LMFDB - Embedded newform 405.5.c.b.161.14

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

$q$-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	$-0.950826\pi$
$2$	− 1.23097i	− 0.307743i	0.988091	−	0.153872i	$-0.0491742\pi$
$3$	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	$0.0364270\pi$
$11$	27.6338i	0.228378i	−0.993459	+	0.114189i	$0.963573\pi$
$12$	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	$0.176603\pi$
$17$	304.483i	1.05357i	−0.849997	+	0.526787i	$0.823397\pi$
$18$	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	$-0.707602\pi$
$23$	− 840.845i	− 1.58950i	0.606937	−	0.794750i	$-0.292398\pi$
$24$	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	$0.152696\pi$
$29$	776.278i	0.923042i	−0.887129	+	0.461521i	$0.847304\pi$
$30$	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	$0.215163\pi$
$41$	2103.41i	1.25128i	−0.780111	+	0.625641i	$0.784837\pi$
$42$	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	$-0.973265\pi$
$47$	− 370.631i	− 0.167782i	0.996475	−	0.0838912i	$-0.0267348\pi$
$48$	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	$0.148510\pi$
$53$	2527.06i	0.899631i	−0.893121	+	0.449816i	$0.851490\pi$
$54$	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	$-0.936645\pi$
$59$	− 1376.56i	− 0.395449i	0.980258	−	0.197724i	$-0.0633551\pi$
$60$	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	$-0.937112\pi$
$71$	− 1978.96i	− 0.392573i	0.980547	−	0.196287i	$-0.0628883\pi$
$72$	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	$-0.996064\pi$
$83$	− 170.358i	− 0.0247290i	0.999924	−	0.0123645i	$-0.00393585\pi$
$84$	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	$-0.750394\pi$
$89$	− 11188.1i	− 1.41246i	0.707982	−	0.706230i	$-0.249606\pi$
$90$	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	$-0.931110\pi$
$101$	− 4381.07i	− 0.429475i	0.976672	−	0.214737i	$-0.0688895\pi$
$102$	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	$0.0246767\pi$
$107$	1773.37i	0.154893i	−0.996997	+	0.0774464i	$0.975323\pi$
$108$	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	$0.442277\pi$
$113$	25119.2i	1.96721i	−0.180349	+	0.983603i	$0.557723\pi$
$114$	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	$-0.720809\pi$
$131$	− 26389.9i	− 1.53778i	0.639380	−	0.768891i	$-0.279191\pi$
$132$	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	$0.339231\pi$
$137$	32851.0i	1.75028i	−0.483869	+	0.875141i	$0.660769\pi$
$138$	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	$-0.667034\pi$
$149$	− 38427.6i	− 1.73089i	0.501000	−	0.865447i	$-0.332966\pi$
$150$	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	$0.0261016\pi$
$167$	4568.71i	0.163818i	−0.996640	+	0.0819088i	$0.973898\pi$
$168$	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	$-0.842515\pi$
$173$	− 28421.5i	− 0.949631i	0.880085	−	0.474815i	$-0.157485\pi$
$174$	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	$0.144838\pi$
$179$	28162.8i	0.878963i	−0.898252	+	0.439481i	$0.855162\pi$
$180$	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	$-0.867066\pi$
$191$	− 29592.7i	− 0.811182i	0.914055	−	0.405591i	$-0.132934\pi$
$192$	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	$0.0900459\pi$
$197$	21665.5i	0.558259i	−0.960253	+	0.279130i	$0.909954\pi$
$198$	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	$0.0489593\pi$
$227$	15789.0i	0.306409i	−0.988195	+	0.153205i	$0.951041\pi$
$228$	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	$0.0151113\pi$
$233$	5152.64i	0.0949114i	−0.998873	+	0.0474557i	$0.984889\pi$
$234$	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	$-0.925653\pi$
$239$	− 26441.4i	− 0.462902i	0.972847	−	0.231451i	$-0.0743472\pi$
$240$	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	$0.196764\pi$
$251$	73021.8i	1.15906i	−0.814952	+	0.579529i	$0.803236\pi$
$252$	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	$0.242825\pi$
$257$	91278.4i	1.38198i	−0.722864	+	0.690990i	$0.757175\pi$
$258$	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	$-0.897598\pi$
$263$	− 43740.6i	− 0.632372i	0.948697	−	0.316186i	$-0.102402\pi$
$264$	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	$-0.751014\pi$
$269$	− 102008.i	− 1.40970i	0.709355	−	0.704852i	$-0.248986\pi$
$270$	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	$0.275502\pi$
$281$	120246.i	1.52286i	−0.648249	+	0.761429i	$0.724498\pi$
$282$	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	$-0.770744\pi$
$293$	− 113244.i	− 1.31911i	0.751655	−	0.659556i	$-0.229256\pi$
$294$	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	$-0.936468\pi$
$311$	− 38353.8i	− 0.396540i	0.980147	−	0.198270i	$-0.0635324\pi$
$312$	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	$0.120185\pi$
$317$	74093.5i	0.737329i	−0.929562	+	0.368665i	$0.879815\pi$
$318$	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	$-0.828342\pi$
$347$	− 123664.i	− 1.02704i	0.858079	−	0.513518i	$-0.171658\pi$
$348$	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	$-0.953112\pi$
$353$	− 36577.6i	− 0.293539i	0.989171	−	0.146770i	$-0.0468876\pi$
$354$	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	$0.0387359\pi$
$359$	31290.3i	0.242785i	−0.992605	+	0.121392i	$0.961264\pi$
$360$	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	$-0.902086\pi$
$383$	− 88828.0i	− 0.605553i	0.953062	−	0.302777i	$-0.0979136\pi$
$384$	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	$-0.690755\pi$
$389$	− 249905.i	− 1.65149i	0.564043	−	0.825746i	$-0.309245\pi$
$390$	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	$0.0921382\pi$
$401$	91796.6i	0.570871i	−0.958398	+	0.285435i	$0.907862\pi$
$402$	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	$0.0971385\pi$
$419$	105496.i	0.600910i	−0.953796	+	0.300455i	$0.902861\pi$
$420$	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	$-0.702539\pi$
$431$	− 298817.i	− 1.60861i	0.594218	−	0.804304i	$-0.297461\pi$
$432$	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	$-0.994366\pi$
$443$	− 6946.22i	− 0.0353949i	0.999843	−	0.0176975i	$-0.00563357\pi$
$444$	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	$-0.817427\pi$
$449$	− 218791.i	− 1.08527i	0.839969	−	0.542635i	$-0.182573\pi$
$450$	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	$0.0999291\pi$
$461$	131255.i	0.617610i	−0.951125	+	0.308805i	$0.900071\pi$
$462$	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	$0.352109\pi$
$467$	389941.i	1.78799i	−0.448076	+	0.893995i	$0.647891\pi$
$468$	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	$-0.891041\pi$
$479$	− 154028.i	− 0.671318i	0.941983	−	0.335659i	$-0.108959\pi$
$480$	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	$-0.797835\pi$
$491$	− 286054.i	− 1.18655i	0.805001	−	0.593273i	$-0.202165\pi$
$492$	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	$0.0304025\pi$
$503$	48257.4i	0.190734i	−0.995442	+	0.0953670i	$0.969598\pi$
$504$	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	$-0.712110\pi$
$509$	− 407314.i	− 1.57215i	0.618131	−	0.786075i	$-0.287890\pi$
$510$	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	$-0.817700\pi$
$521$	− 294195.i	− 1.08382i	0.840435	−	0.541912i	$-0.182300\pi$
$522$	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	$-0.951939\pi$
$557$	− 93331.8i	− 0.300829i	0.988623	−	0.150414i	$-0.0480608\pi$
$558$	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	$-0.820658\pi$
$563$	− 338574.i	− 1.06816i	0.845433	−	0.534081i	$-0.179342\pi$
$564$	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	$0.0182783\pi$
$569$	37162.2i	0.114783i	−0.998352	+	0.0573915i	$0.981722\pi$
$570$	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	$0.182130\pi$
$587$	373143.i	1.08293i	−0.840724	+	0.541464i	$0.817870\pi$
$588$	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	$-0.931488\pi$
$593$	− 150209.i	− 0.427155i	0.976926	−	0.213578i	$-0.0685117\pi$
$594$	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	$-0.762912\pi$
$599$	− 486427.i	− 1.35570i	0.735200	−	0.677851i	$-0.237088\pi$
$600$	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	$-0.928270\pi$
$617$	− 170126.i	− 0.446891i	0.974716	−	0.223445i	$-0.0717304\pi$
$618$	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	$0.253160\pi$
$641$	586812.i	1.42818i	−0.700053	+	0.714090i	$0.746840\pi$
$642$	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	$0.202379\pi$
$647$	497153.i	1.18763i	−0.804601	+	0.593816i	$0.797621\pi$
$648$	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	$0.0278872\pi$
$653$	74620.0i	0.174996i	−0.996165	+	0.0874981i	$0.972113\pi$
$654$	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	$0.262184\pi$
$659$	637218.i	1.46729i	−0.679530	+	0.733647i	$0.737816\pi$
$660$	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	$-0.894230\pi$
$677$	− 299018.i	− 0.652409i	0.945299	−	0.326204i	$-0.105770\pi$
$678$	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	$-0.803379\pi$
$683$	− 540348.i	− 1.15833i	0.815210	−	0.579165i	$-0.196621\pi$
$684$	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	$-0.799864\pi$
$701$	− 578017.i	− 1.17626i	0.808765	−	0.588132i	$-0.200136\pi$
$702$	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	$0.189074\pi$
$719$	578659.i	1.11935i	−0.828713	+	0.559674i	$0.810926\pi$
$720$	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	$0.158782\pi$
$743$	528197.i	0.956794i	−0.878144	+	0.478397i	$0.841218\pi$
$744$	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	$0.257720\pi$
$761$	838622.i	1.44809i	−0.689750	+	0.724047i	$0.742280\pi$
$762$	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	$0.210675\pi$
$773$	734461.i	1.22916i	−0.788853	+	0.614582i	$0.789325\pi$
$774$	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	$0.0186053\pi$
$797$	74214.0i	0.116834i	−0.998292	+	0.0584170i	$0.981395\pi$
$798$	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	$-0.716090\pi$
$809$	− 1.01875e6i	− 1.55657i	0.627911	−	0.778285i	$-0.283910\pi$
$810$	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	$0.0850444\pi$
$821$	355904.i	0.528015i	−0.964521	+	0.264008i	$0.914956\pi$
$822$	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	$0.284017\pi$
$827$	1.06487e6i	1.55699i	−0.627650	+	0.778496i	$0.715983\pi$
$828$	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	$0.116372\pi$
$839$	503310.i	0.715009i	−0.933911	+	0.357505i	$0.883628\pi$
$840$	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	$0.0806233\pi$
$857$	368085.i	0.501172i	−0.968094	+	0.250586i	$0.919377\pi$
$858$	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	$-0.774781\pi$
$863$	− 968158.i	− 1.29994i	0.759958	−	0.649972i	$-0.225219\pi$
$864$	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	$0.0523093\pi$
$881$	253953.i	0.327192i	−0.986527	+	0.163596i	$0.947691\pi$
$882$	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	$0.0291008\pi$
$887$	143657.i	0.182591i	−0.995824	+	0.0912955i	$0.970899\pi$
$888$	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	$0.0951540\pi$
$911$	488828.i	0.589005i	−0.955651	+	0.294503i	$0.904846\pi$
$912$	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	$-0.937118\pi$
$929$	− 338776.i	− 0.392537i	0.980550	−	0.196269i	$-0.0628825\pi$
$930$	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	$0.00807573\pi$
$941$	44925.7i	0.0507359i	−0.999678	+	0.0253679i	$0.991924\pi$
$942$	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	$-0.868193\pi$
$947$	− 721667.i	− 0.804705i	0.915485	−	0.402353i	$-0.131807\pi$
$948$	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	$0.315360\pi$
$953$	1.51930e6i	1.67286i	−0.548077	+	0.836428i	$0.684640\pi$
$954$	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	$-0.798807\pi$
$971$	− 1.11408e6i	− 1.18162i	0.806809	−	0.590812i	$-0.201193\pi$
$972$	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	$-0.979899\pi$
$977$	− 120477.i	− 0.126217i	0.998007	−	0.0631083i	$-0.0201013\pi$
$978$	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	$-0.989323\pi$
$983$	− 64809.9i	− 0.0670709i	0.999438	−	0.0335354i	$-0.0106767\pi$
$984$	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.161.14		32
3.2	odd	2	inner	405.5.c.b.161.19		32
9.2	odd	6		135.5.i.a.71.9		32
9.4	even	3		135.5.i.a.116.9		32
9.5	odd	6		45.5.i.a.11.8	✓	32
9.7	even	3		45.5.i.a.41.8	yes	32

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
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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