.64i	− 0.0949114i	0.998873	−	0.0474557i	$-0.0151113\pi$
$234$	0	0
$235$	4143.78	0.0750346
$236$	19939.0i	0.357997i
$237$	0	0
$238$	15840.6	0.279651
$239$	26441.4i	0.462902i	0.972847	+	0.231451i	$0.0743472\pi$
$239$	26441.4i	0.462902i	−0.972847	+	0.231451i	$0.925653\pi$
$240$	0	0
$241$	12813.4	0.220613	0.110306	−	0.993898i	$-0.464817\pi$
$241$	12813.4	0.220613	0.110306	+	0.993898i	$0.464817\pi$
$242$	17082.7i	0.291692i
$243$	0	0
$244$	11593.6	0.194732
$245$	6874.24i	0.114523i
$246$	0	0
$247$	−74661.8	−1.22378
$248$	62526.9i	1.01663i
$249$	0	0
$250$	−1720.34	−0.0275254
$251$	− 73021.8i	− 1.15906i	−0.814952	−	0.579529i	$-0.803236\pi$
$251$	− 73021.8i	− 1.15906i	0.814952	−	0.579529i	$-0.196764\pi$
$252$	0	0
$253$	23235.7	0.363007
$254$	14316.2i	0.221902i
$255$	0	0
$256$	11902.2	0.181614
$257$	− 91278.4i	− 1.38198i	−0.722864	−	0.690990i	$-0.757175\pi$
$257$	− 91278.4i	− 1.38198i	0.722864	−	0.690990i	$-0.242825\pi$
$258$	0	0
$259$	−8852.98	−0.131974
$260$	− 38318.2i	− 0.566838i
$261$	0	0
$262$	−32485.2	−0.473242
$263$	43740.6i	0.632372i	0.948697	+	0.316186i	$0.102402\pi$
$263$	43740.6i	0.632372i	−0.948697	+	0.316186i	$0.897598\pi$
$264$	0	0
$265$	−28253.4	−0.402327
$266$	− 16415.9i	− 0.232007i
$267$	0	0
$268$	34551.8	0.481061
$269$	102008.i	1.40970i	0.709355	+	0.704852i	$0.248986\pi$
$269$	102008.i	1.40970i	−0.709355	+	0.704852i	$0.751014\pi$
$270$	0	0
$271$	−112531.	−1.53227	−0.766135	−	0.642680i	$-0.777822\pi$
$271$	−112531.	−1.53227	−0.766135	+	0.642680i	$0.777822\pi$
$272$	− 56500.5i	− 0.763685i
$273$	0	0
$274$	40438.7	0.538637
$275$	3454.22i	0.0456757i
$276$	0	0
$277$	−50158.5	−0.653710	−0.326855	−	0.945074i	$-0.605989\pi$
$277$	−50158.5	−0.653710	−0.326855	+	0.945074i	$0.605989\pi$
$278$	14375.7i	0.186012i
$279$	0	0
$280$	17731.5	0.226167
$281$	− 120246.i	− 1.52286i	−0.648249	−	0.761429i	$-0.724498\pi$
$281$	− 120246.i	− 1.52286i	0.648249	−	0.761429i	$-0.275502\pi$
$282$	0	0
$283$	−68232.6	−0.851960	−0.425980	−	0.904733i	$-0.640071\pi$
$283$	−68232.6	−0.851960	−0.425980	+	0.904733i	$0.640071\pi$
$284$	28664.7i	0.355394i
$285$	0	0
$286$	8048.78	0.0984006
$287$	− 88895.9i	− 1.07924i
$288$	0	0
$289$	−9188.85	−0.110018
$290$	− 10683.7i	− 0.127035i
$291$	0	0
$292$	149218.	1.75007
$293$	113244.i	1.31911i	0.751655	+	0.659556i	$0.229256\pi$
$293$	113244.i	1.31911i	−0.751655	+	0.659556i	$0.770744\pi$
$294$	0	0
$295$	15390.4	0.176850
$296$	− 7860.70i	− 0.0897176i
$297$	0	0
$298$	−47303.3	−0.532671
$299$	198956.i	2.22543i
$300$	0	0
$301$	−68086.4	−0.751497
$302$	− 51366.8i	− 0.563207i
$303$	0	0
$304$	−58552.7	−0.633577
$305$	− 8948.75i	− 0.0961972i
$306$	0	0
$307$	−149777.	−1.58916	−0.794579	−	0.607161i	$-0.792308\pi$
$307$	−149777.	−1.58916	−0.794579	+	0.607161i	$0.792308\pi$
$308$	− 16916.4i	− 0.178323i
$309$	0	0
$310$	22931.9	0.238625
$311$	38353.8i	0.396540i	0.980147	+	0.198270i	$0.0635324\pi$
$311$	38353.8i	0.396540i	−0.980147	+	0.198270i	$0.936468\pi$
$312$	0	0
$313$	23160.2	0.236403	0.118202	−	0.992990i	$-0.462287\pi$
$313$	23160.2	0.236403	0.118202	+	0.992990i	$0.462287\pi$
$314$	1046.28i	0.0106118i
$315$	0	0
$316$	84468.2	0.845900
$317$	− 74093.5i	− 0.737329i	−0.929562	−	0.368665i	$-0.879815\pi$
$317$	− 74093.5i	− 0.737329i	0.929562	−	0.368665i	$-0.120185\pi$
$318$	0	0
$319$	−21451.5	−0.210803
$320$	− 21787.3i	− 0.212767i
$321$	0	0
$322$	−43744.5	−0.421902
$323$	96077.3i	0.920907i
$324$	0	0
$325$	−29576.8	−0.280017
$326$	− 17091.8i	− 0.160825i
$327$	0	0
$328$	78932.0	0.733678
$329$	15663.9i	0.144714i
$330$	0	0
$331$	19838.6	0.181074	0.0905369	−	0.995893i	$-0.471142\pi$
$331$	19838.6	0.181074	0.0905369	+	0.995893i	$0.471142\pi$
$332$	2467.59i	0.0223871i
$333$	0	0
$334$	5623.96	0.0504138
$335$	− 26669.5i	− 0.237643i
$336$	0	0
$337$	−44449.2	−0.391385	−0.195692	−	0.980665i	$-0.562695\pi$
$337$	−44449.2	−0.391385	−0.195692	+	0.980665i	$0.562695\pi$
$338$	33759.8i	0.295506i
$339$	0	0
$340$	−49309.2	−0.426550
$341$	− 46044.4i	− 0.395975i
$342$	0	0
$343$	−127458.	−1.08338
$344$	− 60454.9i	− 0.510875i
$345$	0	0
$346$	−34986.1	−0.292242
$347$	123664.i	1.02704i	0.858079	+	0.513518i	$0.171658\pi$
$347$	123664.i	1.02704i	−0.858079	+	0.513518i	$0.828342\pi$
$348$	0	0
$349$	−187550.	−1.53981	−0.769903	−	0.638161i	$-0.779695\pi$
$349$	−187550.	−1.53981	−0.769903	+	0.638161i	$0.779695\pi$
$350$	− 6503.05i	− 0.0530862i
$351$	0	0
$352$	22903.9	0.184852
$353$	36577.6i	0.293539i	0.989171	+	0.146770i	$0.0468876\pi$
$353$	36577.6i	0.293539i	−0.989171	+	0.146770i	$0.953112\pi$
$354$	0	0
$355$	22125.5	0.175564
$356$	162056.i	1.27869i
$357$	0	0
$358$	34667.7	0.270495
$359$	− 31290.3i	− 0.242785i	−0.992605	−	0.121392i	$-0.961264\pi$
$359$	− 31290.3i	− 0.242785i	0.992605	−	0.121392i	$-0.0387359\pi$
$360$	0	0
$361$	−30754.0	−0.235986
$362$	− 54352.0i	− 0.414762i
$363$	0	0
$364$	144847.	1.09322
$365$	− 115177.i	− 0.864533i
$366$	0	0
$367$	137363.	1.01985	0.509927	−	0.860218i	$-0.329673\pi$
$367$	137363.	1.01985	0.509927	+	0.860218i	$0.329673\pi$
$368$	156029.i	1.15215i
$369$	0	0
$370$	−2882.93	−0.0210587
$371$	− 106801.i	− 0.775939i
$372$	0	0
$373$	−37842.9	−0.271999	−0.136000	−	0.990709i	$-0.543425\pi$
$373$	−37842.9	−0.271999	−0.136000	+	0.990709i	$0.543425\pi$
$374$	− 10357.4i	− 0.0740472i
$375$	0	0
$376$	−13908.3	−0.0983778
$377$	− 183679.i	− 1.29234i
$378$	0	0
$379$	257039.	1.78945	0.894727	−	0.446613i	$-0.147370\pi$
$379$	257039.	1.78945	0.894727	+	0.446613i	$0.147370\pi$
$380$	51100.2i	0.353879i
$381$	0	0
$382$	−36427.8	−0.249636
$383$	88828.0i	0.605553i	0.953062	+	0.302777i	$0.0979136\pi$
$383$	88828.0i	0.605553i	−0.953062	+	0.302777i	$0.902086\pi$
$384$	0	0
$385$	−13057.3	−0.0880913
$386$	− 8320.37i	− 0.0558429i
$387$	0	0
$388$	−14961.1	−0.0993803
$389$	249905.i	1.65149i	0.564043	+	0.825746i	$0.309245\pi$
$389$	249905.i	1.65149i	−0.564043	+	0.825746i	$0.690755\pi$
$390$	0	0
$391$	256023.	1.67466
$392$	− 23072.8i	− 0.150151i
$393$	0	0
$394$	26669.6	0.171800
$395$	− 65198.6i	− 0.417873i
$396$	0	0
$397$	253790.	1.61025	0.805124	−	0.593107i	$-0.202099\pi$
$397$	253790.	1.61025	0.805124	+	0.593107i	$0.202099\pi$
$398$	− 68919.5i	− 0.435087i
$399$	0	0
$400$	−23195.3	−0.144970
$401$	− 91796.6i	− 0.570871i	−0.958398	−	0.285435i	$-0.907862\pi$
$401$	− 91796.6i	− 0.570871i	0.958398	−	0.285435i	$-0.0921382\pi$
$402$	0	0
$403$	394255.	2.42754
$404$	63458.5i	0.388801i
$405$	0	0
$406$	40385.4	0.245004
$407$	5788.57i	0.0349448i
$408$	0	0
$409$	13724.4	0.0820443	0.0410221	−	0.999158i	$-0.486939\pi$
$409$	13724.4	0.0820443	0.0410221	+	0.999158i	$0.486939\pi$
$410$	− 28948.5i	− 0.172210i
$411$	0	0
$412$	169632.	0.999340
$413$	58177.3i	0.341078i
$414$	0	0
$415$	1904.66	0.0110592
$416$	196114.i	1.13324i
$417$	0	0
$418$	−10733.6	−0.0614319
$419$	− 105496.i	− 0.600910i	−0.953796	−	0.300455i	$-0.902861\pi$
$419$	− 105496.i	− 0.600910i	0.953796	−	0.300455i	$-0.0971385\pi$
$420$	0	0
$421$	−2607.30	−0.0147105	−0.00735523	−	0.999973i	$-0.502341\pi$
$421$	−2607.30	−0.0147105	−0.00735523	+	0.999973i	$0.502341\pi$
$422$	− 54569.1i	− 0.306423i
$423$	0	0
$424$	94830.2	0.527491
$425$	38060.4i	0.210715i
$426$	0	0
$427$	33827.2	0.185528
$428$	− 25686.7i	− 0.140224i
$429$	0	0
$430$	−22172.0	−0.119913
$431$	298817.i	1.60861i	0.594218	+	0.804304i	$0.297461\pi$
$431$	298817.i	1.60861i	−0.594218	+	0.804304i	$0.702539\pi$
$432$	0	0
$433$	29.3738	0.000156669 0	7.83347e−5	−	1.00000i	$-0.499975\pi$
$433$	29.3738	0.000156669 0	7.83347e−5		1.00000i	$0.499975\pi$
$434$	86685.0i	0.460219i
$435$	0	0
$436$	−109391.	−0.575451
$437$	− 265322.i	− 1.38935i
$438$	0	0
$439$	−23871.7	−0.123867	−0.0619334	−	0.998080i	$-0.519727\pi$
$439$	−23871.7	−0.123867	−0.0619334	+	0.998080i	$0.519727\pi$
$440$	− 11593.8i	− 0.0598854i
$441$	0	0
$442$	88685.4	0.453950
$443$	6946.22i	0.0353949i	0.999843	+	0.0176975i	$0.00563357\pi$
$443$	6946.22i	0.0353949i	−0.999843	+	0.0176975i	$0.994366\pi$
$444$	0	0
$445$	125087.	0.631671
$446$	− 79309.4i	− 0.398708i
$447$	0	0
$448$	82358.4	0.410348
$449$	218791.i	1.08527i	0.839969	+	0.542635i	$0.182573\pi$
$449$	218791.i	1.08527i	−0.839969	+	0.542635i	$0.817427\pi$
$450$	0	0
$451$	−58125.1	−0.285766
$452$	− 363845.i	− 1.78090i
$453$	0	0
$454$	19435.8	0.0942953
$455$	− 111803.i	− 0.540048i
$456$	0	0
$457$	−133229.	−0.637920	−0.318960	−	0.947768i	$-0.603334\pi$
$457$	−133229.	−0.637920	−0.318960	+	0.947768i	$0.603334\pi$
$458$	− 45053.4i	− 0.214781i
$459$	0	0
$460$	136170.	0.643524
$461$	− 131255.i	− 0.617610i	−0.951125	−	0.308805i	$-0.900071\pi$
$461$	− 131255.i	− 0.617610i	0.951125	−	0.308805i	$-0.0999291\pi$
$462$	0	0
$463$	−208001.	−0.970294	−0.485147	−	0.874433i	$-0.661234\pi$
$463$	−208001.	−0.970294	−0.485147	+	0.874433i	$0.661234\pi$
$464$	− 144048.i	− 0.669069i
$465$	0	0
$466$	6342.76	0.0292083
$467$	− 389941.i	− 1.78799i	−0.448076	−	0.893995i	$-0.647891\pi$
$467$	− 389941.i	− 1.78799i	0.448076	−	0.893995i	$-0.352109\pi$
$468$	0	0
$469$	100814.	0.458325
$470$	5100.89i	0.0230914i
$471$	0	0
$472$	−51656.5	−0.231868
$473$	44518.6i	0.198985i
$474$	0	0
$475$	39442.8	0.174816
$476$	− 186394.i	− 0.822655i
$477$	0	0
$478$	−32548.6	−0.142455
$479$	154028.i	0.671318i	0.941983	+	0.335659i	$0.108959\pi$
$479$	154028.i	0.671318i	−0.941983	+	0.335659i	$0.891041\pi$
$480$	0	0
$481$	−49564.6	−0.214230
$482$	15772.9i	0.0678920i
$483$	0	0
$484$	201010.	0.858077
$485$	11548.1i	0.0490937i
$486$	0	0
$487$	−209417.	−0.882987	−0.441493	−	0.897265i	$-0.645551\pi$
$487$	−209417.	−0.882987	−0.441493	+	0.897265i	$0.645551\pi$
$488$	30035.7i	0.126124i
$489$	0	0
$490$	−8462.00	−0.0352436
$491$	286054.i	1.18655i	0.805001	+	0.593273i	$0.202165\pi$
$491$	286054.i	1.18655i	−0.805001	+	0.593273i	$0.797835\pi$
$492$	0	0
$493$	−236364.	−0.972493
$494$	− 91906.7i	− 0.376611i
$495$	0	0
$496$	309190.	1.25679
$497$	83636.6i	0.338597i
$498$	0	0
$499$	28512.5	0.114507	0.0572537	−	0.998360i	$-0.481766\pi$
$499$	28512.5	0.114507	0.0572537	+	0.998360i	$0.481766\pi$
$500$	20243.0i	0.0809720i
$501$	0	0
$502$	89887.9	0.356692
$503$	− 48257.4i	− 0.190734i	−0.995442	−	0.0953670i	$-0.969598\pi$
$503$	− 48257.4i	− 0.190734i	0.995442	−	0.0953670i	$-0.0304025\pi$
$504$	0	0
$505$	48981.9	0.192067
$506$	28602.6i	0.111713i
$507$	0	0
$508$	168457.	0.652774
$509$	407314.i	1.57215i	0.618131	+	0.786075i	$0.287890\pi$
$509$	407314.i	1.57215i	−0.618131	+	0.786075i	$0.712110\pi$
$510$	0	0
$511$	435383.	1.66736
$512$	265214.i	1.01171i
$513$	0	0
$514$	112361.	0.425295
$515$	− 130934.i	− 0.493672i
$516$	0	0
$517$	10242.0	0.0383179
$518$	− 10897.8i	− 0.0406142i
$519$	0	0
$520$	99271.9	0.367130
$521$	294195.i	1.08382i	0.840435	+	0.541912i	$0.182300\pi$
$521$	294195.i	1.08382i	−0.840435	+	0.541912i	$0.817700\pi$
$522$	0	0
$523$	236625.	0.865083	0.432541	−	0.901614i	$-0.357617\pi$
$523$	236625.	0.865083	0.432541	+	0.901614i	$0.357617\pi$
$524$	382250.i	1.39215i
$525$	0	0
$526$	−53843.4	−0.194608
$527$	− 507340.i	− 1.82675i
$528$	0	0
$529$	−427180.	−1.52651
$530$	− 34779.2i	− 0.123814i
$531$	0	0
$532$	−193164.	−0.682501
$533$	− 497696.i	− 1.75190i
$534$	0	0
$535$	−19826.9	−0.0692702
$536$	89514.0i	0.311574i
$537$	0	0
$538$	−125568.	−0.433827
$539$	16990.7i	0.0584834i
$540$	0	0
$541$	−230383.	−0.787147	−0.393573	−	0.919293i	$-0.628761\pi$
$541$	−230383.	−0.787147	−0.393573	+	0.919293i	$0.628761\pi$
$542$	− 138523.i	− 0.471545i
$543$	0	0
$544$	252366.	0.852773
$545$	84435.8i	0.284272i
$546$	0	0
$547$	−248300.	−0.829855	−0.414928	−	0.909854i	$-0.636193\pi$
$547$	−248300.	−0.829855	−0.414928	+	0.909854i	$0.636193\pi$
$548$	− 475837.i	− 1.58452i
$549$	0	0
$550$	−4252.06	−0.0140564
$551$	244949.i	0.806811i
$552$	0	0
$553$	246458.	0.805920
$554$	− 61743.8i	− 0.201175i
$555$	0	0
$556$	169157.	0.547194
$557$	93331.8i	0.300829i	0.988623	+	0.150414i	$0.0480608\pi$
$557$	93331.8i	0.300829i	−0.988623	+	0.150414i	$0.951939\pi$
$558$	0	0
$559$	−381190.	−1.21988
$560$	− 87680.5i	− 0.279593i
$561$	0	0
$562$	148020.	0.468649
$563$	338574.i	1.06816i	0.845433	+	0.534081i	$0.179342\pi$
$563$	338574.i	1.06816i	−0.845433	+	0.534081i	$0.820658\pi$
$564$	0	0
$565$	−280842.	−0.879761
$566$	− 83992.5i	− 0.262185i
$567$	0	0
$568$	−74262.2	−0.230182
$569$	− 37162.2i	− 0.114783i	−0.998352	−	0.0573915i	$-0.981722\pi$
$569$	− 37162.2i	− 0.114783i	0.998352	−	0.0573915i	$-0.0182783\pi$
$570$	0	0
$571$	297866.	0.913586	0.456793	−	0.889573i	$-0.348998\pi$
$571$	297866.	0.913586	0.456793	+	0.889573i	$0.348998\pi$
$572$	− 94709.0i	− 0.289467i
$573$	0	0
$574$	109428.	0.332129
$575$	− 105106.i	− 0.317900i
$576$	0	0
$577$	−54793.5	−0.164580	−0.0822900	−	0.996608i	$-0.526223\pi$
$577$	−54793.5	−0.164580	−0.0822900	+	0.996608i	$0.526223\pi$
$578$	− 11311.2i	− 0.0338574i
$579$	0	0
$580$	−125714.	−0.373703
$581$	7199.83i	0.0213290i
$582$	0	0
$583$	−69832.4	−0.205456
$584$	386583.i	1.13349i
$585$	0	0
$586$	−139401.	−0.405948
$587$	− 373143.i	− 1.08293i	−0.840724	−	0.541464i	$-0.817870\pi$
$587$	− 373143.i	− 1.08293i	0.840724	−	0.541464i	$-0.182130\pi$
$588$	0	0
$589$	−525768.	−1.51553
$590$	18945.1i	0.0544244i
$591$	0	0
$592$	−38870.5	−0.110911
$593$	150209.i	0.427155i	0.976926	+	0.213578i	$0.0685117\pi$
$593$	150209.i	0.427155i	−0.976926	+	0.213578i	$0.931488\pi$
$594$	0	0
$595$	−143872.	−0.406390
$596$	556612.i	1.56697i
$597$	0	0
$598$	−244909.	−0.684862
$599$	486427.i	1.35570i	0.735200	+	0.677851i	$0.237088\pi$
$599$	486427.i	1.35570i	−0.735200	+	0.677851i	$0.762912\pi$
$600$	0	0
$601$	−253411.	−0.701578	−0.350789	−	0.936454i	$-0.614087\pi$
$601$	−253411.	−0.701578	−0.350789	+	0.936454i	$0.614087\pi$
$602$	− 83812.4i	− 0.231268i
$603$	0	0
$604$	−604426.	−1.65680
$605$	− 155154.i	− 0.423888i
$606$	0	0
$607$	53018.4	0.143896	0.0719482	−	0.997408i	$-0.477078\pi$
$607$	53018.4	0.143896	0.0719482	+	0.997408i	$0.477078\pi$
$608$	− 261533.i	− 0.707488i
$609$	0	0
$610$	11015.7	0.0296040
$611$	87696.7i	0.234910i
$612$	0	0
$613$	22974.2	0.0611391	0.0305695	−	0.999533i	$-0.490268\pi$
$613$	22974.2	0.0611391	0.0305695	+	0.999533i	$0.490268\pi$
$614$	− 184371.i	− 0.489052i
$615$	0	0
$616$	43825.8	0.115496
$617$	170126.i	0.446891i	0.974716	+	0.223445i	$0.0717304\pi$
$617$	170126.i	0.446891i	−0.974716	+	0.223445i	$0.928270\pi$
$618$	0	0
$619$	238707.	0.622995	0.311498	−	0.950247i	$-0.399169\pi$
$619$	238707.	0.622995	0.311498	+	0.950247i	$0.399169\pi$
$620$	− 269837.i	− 0.701969i
$621$	0	0
$622$	−47212.5	−0.122033
$623$	472841.i	1.21826i
$624$	0	0
$625$	15625.0	0.0400000
$626$	28509.6i	0.0727515i
$627$	0	0
$628$	12311.5	0.0312170
$629$	63781.3i	0.161210i
$630$	0	0
$631$	−196615.	−0.493809	−0.246904	−	0.969040i	$-0.579413\pi$
$631$	−196615.	−0.493809	−0.246904	+	0.969040i	$0.579413\pi$
$632$	218834.i	0.547873i
$633$	0	0
$634$	91207.1	0.226908
$635$	− 130028.i	− 0.322469i
$636$	0	0
$637$	−145482.	−0.358535
$638$	− 26406.2i	− 0.0648732i
$639$	0	0
$640$	175086.	0.427456
$641$	− 586812.i	− 1.42818i	−0.700053	−	0.714090i	$-0.746840\pi$
$641$	− 586812.i	− 1.42818i	0.700053	−	0.714090i	$-0.253160\pi$
$642$	0	0
$643$	343741.	0.831398	0.415699	−	0.909502i	$-0.363537\pi$
$643$	343741.	0.831398	0.415699	+	0.909502i	$0.363537\pi$
$644$	514736.i	1.24112i
$645$	0	0
$646$	−118268.	−0.283403
$647$	− 497153.i	− 1.18763i	−0.804601	−	0.593816i	$-0.797621\pi$
$647$	− 497153.i	− 1.18763i	0.804601	−	0.593816i	$-0.202379\pi$
$648$	0	0
$649$	38039.5	0.0903120
$650$	− 36408.2i	− 0.0861733i
$651$	0	0
$652$	−201117.	−0.473101
$653$	− 74620.0i	− 0.174996i	−0.996165	−	0.0874981i	$-0.972113\pi$
$653$	− 74620.0i	− 0.174996i	0.996165	−	0.0874981i	$-0.0278872\pi$
$654$	0	0
$655$	295048.	0.687717
$656$	− 390312.i	− 0.906994i
$657$	0	0
$658$	−19281.9	−0.0445346
$659$	− 637218.i	− 1.46729i	−0.679530	−	0.733647i	$-0.737816\pi$
$659$	− 637218.i	− 1.46729i	0.679530	−	0.733647i	$-0.262184\pi$
$660$	0	0
$661$	−411755.	−0.942401	−0.471201	−	0.882026i	$-0.656179\pi$
$661$	−411755.	−0.942401	−0.471201	+	0.882026i	$0.656179\pi$
$662$	24420.8i	0.0557242i
$663$	0	0
$664$	−6392.84	−0.0144997
$665$	149098.i	0.337154i
$666$	0	0
$667$	652730.	1.46717
$668$	− 66176.4i	− 0.148303i
$669$	0	0
$670$	32829.5	0.0731331
$671$	− 22118.1i	− 0.0491250i
$672$	0	0
$673$	−446673.	−0.986187	−0.493094	−	0.869976i	$-0.664134\pi$
$673$	−446673.	−0.986187	−0.493094	+	0.869976i	$0.664134\pi$
$674$	− 54715.7i	− 0.120446i
$675$	0	0
$676$	397247.	0.869296
$677$	299018.i	0.652409i	0.945299	+	0.326204i	$0.105770\pi$
$677$	299018.i	0.652409i	−0.945299	+	0.326204i	$0.894230\pi$
$678$	0	0
$679$	−43652.9	−0.0946833
$680$	− 127746.i	− 0.276268i
$681$	0	0
$682$	56679.4	0.121859
$683$	540348.i	1.15833i	0.815210	+	0.579165i	$0.196621\pi$
$683$	540348.i	1.15833i	−0.815210	+	0.579165i	$0.803379\pi$
$684$	0	0
$685$	−367286.	−0.782749
$686$	− 156898.i	− 0.333403i
$687$	0	0
$688$	−298944.	−0.631558
$689$	− 597939.i	− 1.25956i
$690$	0	0
$691$	−368100.	−0.770921	−0.385460	−	0.922724i	$-0.625957\pi$
$691$	−368100.	−0.770921	−0.385460	+	0.922724i	$0.625957\pi$
$692$	411677.i	0.859695i
$693$	0	0
$694$	−152227.	−0.316063
$695$	− 130568.i	− 0.270313i
$696$	0	0
$697$	−640451.	−1.31832
$698$	− 230869.i	− 0.473865i
$699$	0	0
$700$	−76520.6	−0.156165
$701$	578017.i	1.17626i	0.808765	+	0.588132i	$0.200136\pi$
$701$	578017.i	1.17626i	−0.808765	+	0.588132i	$0.799864\pi$
$702$	0	0
$703$	66098.0	0.133745
$704$	− 53850.5i	− 0.108654i
$705$	0	0
$706$	−45026.1	−0.0903347
$707$	185157.i	0.370425i
$708$	0	0
$709$	561340.	1.11669	0.558346	−	0.829608i	$-0.311436\pi$
$709$	561340.	1.11669	0.558346	+	0.829608i	$0.311436\pi$
$710$	27235.9i	0.0540287i
$711$	0	0
$712$	−419843.	−0.828184
$713$	1.40105e6i	2.75596i
$714$	0	0
$715$	−73103.2	−0.142996
$716$	− 407931.i	− 0.795720i
$717$	0	0
$718$	38517.5	0.0747153
$719$	− 578659.i	− 1.11935i	−0.828713	−	0.559674i	$-0.810926\pi$
$719$	− 578659.i	− 1.11935i	0.828713	−	0.559674i	$-0.189074\pi$
$720$	0	0
$721$	494945.	0.952109
$722$	− 37857.3i	− 0.0726232i
$723$	0	0
$724$	−639553.	−1.22011
$725$	97034.8i	0.184608i
$726$	0	0
$727$	−357109.	−0.675666	−0.337833	−	0.941206i	$-0.609694\pi$
$727$	−357109.	−0.675666	−0.337833	+	0.941206i	$0.609694\pi$
$728$	375258.i	0.708056i
$729$	0	0
$730$	141780.	0.266054
$731$	490528.i	0.917972i
$732$	0	0
$733$	−160129.	−0.298032	−0.149016	−	0.988835i	$-0.547611\pi$
$733$	−160129.	−0.298032	−0.149016	+	0.988835i	$0.547611\pi$
$734$	169090.i	0.313853i
$735$	0	0
$736$	−696922.	−1.28656
$737$	− 65917.5i	− 0.121357i
$738$	0	0
$739$	251930.	0.461307	0.230654	−	0.973036i	$-0.425914\pi$
$739$	251930.	0.461307	0.230654	+	0.973036i	$0.425914\pi$
$740$	33923.1i	0.0619487i
$741$	0	0
$742$	131469.	0.238790
$743$	− 528197.i	− 0.956794i	−0.878144	−	0.478397i	$-0.841218\pi$
$743$	− 528197.i	− 0.956794i	0.878144	−	0.478397i	$-0.158782\pi$
$744$	0	0
$745$	429634.	0.774080
$746$	− 46583.6i	− 0.0837058i
$747$	0	0
$748$	−121875.	−0.217826
$749$	− 74947.6i	− 0.133596i
$750$	0	0
$751$	960470.	1.70296	0.851479	−	0.524389i	$-0.175706\pi$
$751$	960470.	1.70296	0.851479	+	0.524389i	$0.175706\pi$
$752$	68775.1i	0.121617i
$753$	0	0
$754$	226103.	0.397708
$755$	466540.i	0.818455i
$756$	0	0
$757$	−487728.	−0.851111	−0.425555	−	0.904932i	$-0.639921\pi$
$757$	−487728.	−0.851111	−0.425555	+	0.904932i	$0.639921\pi$
$758$	316408.i	0.550692i
$759$	0	0
$760$	−132386.	−0.229201
$761$	− 838622.i	− 1.44809i	−0.689750	−	0.724047i	$-0.742280\pi$
$761$	− 838622.i	− 1.44809i	0.689750	−	0.724047i	$-0.257720\pi$
$762$	0	0
$763$	−319176.	−0.548254
$764$	428642.i	0.734358i
$765$	0	0
$766$	−109345.	−0.186355
$767$	325713.i	0.553662i
$768$	0	0
$769$	−224645.	−0.379878	−0.189939	−	0.981796i	$-0.560829\pi$
$769$	−224645.	−0.379878	−0.189939	+	0.981796i	$0.560829\pi$
$770$	− 16073.2i	− 0.0271095i
$771$	0	0
$772$	−97904.8	−0.164274
$773$	− 734461.i	− 1.22916i	−0.788853	−	0.614582i	$-0.789325\pi$
$773$	− 734461.i	− 1.22916i	0.788853	−	0.614582i	$-0.210675\pi$
$774$	0	0
$775$	−208279.	−0.346771
$776$	− 38760.1i	− 0.0643667i
$777$	0	0
$778$	−307627.	−0.508235
$779$	663714.i	1.09372i
$780$	0	0
$781$	54686.3	0.0896553
$782$	315157.i	0.515364i
$783$	0	0
$784$	−114093.	−0.185621
$785$	− 9502.88i	− 0.0154211i
$786$	0	0
$787$	−284789.	−0.459804	−0.229902	−	0.973214i	$-0.573841\pi$
$787$	−284789.	−0.459804	−0.229902	+	0.973214i	$0.573841\pi$
$788$	− 313818.i	− 0.505389i
$789$	0	0
$790$	80257.7	0.128598
$791$	− 1.06161e6i	− 1.69673i
$792$	0	0
$793$	189386.	0.301163
$794$	312408.i	0.495543i
$795$	0	0
$796$	−810967.	−1.27990
$797$	− 74214.0i	− 0.116834i	−0.998292	−	0.0584170i	$-0.981395\pi$
$797$	− 74214.0i	− 0.116834i	0.998292	−	0.0584170i	$-0.0186053\pi$
$798$	0	0
$799$	112851.	0.176771
$800$	− 103604.i	− 0.161882i
$801$	0	0
$802$	112999.	0.175682
$803$	− 284677.i	− 0.441491i
$804$	0	0
$805$	397310.	0.613110
$806$	485317.i	0.747060i
$807$	0	0
$808$	−164403.	−0.251819
$809$	1.01875e6i	1.55657i	0.627911	+	0.778285i	$0.283910\pi$
$809$	1.01875e6i	1.55657i	−0.627911	+	0.778285i	$0.716090\pi$
$810$	0	0
$811$	63192.0	0.0960772	0.0480386	−	0.998845i	$-0.484703\pi$
$811$	63192.0	0.0960772	0.0480386	+	0.998845i	$0.484703\pi$
$812$	− 475211.i	− 0.720732i
$813$	0	0
$814$	−7125.57	−0.0107540
$815$	155237.i	0.233711i
$816$	0	0
$817$	508345.	0.761579
$818$	16894.4i	0.0252486i
$819$	0	0
$820$	−340634.	−0.506594
$821$	− 355904.i	− 0.528015i	−0.964521	−	0.264008i	$-0.914956\pi$
$821$	− 355904.i	− 0.528015i	0.964521	−	0.264008i	$-0.0850444\pi$
$822$	0	0
$823$	96904.8	0.143069	0.0715345	−	0.997438i	$-0.477210\pi$
$823$	96904.8	0.143069	0.0715345	+	0.997438i	$0.477210\pi$
$824$	439469.i	0.647253i
$825$	0	0
$826$	−71614.6	−0.104964
$827$	− 1.06487e6i	− 1.55699i	−0.627650	−	0.778496i	$-0.715983\pi$
$827$	− 1.06487e6i	− 1.55699i	0.627650	−	0.778496i	$-0.284017\pi$
$828$	0	0
$829$	−733212.	−1.06689	−0.533446	−	0.845834i	$-0.679103\pi$
$829$	−733212.	−1.06689	−0.533446	+	0.845834i	$0.679103\pi$
$830$	2344.59i	0.00340338i
$831$	0	0
$832$	461095.	0.666106
$833$	187211.i	0.269800i
$834$	0	0
$835$	−51079.7	−0.0732615
$836$	126301.i	0.180716i
$837$	0	0
$838$	129863.	0.184926
$839$	− 503310.i	− 0.715009i	−0.933911	−	0.357505i	$-0.883628\pi$
$839$	− 503310.i	− 0.715009i	0.933911	−	0.357505i	$-0.116372\pi$
$840$	0	0
$841$	104673.	0.147993
$842$	− 3209.51i	− 0.00452705i
$843$	0	0
$844$	−642108.	−0.901411
$845$	− 306624.i	− 0.429430i
$846$	0	0
$847$	586498.	0.817522
$848$	− 468927.i	− 0.652099i
$849$	0	0
$850$	−46851.3	−0.0648460
$851$	− 176135.i	− 0.243213i
$852$	0	0
$853$	−494872.	−0.680134	−0.340067	−	0.940401i	$-0.610450\pi$
$853$	−494872.	−0.680134	−0.340067	+	0.940401i	$0.610450\pi$
$854$	41640.4i	0.0570951i
$855$	0	0
$856$	66547.1	0.0908200
$857$	− 368085.i	− 0.501172i	−0.968094	−	0.250586i	$-0.919377\pi$
$857$	− 368085.i	− 0.501172i	0.968094	−	0.250586i	$-0.0806233\pi$
$858$	0	0
$859$	−507112.	−0.687255	−0.343627	−	0.939106i	$-0.611656\pi$
$859$	−507112.	−0.687255	−0.343627	+	0.939106i	$0.611656\pi$
$860$	260895.i	0.352752i
$861$	0	0
$862$	−367835.	−0.495038
$863$	968158.i	1.29994i	0.759958	+	0.649972i	$0.225219\pi$
$863$	968158.i	1.29994i	−0.759958	+	0.649972i	$0.774781\pi$
$864$	0	0
$865$	317762.	0.424688
$866$	36.1583i	0 4.82139e-5i
$867$	0	0
$868$	1.02001e6	1.35383
$869$	− 161148.i	− 0.213395i
$870$	0	0
$871$	564418.	0.743986
$872$	− 283402.i	− 0.372708i
$873$	0	0
$874$	326605.	0.427562
$875$	59064.1i	0.0771450i
$876$	0	0
$877$	209966.	0.272992	0.136496	−	0.990641i	$-0.456416\pi$
$877$	209966.	0.272992	0.136496	+	0.990641i	$0.456416\pi$
$878$	− 29385.5i	− 0.0381192i
$879$	0	0
$880$	−57330.4	−0.0740320
$881$	− 253953.i	− 0.327192i	−0.986527	−	0.163596i	$-0.947691\pi$
$881$	− 253953.i	− 0.327192i	0.986527	−	0.163596i	$-0.0523093\pi$
$882$	0	0
$883$	−396178.	−0.508123	−0.254062	−	0.967188i	$-0.581767\pi$
$883$	−396178.	−0.508123	−0.254062	+	0.967188i	$0.581767\pi$
$884$	− 1.04355e6i	− 1.33539i
$885$	0	0
$886$	−8550.61	−0.0108926
$887$	− 143657.i	− 0.182591i	−0.995824	−	0.0912955i	$-0.970899\pi$
$887$	− 143657.i	− 0.182591i	0.995824	−	0.0912955i	$-0.0291008\pi$
$888$	0	0
$889$	491518.	0.621922
$890$	153978.i	0.194393i
$891$	0	0
$892$	−933225.	−1.17289
$893$	− 116950.i	− 0.146655i
$894$	0	0
$895$	−314870.	−0.393084
$896$	661844.i	0.824403i
$897$	0	0
$898$	−269326.	−0.333984
$899$	− 1.29346e6i	− 1.60042i
$900$	0	0
$901$	−769448.	−0.947828
$902$	− 71550.4i	− 0.0879425i
$903$	0	0
$904$	942621.	1.15345
$905$	493653.i	0.602733i
$906$	0	0
$907$	158048.	0.192121	0.0960603	−	0.995376i	$-0.469376\pi$
$907$	158048.	0.192121	0.0960603	+	0.995376i	$0.469376\pi$
$908$	− 228698.i	− 0.277390i
$909$	0	0
$910$	137627.	0.166196
$911$	− 488828.i	− 0.589005i	−0.955651	−	0.294503i	$-0.904846\pi$
$911$	− 488828.i	− 0.589005i	0.955651	−	0.294503i	$-0.0951540\pi$
$912$	0	0
$913$	4707.65	0.00564758
$914$	− 164001.i	− 0.196316i
$915$	0	0
$916$	−530138.	−0.631826
$917$	1.11531e6i	1.32635i
$918$	0	0
$919$	193983.	0.229685	0.114843	−	0.993384i	$-0.463364\pi$
$919$	193983.	0.229685	0.114843	+	0.993384i	$0.463364\pi$
$920$	352778.i	0.416798i
$921$	0	0
$922$	161572.	0.190065
$923$	468251.i	0.549636i
$924$	0	0
$925$	26184.3	0.0306025
$926$	− 256043.i	− 0.298601i
$927$	0	0
$928$	643407.	0.747119
$929$	338776.i	0.392537i	0.980550	+	0.196269i	$0.0628825\pi$
$929$	338776.i	0.392537i	−0.980550	+	0.196269i	$0.937118\pi$
$930$	0	0
$931$	194011.	0.223835
$932$	− 74634.5i	− 0.0859227i
$933$	0	0
$934$	480007.	0.550242
$935$	94071.6i	0.107606i
$936$	0	0
$937$	576017.	0.656079	0.328039	−	0.944664i	$-0.393612\pi$
$937$	576017.	0.656079	0.328039	+	0.944664i	$0.393612\pi$
$938$	124099.i	0.141046i
$939$	0	0
$940$	60021.5	0.0679284
$941$	− 44925.7i	− 0.0507359i	−0.999678	−	0.0253679i	$-0.991924\pi$
$941$	− 44925.7i	− 0.0507359i	0.999678	−	0.0253679i	$-0.00807573\pi$
$942$	0	0
$943$	1.76864e6	1.98891
$944$	255437.i	0.286642i
$945$	0	0
$946$	−54801.2	−0.0612362
$947$	721667.i	0.804705i	0.915485	+	0.402353i	$0.131807\pi$
$947$	721667.i	0.804705i	−0.915485	+	0.402353i	$0.868193\pi$
$948$	0	0
$949$	2.43755e6	2.70658
$950$	48553.0i	0.0537983i
$951$	0	0
$952$	482895.	0.532817
$953$	− 1.51930e6i	− 1.67286i	−0.548077	−	0.836428i	$-0.684640\pi$
$953$	− 1.51930e6i	− 1.67286i	0.548077	−	0.836428i	$-0.315360\pi$
$954$	0	0
$955$	330857.	0.362772
$956$	382996.i	0.419062i
$957$	0	0
$958$	−189604.	−0.206594
$959$	− 1.38838e6i	− 1.50963i
$960$	0	0
$961$	1.85282e6	2.00626
$962$	− 61012.6i	− 0.0659280i
$963$	0	0
$964$	185598.	0.199719
$965$	75570.0i	0.0811512i
$966$	0	0
$967$	1.59652e6	1.70735	0.853673	−	0.520809i	$-0.174370\pi$
$967$	1.59652e6	1.70735	0.853673	+	0.520809i	$0.174370\pi$
$968$	520760.i	0.555760i
$969$	0	0
$970$	−14215.4	−0.0151083
$971$	1.11408e6i	1.18162i	0.806809	+	0.590812i	$0.201193\pi$
$971$	1.11408e6i	1.18162i	−0.806809	+	0.590812i	$0.798807\pi$
$972$	0	0
$973$	493560.	0.521332
$974$	− 257787.i	− 0.271733i
$975$	0	0
$976$	148524.	0.155918
$977$	120477.i	0.126217i	0.998007	+	0.0631083i	$0.0201013\pi$
$977$	120477.i	0.126217i	−0.998007	+	0.0631083i	$0.979899\pi$
$978$	0	0
$979$	309170.	0.322575
$980$	99571.3i	0.103677i
$981$	0	0
$982$	−352124.	−0.365152
$983$	64809.9i	0.0670709i	0.999438	+	0.0335354i	$0.0106767\pi$
$983$	64809.9i	0.0670709i	−0.999438	+	0.0335354i	$0.989323\pi$
$984$	0	0
$985$	−242227.	−0.249661
$986$	− 290957.i	− 0.299278i
$987$	0	0
$988$	−1.08145e6	−1.10788
$989$	− 1.35462e6i	− 1.38492i
$990$	0	0
$991$	551476.	0.561539	0.280769	−	0.959775i	$-0.409410\pi$
$991$	551476.	0.561539	0.280769	+	0.959775i	$0.409410\pi$
$992$	1.38103e6i	1.40340i
$993$	0	0
$994$	−102954.	−0.104201
$995$	625963.i	0.632270i
$996$	0	0
$997$	690857.	0.695021	0.347511	−	0.937676i	$-0.387027\pi$
$997$	690857.	0.695021	0.347511	+	0.937676i	$0.387027\pi$
$998$	35098.1i	0.0352389i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.c.b.161.19		32
3.2	odd	2	inner	405.5.c.b.161.14		32
9.2	odd	6		45.5.i.a.41.8	yes	32
9.4	even	3		45.5.i.a.11.8	✓	32
9.5	odd	6		135.5.i.a.116.9		32
9.7	even	3		135.5.i.a.71.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.i.a.11.8	✓	32	9.4	even	3
45.5.i.a.41.8	yes	32	9.2	odd	6
135.5.i.a.71.9		32	9.7	even	3
135.5.i.a.116.9		32	9.5	odd	6
405.5.c.b.161.14		32	3.2	odd	2	inner
405.5.c.b.161.19		32	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	405.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.23097i q^{2} +14.4847 q^{4} -11.1803i q^{5} +42.2629 q^{7} +37.5258i q^{8} +13.7627 q^{10} -27.6338i q^{11} +236.614 q^{13} +52.0244i q^{14} +185.562 q^{16} -304.483i q^{17} -315.542 q^{19} -161.944i q^{20} +34.0165 q^{22} +840.845i q^{23} -125.000 q^{25} +291.266i q^{26} +612.165 q^{28} -776.278i q^{29} +1666.24 q^{31} +828.835i q^{32} +374.810 q^{34} -472.513i q^{35} -209.474 q^{37} -388.424i q^{38} +419.552 q^{40} -2103.41i q^{41} -1611.02 q^{43} -400.267i q^{44} -1035.06 q^{46} +370.631i q^{47} -614.850 q^{49} -153.872i q^{50} +3427.29 q^{52} -2527.06i q^{53} -308.955 q^{55} +1585.95i q^{56} +955.577 q^{58} +1376.56i q^{59} +800.400 q^{61} +2051.09i q^{62} +1948.72 q^{64} -2645.43i q^{65} +2385.40 q^{67} -4410.35i q^{68} +581.651 q^{70} +1978.96i q^{71} +10301.8 q^{73} -257.857i q^{74} -4570.54 q^{76} -1167.88i q^{77} +5831.54 q^{79} -2074.65i q^{80} +2589.23 q^{82} +170.358i q^{83} -3404.22 q^{85} -1983.12i q^{86} +1036.98 q^{88} +11188.1i q^{89} +9999.99 q^{91} +12179.4i q^{92} -456.237 q^{94} +3527.87i q^{95} -1032.89 q^{97} -756.864i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 256 q^{4} + 52 q^{7} - 20 q^{13} + 2048 q^{16} + 508 q^{19} - 1344 q^{22} - 4000 q^{25} - 1664 q^{28} + 2944 q^{31} + 1188 q^{34} + 2068 q^{37} - 3300 q^{40} - 1136 q^{43} + 5724 q^{46} + 3348 q^{49}+ \cdots - 46532 q^{97}+O(q^{100}) \) 32 * q - 256 * q^4 + 52 * q^7 - 20 * q^13 + 2048 * q^16 + 508 * q^19 - 1344 * q^22 - 4000 * q^25 - 1664 * q^28 + 2944 * q^31 + 1188 * q^34 + 2068 * q^37 - 3300 * q^40 - 1136 * q^43 + 5724 * q^46 + 3348 * q^49 - 3068 * q^52 + 16380 * q^58 + 8956 * q^61 - 37216 * q^64 - 25676 * q^67 - 4800 * q^70 + 29236 * q^73 - 12380 * q^76 + 7024 * q^79 + 34692 * q^82 + 8700 * q^85 + 32256 * q^88 - 48904 * q^91 - 10236 * q^94 - 46532 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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