LMFDB - Embedded newform 1575.4.a.bq.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)

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:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 $ x^5 - x^4 - 27x^3 + 7x^2 + 120*x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-4.31366$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.31366 q^{2} +20.2350 q^{4} -7.00000 q^{7} +65.0123 q^{8} -25.5420 q^{11} +64.1014 q^{13} -37.1956 q^{14} +183.574 q^{16} -27.6952 q^{17} +0.792436 q^{19} -135.721 q^{22} +108.606 q^{23} +340.613 q^{26} -141.645 q^{28} +234.000 q^{29} +129.204 q^{31} +455.349 q^{32} -147.163 q^{34} -38.3108 q^{37} +4.21073 q^{38} +403.216 q^{41} -172.895 q^{43} -516.840 q^{44} +577.097 q^{46} +206.943 q^{47} +49.0000 q^{49} +1297.09 q^{52} +144.031 q^{53} -455.086 q^{56} +1243.40 q^{58} +679.086 q^{59} -574.717 q^{61} +686.544 q^{62} +950.977 q^{64} -515.640 q^{67} -560.411 q^{68} -556.612 q^{71} +173.243 q^{73} -203.571 q^{74} +16.0349 q^{76} +178.794 q^{77} +79.3290 q^{79} +2142.55 q^{82} +1043.56 q^{83} -918.703 q^{86} -1660.54 q^{88} -652.060 q^{89} -448.710 q^{91} +2197.65 q^{92} +1099.62 q^{94} -515.714 q^{97} +260.369 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 18 q^{4} - 35 q^{7} + 42 q^{8} - 42 q^{11} - 34 q^{13} - 28 q^{14} + 74 q^{16} + 238 q^{17} - 36 q^{19} - 358 q^{22} + 152 q^{23} + 310 q^{26} - 126 q^{28} + 44 q^{29} + 60 q^{31} + 710 q^{32}+ \cdots + 196 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 18 * q^4 - 35 * q^7 + 42 * q^8 - 42 * q^11 - 34 * q^13 - 28 * q^14 + 74 * q^16 + 238 * q^17 - 36 * q^19 - 358 * q^22 + 152 * q^23 + 310 * q^26 - 126 * q^28 + 44 * q^29 + 60 * q^31 + 710 * q^32 - 482 * q^34 - 312 * q^37 + 280 * q^38 + 426 * q^41 - 304 * q^43 - 712 * q^44 + 88 * q^46 + 370 * q^47 + 245 * q^49 + 1156 * q^52 + 976 * q^53 - 294 * q^56 + 2722 * q^58 + 432 * q^59 - 442 * q^61 - 956 * q^62 + 1362 * q^64 + 804 * q^67 - 420 * q^68 - 440 * q^71 - 564 * q^73 + 1512 * q^74 - 1336 * q^76 + 294 * q^77 + 1790 * q^79 + 3480 * q^82 + 1656 * q^83 - 1216 * q^86 + 1092 * q^88 - 746 * q^89 + 238 * q^91 + 572 * q^92 - 826 * q^94 - 518 * q^97 + 196 * q^98

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 $

$2$	5.31366	1.87866	0.939331	−	0.343013i	$-0.111447\pi$
$2$	5.31366	1.87866	0.939331	+	0.343013i	$0.111447\pi$
$3$	0	0
$3$	0	0
$4$	20.2350	2.52937
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	65.0123	2.87317
$9$	0	0
$10$	0	0
$11$	−25.5420	−0.700108	−0.350054	−	0.936730i	$-0.613837\pi$
$11$	−25.5420	−0.700108	−0.350054	+	0.936730i	$0.613837\pi$
$12$	0	0
$13$	64.1014	1.36758	0.683790	−	0.729679i	$-0.260330\pi$
$13$	64.1014	1.36758	0.683790	+	0.729679i	$0.260330\pi$
$14$	−37.1956	−0.710067
$15$	0	0
$16$	183.574	2.86834
$17$	−27.6952	−0.395122	−0.197561	−	0.980291i	$-0.563302\pi$
$17$	−27.6952	−0.395122	−0.197561	+	0.980291i	$0.563302\pi$
$18$	0	0
$19$	0.792436	0.00956828	0.00478414	−	0.999989i	$-0.498477\pi$
$19$	0.792436	0.00956828	0.00478414	+	0.999989i	$0.498477\pi$
$20$	0	0
$21$	0	0
$22$	−135.721	−1.31527
$23$	108.606	0.984609	0.492305	−	0.870423i	$-0.336155\pi$
$23$	108.606	0.984609	0.492305	+	0.870423i	$0.336155\pi$
$24$	0	0
$25$	0	0
$26$	340.613	2.56922
$27$	0	0
$28$	−141.645	−0.956012
$29$	234.000	1.49837	0.749186	−	0.662360i	$-0.230445\pi$
$29$	234.000	1.49837	0.749186	+	0.662360i	$0.230445\pi$
$30$	0	0
$31$	129.204	0.748570	0.374285	−	0.927314i	$-0.377888\pi$
$31$	129.204	0.748570	0.374285	+	0.927314i	$0.377888\pi$
$32$	455.349	2.51547
$33$	0	0
$34$	−147.163	−0.742300
$35$	0	0
$36$	0	0
$37$	−38.3108	−0.170223	−0.0851116	−	0.996371i	$-0.527125\pi$
$37$	−38.3108	−0.170223	−0.0851116	+	0.996371i	$0.527125\pi$
$38$	4.21073	0.0179756
$39$	0	0
$40$	0	0
$41$	403.216	1.53590	0.767949	−	0.640511i	$-0.221278\pi$
$41$	403.216	1.53590	0.767949	+	0.640511i	$0.221278\pi$
$42$	0	0
$43$	−172.895	−0.613167	−0.306584	−	0.951844i	$-0.599186\pi$
$43$	−172.895	−0.613167	−0.306584	+	0.951844i	$0.599186\pi$
$44$	−516.840	−1.77083
$45$	0	0
$46$	577.097	1.84975
$47$	206.943	0.642250	0.321125	−	0.947037i	$-0.395939\pi$
$47$	206.943	0.642250	0.321125	+	0.947037i	$0.395939\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	1297.09	3.45911
$53$	144.031	0.373287	0.186643	−	0.982428i	$-0.440239\pi$
$53$	144.031	0.373287	0.186643	+	0.982428i	$0.440239\pi$
$54$	0	0
$55$	0	0
$56$	−455.086	−1.08595
$57$	0	0
$58$	1243.40	2.81493
$59$	679.086	1.49846	0.749232	−	0.662307i	$-0.230423\pi$
$59$	679.086	1.49846	0.749232	+	0.662307i	$0.230423\pi$
$60$	0	0
$61$	−574.717	−1.20631	−0.603155	−	0.797624i	$-0.706090\pi$
$61$	−574.717	−1.20631	−0.603155	+	0.797624i	$0.706090\pi$
$62$	686.544	1.40631
$63$	0	0
$64$	950.977	1.85738
$65$	0	0
$66$	0	0
$67$	−515.640	−0.940230	−0.470115	−	0.882605i	$-0.655788\pi$
$67$	−515.640	−0.940230	−0.470115	+	0.882605i	$0.655788\pi$
$68$	−560.411	−0.999409
$69$	0	0
$70$	0	0
$71$	−556.612	−0.930391	−0.465195	−	0.885208i	$-0.654016\pi$
$71$	−556.612	−0.930391	−0.465195	+	0.885208i	$0.654016\pi$
$72$	0	0
$73$	173.243	0.277762	0.138881	−	0.990309i	$-0.455650\pi$
$73$	173.243	0.277762	0.138881	+	0.990309i	$0.455650\pi$
$74$	−203.571	−0.319792
$75$	0	0
$76$	16.0349	0.0242017
$77$	178.794	0.264616
$78$	0	0
$79$	79.3290	0.112977	0.0564887	−	0.998403i	$-0.482010\pi$
$79$	79.3290	0.112977	0.0564887	+	0.998403i	$0.482010\pi$
$80$	0	0
$81$	0	0
$82$	2142.55	2.88543
$83$	1043.56	1.38007	0.690034	−	0.723777i	$-0.257596\pi$
$83$	1043.56	1.38007	0.690034	+	0.723777i	$0.257596\pi$
$84$	0	0
$85$	0	0
$86$	−918.703	−1.15193
$87$	0	0
$88$	−1660.54	−2.01153
$89$	−652.060	−0.776609	−0.388304	−	0.921531i	$-0.626939\pi$
$89$	−652.060	−0.776609	−0.388304	+	0.921531i	$0.626939\pi$
$90$	0	0
$91$	−448.710	−0.516897
$92$	2197.65	2.49044
$93$	0	0
$94$	1099.62	1.20657
$95$	0	0
$96$	0	0
$97$	−515.714	−0.539823	−0.269912	−	0.962885i	$-0.586995\pi$
$97$	−515.714	−0.539823	−0.269912	+	0.962885i	$0.586995\pi$
$98$	260.369	0.268380
$99$	0	0
$100$	0	0
$101$	536.339	0.528394	0.264197	−	0.964469i	$-0.414893\pi$
$101$	536.339	0.528394	0.264197	+	0.964469i	$0.414893\pi$
$102$	0	0
$103$	−381.693	−0.365139	−0.182570	−	0.983193i	$-0.558441\pi$
$103$	−381.693	−0.365139	−0.182570	+	0.983193i	$0.558441\pi$
$104$	4167.38	3.92928
$105$	0	0
$106$	765.332	0.701279
$107$	−1381.12	−1.24783	−0.623915	−	0.781492i	$-0.714459\pi$
$107$	−1381.12	−1.24783	−0.623915	+	0.781492i	$0.714459\pi$
$108$	0	0
$109$	−390.582	−0.343220	−0.171610	−	0.985165i	$-0.554897\pi$
$109$	−390.582	−0.343220	−0.171610	+	0.985165i	$0.554897\pi$
$110$	0	0
$111$	0	0
$112$	−1285.02	−1.08413
$113$	1643.15	1.36792	0.683958	−	0.729521i	$-0.260257\pi$
$113$	1643.15	1.36792	0.683958	+	0.729521i	$0.260257\pi$
$114$	0	0
$115$	0	0
$116$	4734.99	3.78994
$117$	0	0
$118$	3608.43	2.81511
$119$	193.866	0.149342
$120$	0	0
$121$	−678.609	−0.509849
$122$	−3053.85	−2.26625
$123$	0	0
$124$	2614.43	1.89341
$125$	0	0
$126$	0	0
$127$	192.032	0.134174	0.0670869	−	0.997747i	$-0.478630\pi$
$127$	192.032	0.134174	0.0670869	+	0.997747i	$0.478630\pi$
$128$	1410.38	0.973914
$129$	0	0
$130$	0	0
$131$	−2082.90	−1.38919	−0.694594	−	0.719402i	$-0.744416\pi$
$131$	−2082.90	−1.38919	−0.694594	+	0.719402i	$0.744416\pi$
$132$	0	0
$133$	−5.54705	−0.00361647
$134$	−2739.93	−1.76637
$135$	0	0
$136$	−1800.53	−1.13525
$137$	−78.1709	−0.0487488	−0.0243744	−	0.999703i	$-0.507759\pi$
$137$	−78.1709	−0.0487488	−0.0243744	+	0.999703i	$0.507759\pi$
$138$	0	0
$139$	1393.67	0.850426	0.425213	−	0.905093i	$-0.360199\pi$
$139$	1393.67	0.850426	0.425213	+	0.905093i	$0.360199\pi$
$140$	0	0
$141$	0	0
$142$	−2957.65	−1.74789
$143$	−1637.28	−0.957454
$144$	0	0
$145$	0	0
$146$	920.556	0.521820
$147$	0	0
$148$	−775.217	−0.430557
$149$	−32.5002	−0.0178693	−0.00893463	−	0.999960i	$-0.502844\pi$
$149$	−32.5002	−0.0178693	−0.00893463	+	0.999960i	$0.502844\pi$
$150$	0	0
$151$	466.762	0.251553	0.125777	−	0.992059i	$-0.459858\pi$
$151$	466.762	0.251553	0.125777	+	0.992059i	$0.459858\pi$
$152$	51.5181	0.0274913
$153$	0	0
$154$	950.048	0.497124
$155$	0	0
$156$	0	0
$157$	−1673.50	−0.850701	−0.425351	−	0.905029i	$-0.639849\pi$
$157$	−1673.50	−0.850701	−0.425351	+	0.905029i	$0.639849\pi$
$158$	421.527	0.212246
$159$	0	0
$160$	0	0
$161$	−760.245	−0.372147
$162$	0	0
$163$	−1869.36	−0.898279	−0.449139	−	0.893462i	$-0.648269\pi$
$163$	−1869.36	−0.898279	−0.449139	+	0.893462i	$0.648269\pi$
$164$	8159.06	3.88485
$165$	0	0
$166$	5545.12	2.59268
$167$	46.5250	0.0215581	0.0107791	−	0.999942i	$-0.496569\pi$
$167$	46.5250	0.0215581	0.0107791	+	0.999942i	$0.496569\pi$
$168$	0	0
$169$	1911.99	0.870275
$170$	0	0
$171$	0	0
$172$	−3498.52	−1.55093
$173$	2496.28	1.09704	0.548521	−	0.836137i	$-0.315191\pi$
$173$	2496.28	1.09704	0.548521	+	0.836137i	$0.315191\pi$
$174$	0	0
$175$	0	0
$176$	−4688.83	−2.00815
$177$	0	0
$178$	−3464.82	−1.45898
$179$	−2975.70	−1.24254	−0.621269	−	0.783598i	$-0.713382\pi$
$179$	−2975.70	−1.24254	−0.621269	+	0.783598i	$0.713382\pi$
$180$	0	0
$181$	966.273	0.396809	0.198405	−	0.980120i	$-0.436424\pi$
$181$	966.273	0.396809	0.198405	+	0.980120i	$0.436424\pi$
$182$	−2384.29	−0.971074
$183$	0	0
$184$	7060.76	2.82895
$185$	0	0
$186$	0	0
$187$	707.389	0.276628
$188$	4187.48	1.62449
$189$	0	0
$190$	0	0
$191$	1545.50	0.585488	0.292744	−	0.956191i	$-0.405432\pi$
$191$	1545.50	0.585488	0.292744	+	0.956191i	$0.405432\pi$
$192$	0	0
$193$	2304.05	0.859322	0.429661	−	0.902990i	$-0.358633\pi$
$193$	2304.05	0.859322	0.429661	+	0.902990i	$0.358633\pi$
$194$	−2740.33	−1.01415
$195$	0	0
$196$	991.513	0.361338
$197$	−222.021	−0.0802960	−0.0401480	−	0.999194i	$-0.512783\pi$
$197$	−222.021	−0.0802960	−0.0401480	+	0.999194i	$0.512783\pi$
$198$	0	0
$199$	3580.56	1.27547	0.637736	−	0.770255i	$-0.279871\pi$
$199$	3580.56	1.27547	0.637736	+	0.770255i	$0.279871\pi$
$200$	0	0
$201$	0	0
$202$	2849.92	0.992673
$203$	−1638.00	−0.566331
$204$	0	0
$205$	0	0
$206$	−2028.19	−0.685973
$207$	0	0
$208$	11767.3	3.92268
$209$	−20.2404	−0.00669883
$210$	0	0
$211$	−4181.04	−1.36415	−0.682073	−	0.731284i	$-0.738921\pi$
$211$	−4181.04	−1.36415	−0.682073	+	0.731284i	$0.738921\pi$
$212$	2914.46	0.944180
$213$	0	0
$214$	−7338.79	−2.34425
$215$	0	0
$216$	0	0
$217$	−904.426	−0.282933
$218$	−2075.42	−0.644794
$219$	0	0
$220$	0	0
$221$	−1775.30	−0.540361
$222$	0	0
$223$	−2361.52	−0.709145	−0.354573	−	0.935028i	$-0.615374\pi$
$223$	−2361.52	−0.709145	−0.354573	+	0.935028i	$0.615374\pi$
$224$	−3187.44	−0.950758
$225$	0	0
$226$	8731.14	2.56985
$227$	586.877	0.171596	0.0857982	−	0.996313i	$-0.472656\pi$
$227$	586.877	0.171596	0.0857982	+	0.996313i	$0.472656\pi$
$228$	0	0
$229$	−4619.55	−1.33305	−0.666526	−	0.745482i	$-0.732219\pi$
$229$	−4619.55	−1.33305	−0.666526	+	0.745482i	$0.732219\pi$
$230$	0	0
$231$	0	0
$232$	15212.9	4.30507
$233$	−5120.44	−1.43971	−0.719853	−	0.694127i	$-0.755791\pi$
$233$	−5120.44	−1.43971	−0.719853	+	0.694127i	$0.755791\pi$
$234$	0	0
$235$	0	0
$236$	13741.3	3.79017
$237$	0	0
$238$	1030.14	0.280563
$239$	−1127.51	−0.305158	−0.152579	−	0.988291i	$-0.548758\pi$
$239$	−1127.51	−0.305158	−0.152579	+	0.988291i	$0.548758\pi$
$240$	0	0
$241$	−3549.53	−0.948736	−0.474368	−	0.880327i	$-0.657323\pi$
$241$	−3549.53	−0.948736	−0.474368	+	0.880327i	$0.657323\pi$
$242$	−3605.89	−0.957833
$243$	0	0
$244$	−11629.4	−3.05120
$245$	0	0
$246$	0	0
$247$	50.7963	0.0130854
$248$	8399.84	2.15077
$249$	0	0
$250$	0	0
$251$	−4717.19	−1.18624	−0.593120	−	0.805114i	$-0.702104\pi$
$251$	−4717.19	−1.18624	−0.593120	+	0.805114i	$0.702104\pi$
$252$	0	0
$253$	−2774.02	−0.689333
$254$	1020.39	0.252067
$255$	0	0
$256$	−113.552	−0.0277227
$257$	6260.31	1.51948	0.759742	−	0.650224i	$-0.225325\pi$
$257$	6260.31	1.51948	0.759742	+	0.650224i	$0.225325\pi$
$258$	0	0
$259$	268.176	0.0643383
$260$	0	0
$261$	0	0
$262$	−11067.8	−2.60981
$263$	−5753.04	−1.34885	−0.674425	−	0.738344i	$-0.735608\pi$
$263$	−5753.04	−1.34885	−0.674425	+	0.738344i	$0.735608\pi$
$264$	0	0
$265$	0	0
$266$	−29.4751	−0.00679412
$267$	0	0
$268$	−10433.9	−2.37819
$269$	7059.21	1.60003	0.800014	−	0.599982i	$-0.204826\pi$
$269$	7059.21	1.60003	0.800014	+	0.599982i	$0.204826\pi$
$270$	0	0
$271$	−8534.52	−1.91305	−0.956523	−	0.291658i	$-0.905793\pi$
$271$	−8534.52	−1.91305	−0.956523	+	0.291658i	$0.905793\pi$
$272$	−5084.11	−1.13334
$273$	0	0
$274$	−415.373	−0.0915826
$275$	0	0
$276$	0	0
$277$	1313.94	0.285008	0.142504	−	0.989794i	$-0.454485\pi$
$277$	1313.94	0.285008	0.142504	+	0.989794i	$0.454485\pi$
$278$	7405.47	1.59766
$279$	0	0
$280$	0	0
$281$	247.229	0.0524856	0.0262428	−	0.999656i	$-0.491646\pi$
$281$	247.229	0.0524856	0.0262428	+	0.999656i	$0.491646\pi$
$282$	0	0
$283$	9074.90	1.90617	0.953087	−	0.302697i	$-0.0978872\pi$
$283$	9074.90	1.90617	0.953087	+	0.302697i	$0.0978872\pi$
$284$	−11263.0	−2.35330
$285$	0	0
$286$	−8699.92	−1.79873
$287$	−2822.51	−0.580515
$288$	0	0
$289$	−4145.98	−0.843879
$290$	0	0
$291$	0	0
$292$	3505.57	0.702562
$293$	−2740.72	−0.546466	−0.273233	−	0.961948i	$-0.588093\pi$
$293$	−2740.72	−0.546466	−0.273233	+	0.961948i	$0.588093\pi$
$294$	0	0
$295$	0	0
$296$	−2490.68	−0.489080
$297$	0	0
$298$	−172.695	−0.0335703
$299$	6961.83	1.34653
$300$	0	0
$301$	1210.26	0.231755
$302$	2480.21	0.472584
$303$	0	0
$304$	145.470	0.0274451
$305$	0	0
$306$	0	0
$307$	6985.46	1.29864	0.649318	−	0.760517i	$-0.275054\pi$
$307$	6985.46	1.29864	0.649318	+	0.760517i	$0.275054\pi$
$308$	3617.88	0.669311
$309$	0	0
$310$	0	0
$311$	356.841	0.0650630	0.0325315	−	0.999471i	$-0.489643\pi$
$311$	356.841	0.0650630	0.0325315	+	0.999471i	$0.489643\pi$
$312$	0	0
$313$	−6630.12	−1.19731	−0.598653	−	0.801009i	$-0.704297\pi$
$313$	−6630.12	−1.19731	−0.598653	+	0.801009i	$0.704297\pi$
$314$	−8892.42	−1.59818
$315$	0	0
$316$	1605.22	0.285762
$317$	2494.05	0.441891	0.220946	−	0.975286i	$-0.429086\pi$
$317$	2494.05	0.441891	0.220946	+	0.975286i	$0.429086\pi$
$318$	0	0
$319$	−5976.83	−1.04902
$320$	0	0
$321$	0	0
$322$	−4039.68	−0.699139
$323$	−21.9467	−0.00378063
$324$	0	0
$325$	0	0
$326$	−9933.13	−1.68756
$327$	0	0
$328$	26214.0	4.41289
$329$	−1448.60	−0.242748
$330$	0	0
$331$	−4682.47	−0.777558	−0.388779	−	0.921331i	$-0.627103\pi$
$331$	−4682.47	−0.777558	−0.388779	+	0.921331i	$0.627103\pi$
$332$	21116.4	3.49070
$333$	0	0
$334$	247.218	0.0405005
$335$	0	0
$336$	0	0
$337$	3596.60	0.581363	0.290681	−	0.956820i	$-0.406118\pi$
$337$	3596.60	0.581363	0.290681	+	0.956820i	$0.406118\pi$
$338$	10159.7	1.63495
$339$	0	0
$340$	0	0
$341$	−3300.12	−0.524080
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−11240.3	−1.76173
$345$	0	0
$346$	13264.4	2.06097
$347$	−1899.51	−0.293865	−0.146932	−	0.989147i	$-0.546940\pi$
$347$	−1899.51	−0.293865	−0.146932	+	0.989147i	$0.546940\pi$
$348$	0	0
$349$	−1037.55	−0.159137	−0.0795683	−	0.996829i	$-0.525354\pi$
$349$	−1037.55	−0.159137	−0.0795683	+	0.996829i	$0.525354\pi$
$350$	0	0
$351$	0	0
$352$	−11630.5	−1.76110
$353$	4087.08	0.616242	0.308121	−	0.951347i	$-0.400300\pi$
$353$	4087.08	0.616242	0.308121	+	0.951347i	$0.400300\pi$
$354$	0	0
$355$	0	0
$356$	−13194.4	−1.96433
$357$	0	0
$358$	−15811.8	−2.33431
$359$	−3472.67	−0.510530	−0.255265	−	0.966871i	$-0.582163\pi$
$359$	−3472.67	−0.510530	−0.255265	+	0.966871i	$0.582163\pi$
$360$	0	0
$361$	−6858.37	−0.999908
$362$	5134.44	0.745470
$363$	0	0
$364$	−9079.63	−1.30742
$365$	0	0
$366$	0	0
$367$	8769.14	1.24726	0.623631	−	0.781719i	$-0.285657\pi$
$367$	8769.14	1.24726	0.623631	+	0.781719i	$0.285657\pi$
$368$	19937.3	2.82419
$369$	0	0
$370$	0	0
$371$	−1008.22	−0.141089
$372$	0	0
$373$	11368.9	1.57817	0.789085	−	0.614284i	$-0.210555\pi$
$373$	11368.9	1.57817	0.789085	+	0.614284i	$0.210555\pi$
$374$	3758.82	0.519690
$375$	0	0
$376$	13453.8	1.84529
$377$	14999.8	2.04914
$378$	0	0
$379$	−12137.4	−1.64500	−0.822501	−	0.568764i	$-0.807422\pi$
$379$	−12137.4	−1.64500	−0.822501	+	0.568764i	$0.807422\pi$
$380$	0	0
$381$	0	0
$382$	8212.24	1.09993
$383$	9869.61	1.31675	0.658373	−	0.752692i	$-0.271245\pi$
$383$	9869.61	1.31675	0.658373	+	0.752692i	$0.271245\pi$
$384$	0	0
$385$	0	0
$386$	12242.9	1.61438
$387$	0	0
$388$	−10435.5	−1.36541
$389$	−57.1166	−0.00744454	−0.00372227	−	0.999993i	$-0.501185\pi$
$389$	−57.1166	−0.00744454	−0.00372227	+	0.999993i	$0.501185\pi$
$390$	0	0
$391$	−3007.88	−0.389041
$392$	3185.60	0.410452
$393$	0	0
$394$	−1179.74	−0.150849
$395$	0	0
$396$	0	0
$397$	−7436.17	−0.940078	−0.470039	−	0.882646i	$-0.655760\pi$
$397$	−7436.17	−0.940078	−0.470039	+	0.882646i	$0.655760\pi$
$398$	19025.8	2.39618
$399$	0	0
$400$	0	0
$401$	12465.0	1.55230	0.776149	−	0.630550i	$-0.217171\pi$
$401$	12465.0	1.55230	0.776149	+	0.630550i	$0.217171\pi$
$402$	0	0
$403$	8282.15	1.02373
$404$	10852.8	1.33650
$405$	0	0
$406$	−8703.79	−1.06394
$407$	978.533	0.119175
$408$	0	0
$409$	−1708.72	−0.206578	−0.103289	−	0.994651i	$-0.532937\pi$
$409$	−1708.72	−0.206578	−0.103289	+	0.994651i	$0.532937\pi$
$410$	0	0
$411$	0	0
$412$	−7723.54	−0.923571
$413$	−4753.60	−0.566366
$414$	0	0
$415$	0	0
$416$	29188.5	3.44011
$417$	0	0
$418$	−107.550	−0.0125848
$419$	−10618.8	−1.23810	−0.619050	−	0.785352i	$-0.712482\pi$
$419$	−10618.8	−1.23810	−0.619050	+	0.785352i	$0.712482\pi$
$420$	0	0
$421$	13273.5	1.53661	0.768304	−	0.640085i	$-0.221101\pi$
$421$	13273.5	1.53661	0.768304	+	0.640085i	$0.221101\pi$
$422$	−22216.6	−2.56277
$423$	0	0
$424$	9363.80	1.07251
$425$	0	0
$426$	0	0
$427$	4023.02	0.455943
$428$	−27946.9	−3.15622
$429$	0	0
$430$	0	0
$431$	−7918.20	−0.884934	−0.442467	−	0.896785i	$-0.645897\pi$
$431$	−7918.20	−0.884934	−0.442467	+	0.896785i	$0.645897\pi$
$432$	0	0
$433$	−4433.34	−0.492038	−0.246019	−	0.969265i	$-0.579123\pi$
$433$	−4433.34	−0.492038	−0.246019	+	0.969265i	$0.579123\pi$
$434$	−4805.81	−0.531535
$435$	0	0
$436$	−7903.40	−0.868129
$437$	86.0637	0.00942102
$438$	0	0
$439$	12958.4	1.40882	0.704408	−	0.709796i	$-0.251213\pi$
$439$	12958.4	1.40882	0.704408	+	0.709796i	$0.251213\pi$
$440$	0	0
$441$	0	0
$442$	−9433.34	−1.01515
$443$	−12040.4	−1.29133	−0.645664	−	0.763621i	$-0.723420\pi$
$443$	−12040.4	−1.29133	−0.645664	+	0.763621i	$0.723420\pi$
$444$	0	0
$445$	0	0
$446$	−12548.3	−1.33224
$447$	0	0
$448$	−6656.84	−0.702023
$449$	−11586.3	−1.21780	−0.608899	−	0.793247i	$-0.708389\pi$
$449$	−11586.3	−1.21780	−0.608899	+	0.793247i	$0.708389\pi$
$450$	0	0
$451$	−10298.9	−1.07529
$452$	33249.1	3.45997
$453$	0	0
$454$	3118.46	0.322371
$455$	0	0
$456$	0	0
$457$	−9734.34	−0.996396	−0.498198	−	0.867063i	$-0.666005\pi$
$457$	−9734.34	−0.996396	−0.498198	+	0.867063i	$0.666005\pi$
$458$	−24546.7	−2.50435
$459$	0	0
$460$	0	0
$461$	1343.41	0.135724	0.0678621	−	0.997695i	$-0.478382\pi$
$461$	1343.41	0.135724	0.0678621	+	0.997695i	$0.478382\pi$
$462$	0	0
$463$	6613.72	0.663857	0.331929	−	0.943305i	$-0.392301\pi$
$463$	6613.72	0.663857	0.331929	+	0.943305i	$0.392301\pi$
$464$	42956.3	4.29784
$465$	0	0
$466$	−27208.3	−2.70472
$467$	−14688.9	−1.45551	−0.727755	−	0.685838i	$-0.759436\pi$
$467$	−14688.9	−1.45551	−0.727755	+	0.685838i	$0.759436\pi$
$468$	0	0
$469$	3609.48	0.355374
$470$	0	0
$471$	0	0
$472$	44148.9	4.30534
$473$	4416.07	0.429283
$474$	0	0
$475$	0	0
$476$	3922.88	0.377741
$477$	0	0
$478$	−5991.22	−0.573288
$479$	15298.6	1.45931	0.729657	−	0.683813i	$-0.239680\pi$
$479$	15298.6	1.45931	0.729657	+	0.683813i	$0.239680\pi$
$480$	0	0
$481$	−2455.78	−0.232794
$482$	−18861.0	−1.78235
$483$	0	0
$484$	−13731.6	−1.28960
$485$	0	0
$486$	0	0
$487$	9653.80	0.898266	0.449133	−	0.893465i	$-0.351733\pi$
$487$	9653.80	0.898266	0.449133	+	0.893465i	$0.351733\pi$
$488$	−37363.7	−3.46593
$489$	0	0
$490$	0	0
$491$	−20142.6	−1.85137	−0.925684	−	0.378297i	$-0.876510\pi$
$491$	−20142.6	−1.85137	−0.925684	+	0.378297i	$0.876510\pi$
$492$	0	0
$493$	−6480.69	−0.592039
$494$	269.914	0.0245830
$495$	0	0
$496$	23718.4	2.14715
$497$	3896.29	0.351655
$498$	0	0
$499$	−1309.29	−0.117459	−0.0587293	−	0.998274i	$-0.518705\pi$
$499$	−1309.29	−0.117459	−0.0587293	+	0.998274i	$0.518705\pi$
$500$	0	0
$501$	0	0
$502$	−25065.5	−2.22855
$503$	2186.17	0.193791	0.0968953	−	0.995295i	$-0.469109\pi$
$503$	2186.17	0.193791	0.0968953	+	0.995295i	$0.469109\pi$
$504$	0	0
$505$	0	0
$506$	−14740.2	−1.29502
$507$	0	0
$508$	3885.76	0.339375
$509$	−3591.11	−0.312718	−0.156359	−	0.987700i	$-0.549976\pi$
$509$	−3591.11	−0.312718	−0.156359	+	0.987700i	$0.549976\pi$
$510$	0	0
$511$	−1212.70	−0.104984
$512$	−11886.4	−1.02600
$513$	0	0
$514$	33265.2	2.85460
$515$	0	0
$516$	0	0
$517$	−5285.73	−0.449644
$518$	1424.99	0.120870
$519$	0	0
$520$	0	0
$521$	8605.34	0.723621	0.361811	−	0.932252i	$-0.382159\pi$
$521$	8605.34	0.723621	0.361811	+	0.932252i	$0.382159\pi$
$522$	0	0
$523$	−22536.4	−1.88423	−0.942113	−	0.335297i	$-0.891163\pi$
$523$	−22536.4	−1.88423	−0.942113	+	0.335297i	$0.891163\pi$
$524$	−42147.3	−3.51377
$525$	0	0
$526$	−30569.7	−2.53403
$527$	−3578.32	−0.295776
$528$	0	0
$529$	−371.637	−0.0305447
$530$	0	0
$531$	0	0
$532$	−112.244	−0.00914738
$533$	25846.7	2.10046
$534$	0	0
$535$	0	0
$536$	−33522.9	−2.70144
$537$	0	0
$538$	37510.2	3.00591
$539$	−1251.56	−0.100015
$540$	0	0
$541$	8782.44	0.697942	0.348971	−	0.937134i	$-0.386531\pi$
$541$	8782.44	0.697942	0.348971	+	0.937134i	$0.386531\pi$
$542$	−45349.5	−3.59396
$543$	0	0
$544$	−12611.0	−0.993917
$545$	0	0
$546$	0	0
$547$	−22593.0	−1.76601	−0.883004	−	0.469366i	$-0.844482\pi$
$547$	−22593.0	−1.76601	−0.883004	+	0.469366i	$0.844482\pi$
$548$	−1581.78	−0.123304
$549$	0	0
$550$	0	0
$551$	185.430	0.0143368
$552$	0	0
$553$	−555.303	−0.0427014
$554$	6981.84	0.535433
$555$	0	0
$556$	28200.8	2.15104
$557$	−7264.13	−0.552587	−0.276294	−	0.961073i	$-0.589106\pi$
$557$	−7264.13	−0.552587	−0.276294	+	0.961073i	$0.589106\pi$
$558$	0	0
$559$	−11082.8	−0.838555
$560$	0	0
$561$	0	0
$562$	1313.69	0.0986027
$563$	3851.42	0.288309	0.144154	−	0.989555i	$-0.453954\pi$
$563$	3851.42	0.288309	0.144154	+	0.989555i	$0.453954\pi$
$564$	0	0
$565$	0	0
$566$	48220.9	3.58105
$567$	0	0
$568$	−36186.7	−2.67317
$569$	−16580.3	−1.22158	−0.610792	−	0.791791i	$-0.709149\pi$
$569$	−16580.3	−1.22158	−0.610792	+	0.791791i	$0.709149\pi$
$570$	0	0
$571$	6385.86	0.468021	0.234010	−	0.972234i	$-0.424815\pi$
$571$	6385.86	0.468021	0.234010	+	0.972234i	$0.424815\pi$
$572$	−33130.2	−2.42175
$573$	0	0
$574$	−14997.9	−1.09059
$575$	0	0
$576$	0	0
$577$	11059.1	0.797912	0.398956	−	0.916970i	$-0.369373\pi$
$577$	11059.1	0.797912	0.398956	+	0.916970i	$0.369373\pi$
$578$	−22030.3	−1.58536
$579$	0	0
$580$	0	0
$581$	−7304.92	−0.521617
$582$	0	0
$583$	−3678.84	−0.261341
$584$	11263.0	0.798055
$585$	0	0
$586$	−14563.2	−1.02662
$587$	−7871.25	−0.553461	−0.276730	−	0.960948i	$-0.589251\pi$
$587$	−7871.25	−0.553461	−0.276730	+	0.960948i	$0.589251\pi$
$588$	0	0
$589$	102.386	0.00716253
$590$	0	0
$591$	0	0
$592$	−7032.85	−0.488258
$593$	−2018.06	−0.139750	−0.0698750	−	0.997556i	$-0.522260\pi$
$593$	−2018.06	−0.139750	−0.0698750	+	0.997556i	$0.522260\pi$
$594$	0	0
$595$	0	0
$596$	−657.640	−0.0451980
$597$	0	0
$598$	36992.8	2.52968
$599$	1356.67	0.0925409	0.0462705	−	0.998929i	$-0.485266\pi$
$599$	1356.67	0.0925409	0.0462705	+	0.998929i	$0.485266\pi$
$600$	0	0
$601$	11178.7	0.758715	0.379358	−	0.925250i	$-0.376145\pi$
$601$	11178.7	0.758715	0.379358	+	0.925250i	$0.376145\pi$
$602$	6430.92	0.435390
$603$	0	0
$604$	9444.91	0.636272
$605$	0	0
$606$	0	0
$607$	−9404.67	−0.628870	−0.314435	−	0.949279i	$-0.601815\pi$
$607$	−9404.67	−0.628870	−0.314435	+	0.949279i	$0.601815\pi$
$608$	360.835	0.0240687
$609$	0	0
$610$	0	0
$611$	13265.3	0.878328
$612$	0	0
$613$	18938.0	1.24779	0.623897	−	0.781507i	$-0.285548\pi$
$613$	18938.0	1.24779	0.623897	+	0.781507i	$0.285548\pi$
$614$	37118.4	2.43970
$615$	0	0
$616$	11623.8	0.760286
$617$	17716.9	1.15600	0.578001	−	0.816036i	$-0.303833\pi$
$617$	17716.9	1.15600	0.578001	+	0.816036i	$0.303833\pi$
$618$	0	0
$619$	−6240.33	−0.405202	−0.202601	−	0.979261i	$-0.564939\pi$
$619$	−6240.33	−0.405202	−0.202601	+	0.979261i	$0.564939\pi$
$620$	0	0
$621$	0	0
$622$	1896.13	0.122231
$623$	4564.42	0.293531
$624$	0	0
$625$	0	0
$626$	−35230.2	−2.24933
$627$	0	0
$628$	−33863.2	−2.15174
$629$	1061.03	0.0672589
$630$	0	0
$631$	−25887.7	−1.63323	−0.816617	−	0.577179i	$-0.804153\pi$
$631$	−25887.7	−1.63323	−0.816617	+	0.577179i	$0.804153\pi$
$632$	5157.37	0.324603
$633$	0	0
$634$	13252.5	0.830165
$635$	0	0
$636$	0	0
$637$	3140.97	0.195369
$638$	−31758.8	−1.97076
$639$	0	0
$640$	0	0
$641$	−18798.4	−1.15834	−0.579168	−	0.815208i	$-0.696622\pi$
$641$	−18798.4	−1.15834	−0.579168	+	0.815208i	$0.696622\pi$
$642$	0	0
$643$	2287.70	0.140308	0.0701541	−	0.997536i	$-0.477651\pi$
$643$	2287.70	0.140308	0.0701541	+	0.997536i	$0.477651\pi$
$644$	−15383.5	−0.941298
$645$	0	0
$646$	−116.617	−0.00710253
$647$	1769.31	0.107510	0.0537548	−	0.998554i	$-0.482881\pi$
$647$	1769.31	0.107510	0.0537548	+	0.998554i	$0.482881\pi$
$648$	0	0
$649$	−17345.2	−1.04909
$650$	0	0
$651$	0	0
$652$	−37826.4	−2.27208
$653$	3891.48	0.233209	0.116604	−	0.993178i	$-0.462799\pi$
$653$	3891.48	0.233209	0.116604	+	0.993178i	$0.462799\pi$
$654$	0	0
$655$	0	0
$656$	74019.8	4.40547
$657$	0	0
$658$	−7697.37	−0.456040
$659$	−20097.6	−1.18800	−0.594001	−	0.804465i	$-0.702452\pi$
$659$	−20097.6	−1.18800	−0.594001	+	0.804465i	$0.702452\pi$
$660$	0	0
$661$	−27167.9	−1.59865	−0.799326	−	0.600898i	$-0.794810\pi$
$661$	−27167.9	−1.59865	−0.799326	+	0.600898i	$0.794810\pi$
$662$	−24881.0	−1.46077
$663$	0	0
$664$	67844.3	3.96516
$665$	0	0
$666$	0	0
$667$	25414.0	1.47531
$668$	941.430	0.0545285
$669$	0	0
$670$	0	0
$671$	14679.4	0.844548
$672$	0	0
$673$	−25909.7	−1.48402	−0.742010	−	0.670389i	$-0.766127\pi$
$673$	−25909.7	−1.48402	−0.742010	+	0.670389i	$0.766127\pi$
$674$	19111.1	1.09218
$675$	0	0
$676$	38689.1	2.20125
$677$	4359.22	0.247472	0.123736	−	0.992315i	$-0.460512\pi$
$677$	4359.22	0.247472	0.123736	+	0.992315i	$0.460512\pi$
$678$	0	0
$679$	3610.00	0.204034
$680$	0	0
$681$	0	0
$682$	−17535.7	−0.984569
$683$	29721.9	1.66512	0.832559	−	0.553937i	$-0.186875\pi$
$683$	29721.9	1.66512	0.832559	+	0.553937i	$0.186875\pi$
$684$	0	0
$685$	0	0
$686$	−1822.58	−0.101438
$687$	0	0
$688$	−31738.9	−1.75877
$689$	9232.60	0.510499
$690$	0	0
$691$	−4929.13	−0.271365	−0.135682	−	0.990752i	$-0.543323\pi$
$691$	−4929.13	−0.271365	−0.135682	+	0.990752i	$0.543323\pi$
$692$	50512.0	2.77482
$693$	0	0
$694$	−10093.4	−0.552073
$695$	0	0
$696$	0	0
$697$	−11167.1	−0.606866
$698$	−5513.18	−0.298964
$699$	0	0
$700$	0	0
$701$	19358.8	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$701$	19358.8	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$702$	0	0
$703$	−30.3589	−0.00162874
$704$	−24289.8	−1.30036
$705$	0	0
$706$	21717.3	1.15771
$707$	−3754.38	−0.199714
$708$	0	0
$709$	−17186.3	−0.910359	−0.455180	−	0.890400i	$-0.650425\pi$
$709$	−17186.3	−0.910359	−0.455180	+	0.890400i	$0.650425\pi$
$710$	0	0
$711$	0	0
$712$	−42391.9	−2.23133
$713$	14032.4	0.737049
$714$	0	0
$715$	0	0
$716$	−60213.1	−3.14284
$717$	0	0
$718$	−18452.6	−0.959113
$719$	15107.3	0.783598	0.391799	−	0.920051i	$-0.371853\pi$
$719$	15107.3	0.783598	0.391799	+	0.920051i	$0.371853\pi$
$720$	0	0
$721$	2671.85	0.138010
$722$	−36443.0	−1.87849
$723$	0	0
$724$	19552.5	1.00368
$725$	0	0
$726$	0	0
$727$	15840.9	0.808124	0.404062	−	0.914732i	$-0.367598\pi$
$727$	15840.9	0.808124	0.404062	+	0.914732i	$0.367598\pi$
$728$	−29171.7	−1.48513
$729$	0	0
$730$	0	0
$731$	4788.35	0.242276
$732$	0	0
$733$	27639.7	1.39276	0.696382	−	0.717671i	$-0.254792\pi$
$733$	27639.7	1.39276	0.696382	+	0.717671i	$0.254792\pi$
$734$	46596.2	2.34318
$735$	0	0
$736$	49453.8	2.47675
$737$	13170.4	0.658263
$738$	0	0
$739$	34874.4	1.73596	0.867982	−	0.496597i	$-0.165417\pi$
$739$	34874.4	1.73596	0.867982	+	0.496597i	$0.165417\pi$
$740$	0	0
$741$	0	0
$742$	−5357.32	−0.265059
$743$	−27686.7	−1.36706	−0.683530	−	0.729922i	$-0.739556\pi$
$743$	−27686.7	−1.36706	−0.683530	+	0.729922i	$0.739556\pi$
$744$	0	0
$745$	0	0
$746$	60410.2	2.96485
$747$	0	0
$748$	14314.0	0.699694
$749$	9667.83	0.471635
$750$	0	0
$751$	4806.85	0.233561	0.116781	−	0.993158i	$-0.462743\pi$
$751$	4806.85	0.233561	0.116781	+	0.993158i	$0.462743\pi$
$752$	37989.3	1.84219
$753$	0	0
$754$	79703.6	3.84965
$755$	0	0
$756$	0	0
$757$	40166.6	1.92851	0.964253	−	0.264983i	$-0.0853664\pi$
$757$	40166.6	1.92851	0.964253	+	0.264983i	$0.0853664\pi$
$758$	−64493.9	−3.09040
$759$	0	0
$760$	0	0
$761$	−25912.3	−1.23432	−0.617162	−	0.786836i	$-0.711718\pi$
$761$	−25912.3	−1.23432	−0.617162	+	0.786836i	$0.711718\pi$
$762$	0	0
$763$	2734.07	0.129725
$764$	31273.1	1.48092
$765$	0	0
$766$	52443.7	2.47372
$767$	43530.4	2.04927
$768$	0	0
$769$	−23231.3	−1.08939	−0.544697	−	0.838633i	$-0.683355\pi$
$769$	−23231.3	−1.08939	−0.544697	+	0.838633i	$0.683355\pi$
$770$	0	0
$771$	0	0
$772$	46622.4	2.17354
$773$	836.306	0.0389131	0.0194566	−	0.999811i	$-0.493806\pi$
$773$	836.306	0.0389131	0.0194566	+	0.999811i	$0.493806\pi$
$774$	0	0
$775$	0	0
$776$	−33527.8	−1.55100
$777$	0	0
$778$	−303.498	−0.0139858
$779$	319.523	0.0146959
$780$	0	0
$781$	14217.0	0.651374
$782$	−15982.8	−0.730875
$783$	0	0
$784$	8995.11	0.409763
$785$	0	0
$786$	0	0
$787$	−31945.6	−1.44693	−0.723467	−	0.690359i	$-0.757453\pi$
$787$	−31945.6	−1.44693	−0.723467	+	0.690359i	$0.757453\pi$
$788$	−4492.58	−0.203098
$789$	0	0
$790$	0	0
$791$	−11502.1	−0.517024
$792$	0	0
$793$	−36840.2	−1.64973
$794$	−39513.3	−1.76609
$795$	0	0
$796$	72452.4	3.22614
$797$	5628.83	0.250168	0.125084	−	0.992146i	$-0.460080\pi$
$797$	5628.83	0.250168	0.125084	+	0.992146i	$0.460080\pi$
$798$	0	0
$799$	−5731.32	−0.253767
$800$	0	0
$801$	0	0
$802$	66234.6	2.91624
$803$	−4424.97	−0.194463
$804$	0	0
$805$	0	0
$806$	44008.5	1.92324
$807$	0	0
$808$	34868.7	1.51816
$809$	27890.7	1.21210	0.606048	−	0.795428i	$-0.292754\pi$
$809$	27890.7	1.21210	0.606048	+	0.795428i	$0.292754\pi$
$810$	0	0
$811$	−12170.3	−0.526949	−0.263474	−	0.964666i	$-0.584868\pi$
$811$	−12170.3	−0.526949	−0.263474	+	0.964666i	$0.584868\pi$
$812$	−33144.9	−1.43246
$813$	0	0
$814$	5199.59	0.223889
$815$	0	0
$816$	0	0
$817$	−137.008	−0.00586696
$818$	−9079.53	−0.388091
$819$	0	0
$820$	0	0
$821$	30932.3	1.31491	0.657457	−	0.753492i	$-0.271632\pi$
$821$	30932.3	1.31491	0.657457	+	0.753492i	$0.271632\pi$
$822$	0	0
$823$	25949.0	1.09906	0.549528	−	0.835475i	$-0.314807\pi$
$823$	25949.0	1.09906	0.549528	+	0.835475i	$0.314807\pi$
$824$	−24814.7	−1.04911
$825$	0	0
$826$	−25259.0	−1.06401
$827$	27728.5	1.16592	0.582959	−	0.812502i	$-0.301895\pi$
$827$	27728.5	1.16592	0.582959	+	0.812502i	$0.301895\pi$
$828$	0	0
$829$	−10650.1	−0.446194	−0.223097	−	0.974796i	$-0.571617\pi$
$829$	−10650.1	−0.446194	−0.223097	+	0.974796i	$0.571617\pi$
$830$	0	0
$831$	0	0
$832$	60959.0	2.54011
$833$	−1357.06	−0.0564460
$834$	0	0
$835$	0	0
$836$	−409.563	−0.0169438
$837$	0	0
$838$	−56424.8	−2.32597
$839$	−29761.9	−1.22466	−0.612332	−	0.790601i	$-0.709768\pi$
$839$	−29761.9	−1.22466	−0.612332	+	0.790601i	$0.709768\pi$
$840$	0	0
$841$	30367.2	1.24512
$842$	70531.0	2.88677
$843$	0	0
$844$	−84603.2	−3.45043
$845$	0	0
$846$	0	0
$847$	4750.26	0.192705
$848$	26440.3	1.07071
$849$	0	0
$850$	0	0
$851$	−4160.80	−0.167603
$852$	0	0
$853$	24410.2	0.979824	0.489912	−	0.871772i	$-0.337029\pi$
$853$	24410.2	0.979824	0.489912	+	0.871772i	$0.337029\pi$
$854$	21376.9	0.856562
$855$	0	0
$856$	−89789.8	−3.58522
$857$	15189.3	0.605433	0.302717	−	0.953081i	$-0.402106\pi$
$857$	15189.3	0.605433	0.302717	+	0.953081i	$0.402106\pi$
$858$	0	0
$859$	−10339.1	−0.410668	−0.205334	−	0.978692i	$-0.565828\pi$
$859$	−10339.1	−0.410668	−0.205334	+	0.978692i	$0.565828\pi$
$860$	0	0
$861$	0	0
$862$	−42074.6	−1.66249
$863$	27566.6	1.08734	0.543671	−	0.839298i	$-0.317034\pi$
$863$	27566.6	1.08734	0.543671	+	0.839298i	$0.317034\pi$
$864$	0	0
$865$	0	0
$866$	−23557.2	−0.924373
$867$	0	0
$868$	−18301.0	−0.715642
$869$	−2026.22	−0.0790964
$870$	0	0
$871$	−33053.2	−1.28584
$872$	−25392.6	−0.986127
$873$	0	0
$874$	457.313	0.0176989
$875$	0	0
$876$	0	0
$877$	6032.49	0.232272	0.116136	−	0.993233i	$-0.462949\pi$
$877$	6032.49	0.232272	0.116136	+	0.993233i	$0.462949\pi$
$878$	68856.4	2.64669
$879$	0	0
$880$	0	0
$881$	−44253.2	−1.69231	−0.846156	−	0.532935i	$-0.821089\pi$
$881$	−44253.2	−1.69231	−0.846156	+	0.532935i	$0.821089\pi$
$882$	0	0
$883$	7744.13	0.295142	0.147571	−	0.989051i	$-0.452854\pi$
$883$	7744.13	0.295142	0.147571	+	0.989051i	$0.452854\pi$
$884$	−35923.1	−1.36677
$885$	0	0
$886$	−63978.8	−2.42597
$887$	18632.5	0.705318	0.352659	−	0.935752i	$-0.385278\pi$
$887$	18632.5	0.705318	0.352659	+	0.935752i	$0.385278\pi$
$888$	0	0
$889$	−1344.22	−0.0507130
$890$	0	0
$891$	0	0
$892$	−47785.3	−1.79369
$893$	163.989	0.00614522
$894$	0	0
$895$	0	0
$896$	−9872.65	−0.368105
$897$	0	0
$898$	−61565.7	−2.28783
$899$	30233.7	1.12164
$900$	0	0
$901$	−3988.97	−0.147494
$902$	−54725.0	−2.02011
$903$	0	0
$904$	106825.	3.93025
$905$	0	0
$906$	0	0
$907$	−1258.19	−0.0460612	−0.0230306	−	0.999735i	$-0.507332\pi$
$907$	−1258.19	−0.0460612	−0.0230306	+	0.999735i	$0.507332\pi$
$908$	11875.4	0.434030
$909$	0	0
$910$	0	0
$911$	−27133.6	−0.986800	−0.493400	−	0.869802i	$-0.664246\pi$
$911$	−27133.6	−0.986800	−0.493400	+	0.869802i	$0.664246\pi$
$912$	0	0
$913$	−26654.6	−0.966197
$914$	−51724.9	−1.87189
$915$	0	0
$916$	−93476.5	−3.37178
$917$	14580.3	0.525064
$918$	0	0
$919$	42534.1	1.52673	0.763367	−	0.645965i	$-0.223545\pi$
$919$	42534.1	1.52673	0.763367	+	0.645965i	$0.223545\pi$
$920$	0	0
$921$	0	0
$922$	7138.42	0.254980
$923$	−35679.7	−1.27238
$924$	0	0
$925$	0	0
$926$	35143.1	1.24716
$927$	0	0
$928$	106552.	3.76911
$929$	−41849.0	−1.47795	−0.738977	−	0.673730i	$-0.764691\pi$
$929$	−41849.0	−1.47795	−0.738977	+	0.673730i	$0.764691\pi$
$930$	0	0
$931$	38.8294	0.00136690
$932$	−103612.	−3.64155
$933$	0	0
$934$	−78051.9	−2.73441
$935$	0	0
$936$	0	0
$937$	33281.9	1.16038	0.580188	−	0.814482i	$-0.302979\pi$
$937$	33281.9	1.16038	0.580188	+	0.814482i	$0.302979\pi$
$938$	19179.5	0.667627
$939$	0	0
$940$	0	0
$941$	−12740.6	−0.441373	−0.220687	−	0.975345i	$-0.570830\pi$
$941$	−12740.6	−0.441373	−0.220687	+	0.975345i	$0.570830\pi$
$942$	0	0
$943$	43791.9	1.51226
$944$	124662.	4.29810
$945$	0	0
$946$	23465.5	0.806478
$947$	20681.5	0.709670	0.354835	−	0.934929i	$-0.384537\pi$
$947$	20681.5	0.709670	0.354835	+	0.934929i	$0.384537\pi$
$948$	0	0
$949$	11105.1	0.379861
$950$	0	0
$951$	0	0
$952$	12603.7	0.429084
$953$	6702.50	0.227823	0.113912	−	0.993491i	$-0.463662\pi$
$953$	6702.50	0.227823	0.113912	+	0.993491i	$0.463662\pi$
$954$	0	0
$955$	0	0
$956$	−22815.2	−0.771857
$957$	0	0
$958$	81291.6	2.74156
$959$	547.196	0.0184253
$960$	0	0
$961$	−13097.4	−0.439643
$962$	−13049.2	−0.437341
$963$	0	0
$964$	−71824.6	−2.39970
$965$	0	0
$966$	0	0
$967$	36459.9	1.21248	0.606242	−	0.795280i	$-0.292676\pi$
$967$	36459.9	1.21248	0.606242	+	0.795280i	$0.292676\pi$
$968$	−44117.9	−1.46488
$969$	0	0
$970$	0	0
$971$	15789.7	0.521850	0.260925	−	0.965359i	$-0.415973\pi$
$971$	15789.7	0.521850	0.260925	+	0.965359i	$0.415973\pi$
$972$	0	0
$973$	−9755.67	−0.321431
$974$	51297.0	1.68754
$975$	0	0
$976$	−105503.	−3.46011
$977$	−34429.2	−1.12742	−0.563709	−	0.825974i	$-0.690626\pi$
$977$	−34429.2	−1.12742	−0.563709	+	0.825974i	$0.690626\pi$
$978$	0	0
$979$	16654.9	0.543710
$980$	0	0
$981$	0	0
$982$	−107031.	−3.47809
$983$	−21047.5	−0.682922	−0.341461	−	0.939896i	$-0.610922\pi$
$983$	−21047.5	−0.682922	−0.341461	+	0.939896i	$0.610922\pi$
$984$	0	0
$985$	0	0
$986$	−34436.1	−1.11224
$987$	0	0
$988$	1027.86	0.0330978
$989$	−18777.5	−0.603730
$990$	0	0
$991$	23839.1	0.764150	0.382075	−	0.924131i	$-0.375210\pi$
$991$	23839.1	0.764150	0.382075	+	0.924131i	$0.375210\pi$
$992$	58832.7	1.88301
$993$	0	0
$994$	20703.5	0.660640
$995$	0	0
$996$	0	0
$997$	−7853.62	−0.249475	−0.124738	−	0.992190i	$-0.539809\pi$
$997$	−7853.62	−0.249475	−0.124738	+	0.992190i	$0.539809\pi$
$998$	−6957.12	−0.220665
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bq.1.5		5
3.2	odd	2		175.4.a.i.1.1		5
5.2	odd	4		315.4.d.c.64.10		10
5.3	odd	4		315.4.d.c.64.1		10
5.4	even	2		1575.4.a.bn.1.1		5
15.2	even	4		35.4.b.a.29.1	✓	10
15.8	even	4		35.4.b.a.29.10	yes	10
15.14	odd	2		175.4.a.j.1.5		5
21.20	even	2		1225.4.a.be.1.1		5
60.23	odd	4		560.4.g.f.449.6		10
60.47	odd	4		560.4.g.f.449.5		10
105.2	even	12		245.4.j.e.214.1		20
105.17	odd	12		245.4.j.f.79.10		20
105.23	even	12		245.4.j.e.214.10		20
105.32	even	12		245.4.j.e.79.10		20
105.38	odd	12		245.4.j.f.79.1		20
105.47	odd	12		245.4.j.f.214.1		20
105.53	even	12		245.4.j.e.79.1		20
105.62	odd	4		245.4.b.d.99.1		10
105.68	odd	12		245.4.j.f.214.10		20
105.83	odd	4		245.4.b.d.99.10		10
105.104	even	2		1225.4.a.bh.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.1	✓	10	15.2	even	4
35.4.b.a.29.10	yes	10	15.8	even	4
175.4.a.i.1.1		5	3.2	odd	2
175.4.a.j.1.5		5	15.14	odd	2
245.4.b.d.99.1		10	105.62	odd	4
245.4.b.d.99.10		10	105.83	odd	4
245.4.j.e.79.1		20	105.53	even	12
245.4.j.e.79.10		20	105.32	even	12
245.4.j.e.214.1		20	105.2	even	12
245.4.j.e.214.10		20	105.23	even	12
245.4.j.f.79.1		20	105.38	odd	12
245.4.j.f.79.10		20	105.17	odd	12
245.4.j.f.214.1		20	105.47	odd	12
245.4.j.f.214.10		20	105.68	odd	12
315.4.d.c.64.1		10	5.3	odd	4
315.4.d.c.64.10		10	5.2	odd	4
560.4.g.f.449.5		10	60.47	odd	4
560.4.g.f.449.6		10	60.23	odd	4
1225.4.a.be.1.1		5	21.20	even	2
1225.4.a.bh.1.5		5	105.104	even	2
1575.4.a.bn.1.1		5	5.4	even	2
1575.4.a.bq.1.5		5	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+5.31366 q^{2} +20.2350 q^{4} -7.00000 q^{7} +65.0123 q^{8} -25.5420 q^{11} +64.1014 q^{13} -37.1956 q^{14} +183.574 q^{16} -27.6952 q^{17} +0.792436 q^{19} -135.721 q^{22} +108.606 q^{23} +340.613 q^{26} -141.645 q^{28} +234.000 q^{29} +129.204 q^{31} +455.349 q^{32} -147.163 q^{34} -38.3108 q^{37} +4.21073 q^{38} +403.216 q^{41} -172.895 q^{43} -516.840 q^{44} +577.097 q^{46} +206.943 q^{47} +49.0000 q^{49} +1297.09 q^{52} +144.031 q^{53} -455.086 q^{56} +1243.40 q^{58} +679.086 q^{59} -574.717 q^{61} +686.544 q^{62} +950.977 q^{64} -515.640 q^{67} -560.411 q^{68} -556.612 q^{71} +173.243 q^{73} -203.571 q^{74} +16.0349 q^{76} +178.794 q^{77} +79.3290 q^{79} +2142.55 q^{82} +1043.56 q^{83} -918.703 q^{86} -1660.54 q^{88} -652.060 q^{89} -448.710 q^{91} +2197.65 q^{92} +1099.62 q^{94} -515.714 q^{97} +260.369 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 18 q^{4} - 35 q^{7} + 42 q^{8} - 42 q^{11} - 34 q^{13} - 28 q^{14} + 74 q^{16} + 238 q^{17} - 36 q^{19} - 358 q^{22} + 152 q^{23} + 310 q^{26} - 126 q^{28} + 44 q^{29} + 60 q^{31} + 710 q^{32}+ \cdots + 196 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 18 * q^4 - 35 * q^7 + 42 * q^8 - 42 * q^11 - 34 * q^13 - 28 * q^14 + 74 * q^16 + 238 * q^17 - 36 * q^19 - 358 * q^22 + 152 * q^23 + 310 * q^26 - 126 * q^28 + 44 * q^29 + 60 * q^31 + 710 * q^32 - 482 * q^34 - 312 * q^37 + 280 * q^38 + 426 * q^41 - 304 * q^43 - 712 * q^44 + 88 * q^46 + 370 * q^47 + 245 * q^49 + 1156 * q^52 + 976 * q^53 - 294 * q^56 + 2722 * q^58 + 432 * q^59 - 442 * q^61 - 956 * q^62 + 1362 * q^64 + 804 * q^67 - 420 * q^68 - 440 * q^71 - 564 * q^73 + 1512 * q^74 - 1336 * q^76 + 294 * q^77 + 1790 * q^79 + 3480 * q^82 + 1656 * q^83 - 1216 * q^86 + 1092 * q^88 - 746 * q^89 + 238 * q^91 + 572 * q^92 - 826 * q^94 - 518 * q^97 + 196 * q^98

Introduction

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