LMFDB - Embedded newform 2475.4.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.91035289.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 285x^{2} + 286x + 19616 $ x^4 - 2x^3 - 285x^2 + 286*x + 19616
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-12.6191$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{2} -5.56155 q^{4} +24.2670 q^{7} -21.1771 q^{8} +O(q^{10})$	$q+1.56155 q^{2} -5.56155 q^{4} +24.2670 q^{7} -21.1771 q^{8} +11.0000 q^{11} +84.2360 q^{13} +37.8942 q^{14} +11.4233 q^{16} +40.9641 q^{17} +120.887 q^{19} +17.1771 q^{22} +9.94182 q^{23} +131.539 q^{26} -134.962 q^{28} -196.860 q^{29} +151.123 q^{31} +187.255 q^{32} +63.9675 q^{34} -253.477 q^{37} +188.772 q^{38} +179.351 q^{41} -90.4851 q^{43} -61.1771 q^{44} +15.5247 q^{46} +483.434 q^{47} +245.887 q^{49} -468.483 q^{52} -567.348 q^{53} -513.904 q^{56} -307.407 q^{58} +491.696 q^{59} +127.008 q^{61} +235.986 q^{62} +201.022 q^{64} +628.226 q^{67} -227.824 q^{68} -309.976 q^{71} -1144.85 q^{73} -395.817 q^{74} -672.322 q^{76} +266.937 q^{77} -87.4620 q^{79} +280.067 q^{82} -1040.13 q^{83} -141.297 q^{86} -232.948 q^{88} -390.723 q^{89} +2044.15 q^{91} -55.2920 q^{92} +754.908 q^{94} +165.548 q^{97} +383.966 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 14 q^{4} + 11 q^{7} + 6 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 14 * q^4 + 11 * q^7 + 6 * q^8	$ 4 q - 2 q^{2} - 14 q^{4} + 11 q^{7} + 6 q^{8} + 44 q^{11} - 25 q^{13} + 3 q^{14} - 78 q^{16} + 85 q^{17} + 118 q^{19} - 22 q^{22} - 10 q^{23} + 157 q^{26} - 47 q^{28} - 251 q^{29} - 135 q^{31} + 246 q^{32} + 289 q^{34} - 419 q^{37} - 76 q^{38} + 103 q^{41} - 15 q^{43} - 154 q^{44} - 420 q^{46} + 665 q^{47} + 1003 q^{49} - 57 q^{52} - 116 q^{53} - 77 q^{56} - 189 q^{58} - 951 q^{59} - 350 q^{61} - 213 q^{62} + 1538 q^{64} - 266 q^{67} - 629 q^{68} - 1526 q^{71} + 566 q^{73} - 1091 q^{74} - 396 q^{76} + 121 q^{77} + 2567 q^{79} - 9 q^{82} - 961 q^{83} - 1837 q^{86} + 66 q^{88} - 1053 q^{89} + 2150 q^{91} + 460 q^{92} + 2209 q^{94} + 172 q^{97} - 1819 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 14 * q^4 + 11 * q^7 + 6 * q^8 + 44 * q^11 - 25 * q^13 + 3 * q^14 - 78 * q^16 + 85 * q^17 + 118 * q^19 - 22 * q^22 - 10 * q^23 + 157 * q^26 - 47 * q^28 - 251 * q^29 - 135 * q^31 + 246 * q^32 + 289 * q^34 - 419 * q^37 - 76 * q^38 + 103 * q^41 - 15 * q^43 - 154 * q^44 - 420 * q^46 + 665 * q^47 + 1003 * q^49 - 57 * q^52 - 116 * q^53 - 77 * q^56 - 189 * q^58 - 951 * q^59 - 350 * q^61 - 213 * q^62 + 1538 * q^64 - 266 * q^67 - 629 * q^68 - 1526 * q^71 + 566 * q^73 - 1091 * q^74 - 396 * q^76 + 121 * q^77 + 2567 * q^79 - 9 * q^82 - 961 * q^83 - 1837 * q^86 + 66 * q^88 - 1053 * q^89 + 2150 * q^91 + 460 * q^92 + 2209 * q^94 + 172 * q^97 - 1819 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.56155	0.552092	0.276046	−	0.961144i	$-0.410976\pi$
$2$	1.56155	0.552092	0.276046	+	0.961144i	$0.410976\pi$
$3$	0	0
$3$	0	0
$4$	−5.56155	−0.695194
$5$	0	0
$5$	0	0
$6$	0	0
$7$	24.2670	1.31029	0.655147	−	0.755501i	$-0.272606\pi$
$7$	24.2670	1.31029	0.655147	+	0.755501i	$0.272606\pi$
$8$	−21.1771	−0.935904
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	84.2360	1.79714	0.898571	−	0.438828i	$-0.144606\pi$
$13$	84.2360	1.79714	0.898571	+	0.438828i	$0.144606\pi$
$14$	37.8942	0.723404
$15$	0	0
$16$	11.4233	0.178489
$17$	40.9641	0.584426	0.292213	−	0.956353i	$-0.405608\pi$
$17$	40.9641	0.584426	0.292213	+	0.956353i	$0.405608\pi$
$18$	0	0
$19$	120.887	1.45966	0.729828	−	0.683630i	$-0.239600\pi$
$19$	120.887	1.45966	0.729828	+	0.683630i	$0.239600\pi$
$20$	0	0
$21$	0	0
$22$	17.1771	0.166462
$23$	9.94182	0.0901310	0.0450655	−	0.998984i	$-0.485650\pi$
$23$	9.94182	0.0901310	0.0450655	+	0.998984i	$0.485650\pi$
$24$	0	0
$25$	0	0
$26$	131.539	0.992189
$27$	0	0
$28$	−134.962	−0.910909
$29$	−196.860	−1.26055	−0.630275	−	0.776372i	$-0.717058\pi$
$29$	−196.860	−1.26055	−0.630275	+	0.776372i	$0.717058\pi$
$30$	0	0
$31$	151.123	0.875562	0.437781	−	0.899082i	$-0.355764\pi$
$31$	151.123	0.875562	0.437781	+	0.899082i	$0.355764\pi$
$32$	187.255	1.03445
$33$	0	0
$34$	63.9675	0.322657
$35$	0	0
$36$	0	0
$37$	−253.477	−1.12625	−0.563126	−	0.826371i	$-0.690401\pi$
$37$	−253.477	−1.12625	−0.563126	+	0.826371i	$0.690401\pi$
$38$	188.772	0.805865
$39$	0	0
$40$	0	0
$41$	179.351	0.683170	0.341585	−	0.939851i	$-0.389036\pi$
$41$	179.351	0.683170	0.341585	+	0.939851i	$0.389036\pi$
$42$	0	0
$43$	−90.4851	−0.320903	−0.160452	−	0.987044i	$-0.551295\pi$
$43$	−90.4851	−0.320903	−0.160452	+	0.987044i	$0.551295\pi$
$44$	−61.1771	−0.209609
$45$	0	0
$46$	15.5247	0.0497606
$47$	483.434	1.50034	0.750172	−	0.661243i	$-0.229971\pi$
$47$	483.434	1.50034	0.750172	+	0.661243i	$0.229971\pi$
$48$	0	0
$49$	245.887	0.716873
$50$	0	0
$51$	0	0
$52$	−468.483	−1.24936
$53$	−567.348	−1.47040	−0.735200	−	0.677850i	$-0.762912\pi$
$53$	−567.348	−1.47040	−0.735200	+	0.677850i	$0.762912\pi$
$54$	0	0
$55$	0	0
$56$	−513.904	−1.22631
$57$	0	0
$58$	−307.407	−0.695939
$59$	491.696	1.08497	0.542487	−	0.840064i	$-0.317483\pi$
$59$	491.696	1.08497	0.542487	+	0.840064i	$0.317483\pi$
$60$	0	0
$61$	127.008	0.266585	0.133292	−	0.991077i	$-0.457445\pi$
$61$	127.008	0.266585	0.133292	+	0.991077i	$0.457445\pi$
$62$	235.986	0.483391
$63$	0	0
$64$	201.022	0.392621
$65$	0	0
$66$	0	0
$67$	628.226	1.14552	0.572762	−	0.819722i	$-0.305872\pi$
$67$	628.226	1.14552	0.572762	+	0.819722i	$0.305872\pi$
$68$	−227.824	−0.406290
$69$	0	0
$70$	0	0
$71$	−309.976	−0.518133	−0.259066	−	0.965859i	$-0.583415\pi$
$71$	−309.976	−0.518133	−0.259066	+	0.965859i	$0.583415\pi$
$72$	0	0
$73$	−1144.85	−1.83554	−0.917772	−	0.397109i	$-0.870014\pi$
$73$	−1144.85	−1.83554	−0.917772	+	0.397109i	$0.870014\pi$
$74$	−395.817	−0.621795
$75$	0	0
$76$	−672.322	−1.01474
$77$	266.937	0.395069
$78$	0	0
$79$	−87.4620	−0.124560	−0.0622800	−	0.998059i	$-0.519837\pi$
$79$	−87.4620	−0.124560	−0.0622800	+	0.998059i	$0.519837\pi$
$80$	0	0
$81$	0	0
$82$	280.067	0.377173
$83$	−1040.13	−1.37553	−0.687764	−	0.725934i	$-0.741408\pi$
$83$	−1040.13	−1.37553	−0.687764	+	0.725934i	$0.741408\pi$
$84$	0	0
$85$	0	0
$86$	−141.297	−0.177168
$87$	0	0
$88$	−232.948	−0.282186
$89$	−390.723	−0.465355	−0.232677	−	0.972554i	$-0.574749\pi$
$89$	−390.723	−0.465355	−0.232677	+	0.972554i	$0.574749\pi$
$90$	0	0
$91$	2044.15	2.35479
$92$	−55.2920	−0.0626585
$93$	0	0
$94$	754.908	0.828328
$95$	0	0
$96$	0	0
$97$	165.548	0.173287	0.0866433	−	0.996239i	$-0.472386\pi$
$97$	165.548	0.173287	0.0866433	+	0.996239i	$0.472386\pi$
$98$	383.966	0.395780
$99$	0	0
$100$	0	0
$101$	−1510.35	−1.48798	−0.743990	−	0.668191i	$-0.767069\pi$
$101$	−1510.35	−1.48798	−0.743990	+	0.668191i	$0.767069\pi$
$102$	0	0
$103$	1245.78	1.19175	0.595876	−	0.803076i	$-0.296805\pi$
$103$	1245.78	1.19175	0.595876	+	0.803076i	$0.296805\pi$
$104$	−1783.87	−1.68195
$105$	0	0
$106$	−885.944	−0.811797
$107$	550.568	0.497434	0.248717	−	0.968576i	$-0.419991\pi$
$107$	550.568	0.497434	0.248717	+	0.968576i	$0.419991\pi$
$108$	0	0
$109$	1380.30	1.21293	0.606463	−	0.795112i	$-0.292588\pi$
$109$	1380.30	1.21293	0.606463	+	0.795112i	$0.292588\pi$
$110$	0	0
$111$	0	0
$112$	277.209	0.233873
$113$	−1017.96	−0.847445	−0.423723	−	0.905792i	$-0.639277\pi$
$113$	−1017.96	−0.847445	−0.423723	+	0.905792i	$0.639277\pi$
$114$	0	0
$115$	0	0
$116$	1094.85	0.876326
$117$	0	0
$118$	767.810	0.599005
$119$	994.075	0.765770
$120$	0	0
$121$	121.000	0.0909091
$122$	198.329	0.147179
$123$	0	0
$124$	−840.477	−0.608686
$125$	0	0
$126$	0	0
$127$	−1181.86	−0.825770	−0.412885	−	0.910783i	$-0.635479\pi$
$127$	−1181.86	−0.825770	−0.412885	+	0.910783i	$0.635479\pi$
$128$	−1184.13	−0.817683
$129$	0	0
$130$	0	0
$131$	2615.26	1.74425	0.872124	−	0.489284i	$-0.162742\pi$
$131$	2615.26	1.74425	0.872124	+	0.489284i	$0.162742\pi$
$132$	0	0
$133$	2933.58	1.91258
$134$	981.009	0.632435
$135$	0	0
$136$	−867.499	−0.546966
$137$	465.369	0.290213	0.145106	−	0.989416i	$-0.453648\pi$
$137$	465.369	0.290213	0.145106	+	0.989416i	$0.453648\pi$
$138$	0	0
$139$	2809.52	1.71439	0.857196	−	0.514990i	$-0.172204\pi$
$139$	2809.52	1.71439	0.857196	+	0.514990i	$0.172204\pi$
$140$	0	0
$141$	0	0
$142$	−484.045	−0.286057
$143$	926.596	0.541859
$144$	0	0
$145$	0	0
$146$	−1787.75	−1.01339
$147$	0	0
$148$	1409.72	0.782963
$149$	583.496	0.320818	0.160409	−	0.987051i	$-0.448719\pi$
$149$	583.496	0.320818	0.160409	+	0.987051i	$0.448719\pi$
$150$	0	0
$151$	1244.33	0.670610	0.335305	−	0.942110i	$-0.391161\pi$
$151$	1244.33	0.670610	0.335305	+	0.942110i	$0.391161\pi$
$152$	−2560.04	−1.36610
$153$	0	0
$154$	416.836	0.218114
$155$	0	0
$156$	0	0
$157$	−303.991	−0.154529	−0.0772647	−	0.997011i	$-0.524619\pi$
$157$	−303.991	−0.154529	−0.0772647	+	0.997011i	$0.524619\pi$
$158$	−136.577	−0.0687687
$159$	0	0
$160$	0	0
$161$	241.258	0.118098
$162$	0	0
$163$	259.982	0.124929	0.0624643	−	0.998047i	$-0.480104\pi$
$163$	259.982	0.124929	0.0624643	+	0.998047i	$0.480104\pi$
$164$	−997.472	−0.474936
$165$	0	0
$166$	−1624.21	−0.759419
$167$	3468.36	1.60712	0.803562	−	0.595222i	$-0.202936\pi$
$167$	3468.36	1.60712	0.803562	+	0.595222i	$0.202936\pi$
$168$	0	0
$169$	4898.70	2.22972
$170$	0	0
$171$	0	0
$172$	503.237	0.223090
$173$	3864.34	1.69827	0.849134	−	0.528177i	$-0.177124\pi$
$173$	3864.34	1.69827	0.849134	+	0.528177i	$0.177124\pi$
$174$	0	0
$175$	0	0
$176$	125.656	0.0538164
$177$	0	0
$178$	−610.135	−0.256919
$179$	−163.870	−0.0684258	−0.0342129	−	0.999415i	$-0.510892\pi$
$179$	−163.870	−0.0684258	−0.0342129	+	0.999415i	$0.510892\pi$
$180$	0	0
$181$	−3200.77	−1.31443	−0.657214	−	0.753704i	$-0.728265\pi$
$181$	−3200.77	−1.31443	−0.657214	+	0.753704i	$0.728265\pi$
$182$	3192.05	1.30006
$183$	0	0
$184$	−210.539	−0.0843539
$185$	0	0
$186$	0	0
$187$	450.605	0.176211
$188$	−2688.65	−1.04303
$189$	0	0
$190$	0	0
$191$	−2421.03	−0.917171	−0.458585	−	0.888650i	$-0.651644\pi$
$191$	−2421.03	−0.917171	−0.458585	+	0.888650i	$0.651644\pi$
$192$	0	0
$193$	−2275.44	−0.848651	−0.424326	−	0.905510i	$-0.639489\pi$
$193$	−2275.44	−0.848651	−0.424326	+	0.905510i	$0.639489\pi$
$194$	258.511	0.0956702
$195$	0	0
$196$	−1367.52	−0.498366
$197$	−1957.94	−0.708107	−0.354054	−	0.935225i	$-0.615197\pi$
$197$	−1957.94	−0.708107	−0.354054	+	0.935225i	$0.615197\pi$
$198$	0	0
$199$	−1975.36	−0.703665	−0.351833	−	0.936063i	$-0.614441\pi$
$199$	−1975.36	−0.703665	−0.351833	+	0.936063i	$0.614441\pi$
$200$	0	0
$201$	0	0
$202$	−2358.50	−0.821502
$203$	−4777.19	−1.65169
$204$	0	0
$205$	0	0
$206$	1945.35	0.657958
$207$	0	0
$208$	962.252	0.320770
$209$	1329.76	0.440103
$210$	0	0
$211$	1179.92	0.384973	0.192486	−	0.981300i	$-0.438345\pi$
$211$	1179.92	0.384973	0.192486	+	0.981300i	$0.438345\pi$
$212$	3155.34	1.02221
$213$	0	0
$214$	859.741	0.274629
$215$	0	0
$216$	0	0
$217$	3667.29	1.14724
$218$	2155.41	0.669647
$219$	0	0
$220$	0	0
$221$	3450.65	1.05030
$222$	0	0
$223$	5742.73	1.72449	0.862247	−	0.506489i	$-0.169057\pi$
$223$	5742.73	1.72449	0.862247	+	0.506489i	$0.169057\pi$
$224$	4544.11	1.35543
$225$	0	0
$226$	−1589.59	−0.467868
$227$	−713.160	−0.208520	−0.104260	−	0.994550i	$-0.533247\pi$
$227$	−713.160	−0.208520	−0.104260	+	0.994550i	$0.533247\pi$
$228$	0	0
$229$	4438.09	1.28069	0.640343	−	0.768089i	$-0.278792\pi$
$229$	4438.09	1.28069	0.640343	+	0.768089i	$0.278792\pi$
$230$	0	0
$231$	0	0
$232$	4168.91	1.17975
$233$	614.179	0.172688	0.0863438	−	0.996265i	$-0.472482\pi$
$233$	614.179	0.172688	0.0863438	+	0.996265i	$0.472482\pi$
$234$	0	0
$235$	0	0
$236$	−2734.60	−0.754267
$237$	0	0
$238$	1552.30	0.422776
$239$	−5191.00	−1.40493	−0.702464	−	0.711720i	$-0.747917\pi$
$239$	−5191.00	−1.40493	−0.702464	+	0.711720i	$0.747917\pi$
$240$	0	0
$241$	−353.708	−0.0945408	−0.0472704	−	0.998882i	$-0.515052\pi$
$241$	−353.708	−0.0945408	−0.0472704	+	0.998882i	$0.515052\pi$
$242$	188.948	0.0501902
$243$	0	0
$244$	−706.360	−0.185328
$245$	0	0
$246$	0	0
$247$	10183.1	2.62321
$248$	−3200.34	−0.819442
$249$	0	0
$250$	0	0
$251$	−2612.91	−0.657073	−0.328537	−	0.944491i	$-0.606555\pi$
$251$	−2612.91	−0.657073	−0.328537	+	0.944491i	$0.606555\pi$
$252$	0	0
$253$	109.360	0.0271755
$254$	−1845.53	−0.455901
$255$	0	0
$256$	−3457.26	−0.844057
$257$	−2434.38	−0.590866	−0.295433	−	0.955364i	$-0.595464\pi$
$257$	−2434.38	−0.590866	−0.295433	+	0.955364i	$0.595464\pi$
$258$	0	0
$259$	−6151.12	−1.47572
$260$	0	0
$261$	0	0
$262$	4083.87	0.962986
$263$	7204.41	1.68914	0.844568	−	0.535449i	$-0.179857\pi$
$263$	7204.41	1.68914	0.844568	+	0.535449i	$0.179857\pi$
$264$	0	0
$265$	0	0
$266$	4580.93	1.05592
$267$	0	0
$268$	−3493.91	−0.796361
$269$	−5232.32	−1.18595	−0.592975	−	0.805221i	$-0.702047\pi$
$269$	−5232.32	−1.18595	−0.592975	+	0.805221i	$0.702047\pi$
$270$	0	0
$271$	6231.13	1.39673	0.698366	−	0.715741i	$-0.253911\pi$
$271$	6231.13	1.39673	0.698366	+	0.715741i	$0.253911\pi$
$272$	467.944	0.104314
$273$	0	0
$274$	726.698	0.160224
$275$	0	0
$276$	0	0
$277$	−4940.23	−1.07159	−0.535793	−	0.844349i	$-0.679987\pi$
$277$	−4940.23	−1.07159	−0.535793	+	0.844349i	$0.679987\pi$
$278$	4387.22	0.946503
$279$	0	0
$280$	0	0
$281$	1766.97	0.375119	0.187559	−	0.982253i	$-0.439942\pi$
$281$	1766.97	0.375119	0.187559	+	0.982253i	$0.439942\pi$
$282$	0	0
$283$	−5304.29	−1.11416	−0.557080	−	0.830459i	$-0.688078\pi$
$283$	−5304.29	−1.11416	−0.557080	+	0.830459i	$0.688078\pi$
$284$	1723.95	0.360203
$285$	0	0
$286$	1446.93	0.299156
$287$	4352.32	0.895155
$288$	0	0
$289$	−3234.95	−0.658446
$290$	0	0
$291$	0	0
$292$	6367.15	1.27606
$293$	−8356.39	−1.66616	−0.833081	−	0.553151i	$-0.813425\pi$
$293$	−8356.39	−1.66616	−0.833081	+	0.553151i	$0.813425\pi$
$294$	0	0
$295$	0	0
$296$	5367.90	1.05406
$297$	0	0
$298$	911.160	0.177121
$299$	837.459	0.161978
$300$	0	0
$301$	−2195.80	−0.420478
$302$	1943.09	0.370238
$303$	0	0
$304$	1380.93	0.260533
$305$	0	0
$306$	0	0
$307$	8558.29	1.59103	0.795517	−	0.605932i	$-0.207200\pi$
$307$	8558.29	1.59103	0.795517	+	0.605932i	$0.207200\pi$
$308$	−1484.58	−0.274649
$309$	0	0
$310$	0	0
$311$	−6135.19	−1.11863	−0.559316	−	0.828954i	$-0.688936\pi$
$311$	−6135.19	−1.11863	−0.559316	+	0.828954i	$0.688936\pi$
$312$	0	0
$313$	−4088.62	−0.738346	−0.369173	−	0.929361i	$-0.620359\pi$
$313$	−4088.62	−0.738346	−0.369173	+	0.929361i	$0.620359\pi$
$314$	−474.698	−0.0853145
$315$	0	0
$316$	486.425	0.0865934
$317$	7957.42	1.40988	0.704942	−	0.709265i	$-0.250973\pi$
$317$	7957.42	1.40988	0.704942	+	0.709265i	$0.250973\pi$
$318$	0	0
$319$	−2165.46	−0.380070
$320$	0	0
$321$	0	0
$322$	376.737	0.0652011
$323$	4952.04	0.853062
$324$	0	0
$325$	0	0
$326$	405.976	0.0689721
$327$	0	0
$328$	−3798.14	−0.639382
$329$	11731.5	1.96589
$330$	0	0
$331$	−3113.97	−0.517097	−0.258549	−	0.965998i	$-0.583244\pi$
$331$	−3113.97	−0.517097	−0.258549	+	0.965998i	$0.583244\pi$
$332$	5784.72	0.956259
$333$	0	0
$334$	5416.02	0.887280
$335$	0	0
$336$	0	0
$337$	8267.16	1.33632	0.668161	−	0.744016i	$-0.267082\pi$
$337$	8267.16	1.33632	0.668161	+	0.744016i	$0.267082\pi$
$338$	7649.58	1.23101
$339$	0	0
$340$	0	0
$341$	1662.35	0.263992
$342$	0	0
$343$	−2356.63	−0.370980
$344$	1916.21	0.300335
$345$	0	0
$346$	6034.37	0.937601
$347$	8673.03	1.34177	0.670883	−	0.741563i	$-0.265915\pi$
$347$	8673.03	1.34177	0.670883	+	0.741563i	$0.265915\pi$
$348$	0	0
$349$	3351.21	0.514000	0.257000	−	0.966411i	$-0.417266\pi$
$349$	3351.21	0.514000	0.257000	+	0.966411i	$0.417266\pi$
$350$	0	0
$351$	0	0
$352$	2059.80	0.311897
$353$	36.5920	0.00551727	0.00275864	−	0.999996i	$-0.499122\pi$
$353$	36.5920	0.00551727	0.00275864	+	0.999996i	$0.499122\pi$
$354$	0	0
$355$	0	0
$356$	2173.03	0.323512
$357$	0	0
$358$	−255.892	−0.0377773
$359$	12789.2	1.88019	0.940095	−	0.340912i	$-0.110736\pi$
$359$	12789.2	1.88019	0.940095	+	0.340912i	$0.110736\pi$
$360$	0	0
$361$	7754.77	1.13060
$362$	−4998.17	−0.725686
$363$	0	0
$364$	−11368.7	−1.63703
$365$	0	0
$366$	0	0
$367$	−1293.66	−0.184001	−0.0920005	−	0.995759i	$-0.529326\pi$
$367$	−1293.66	−0.184001	−0.0920005	+	0.995759i	$0.529326\pi$
$368$	113.568	0.0160874
$369$	0	0
$370$	0	0
$371$	−13767.8	−1.92666
$372$	0	0
$373$	8638.61	1.19917	0.599585	−	0.800311i	$-0.295332\pi$
$373$	8638.61	1.19917	0.599585	+	0.800311i	$0.295332\pi$
$374$	703.643	0.0972848
$375$	0	0
$376$	−10237.7	−1.40418
$377$	−16582.7	−2.26539
$378$	0	0
$379$	1853.53	0.251212	0.125606	−	0.992080i	$-0.459912\pi$
$379$	1853.53	0.251212	0.125606	+	0.992080i	$0.459912\pi$
$380$	0	0
$381$	0	0
$382$	−3780.57	−0.506363
$383$	3888.44	0.518773	0.259387	−	0.965774i	$-0.416480\pi$
$383$	3888.44	0.518773	0.259387	+	0.965774i	$0.416480\pi$
$384$	0	0
$385$	0	0
$386$	−3553.22	−0.468534
$387$	0	0
$388$	−920.701	−0.120468
$389$	10791.1	1.40651	0.703255	−	0.710938i	$-0.251729\pi$
$389$	10791.1	1.40651	0.703255	+	0.710938i	$0.251729\pi$
$390$	0	0
$391$	407.257	0.0526749
$392$	−5207.18	−0.670924
$393$	0	0
$394$	−3057.42	−0.390941
$395$	0	0
$396$	0	0
$397$	2875.92	0.363573	0.181787	−	0.983338i	$-0.441812\pi$
$397$	2875.92	0.363573	0.181787	+	0.983338i	$0.441812\pi$
$398$	−3084.63	−0.388488
$399$	0	0
$400$	0	0
$401$	−7615.88	−0.948426	−0.474213	−	0.880410i	$-0.657267\pi$
$401$	−7615.88	−0.948426	−0.474213	+	0.880410i	$0.657267\pi$
$402$	0	0
$403$	12730.0	1.57351
$404$	8399.92	1.03443
$405$	0	0
$406$	−7459.84	−0.911886
$407$	−2788.24	−0.339578
$408$	0	0
$409$	−3791.51	−0.458382	−0.229191	−	0.973381i	$-0.573608\pi$
$409$	−3791.51	−0.458382	−0.229191	+	0.973381i	$0.573608\pi$
$410$	0	0
$411$	0	0
$412$	−6928.48	−0.828500
$413$	11932.0	1.42163
$414$	0	0
$415$	0	0
$416$	15773.6	1.85905
$417$	0	0
$418$	2076.49	0.242978
$419$	−8806.66	−1.02681	−0.513405	−	0.858146i	$-0.671616\pi$
$419$	−8806.66	−1.02681	−0.513405	+	0.858146i	$0.671616\pi$
$420$	0	0
$421$	−7730.71	−0.894945	−0.447473	−	0.894298i	$-0.647676\pi$
$421$	−7730.71	−0.894945	−0.447473	+	0.894298i	$0.647676\pi$
$422$	1842.51	0.212541
$423$	0	0
$424$	12014.8	1.37615
$425$	0	0
$426$	0	0
$427$	3082.10	0.349305
$428$	−3062.01	−0.345813
$429$	0	0
$430$	0	0
$431$	−9017.97	−1.00784	−0.503922	−	0.863749i	$-0.668110\pi$
$431$	−9017.97	−1.00784	−0.503922	+	0.863749i	$0.668110\pi$
$432$	0	0
$433$	13799.1	1.53150	0.765751	−	0.643137i	$-0.222367\pi$
$433$	13799.1	1.53150	0.765751	+	0.643137i	$0.222367\pi$
$434$	5726.67	0.633385
$435$	0	0
$436$	−7676.62	−0.843219
$437$	1201.84	0.131560
$438$	0	0
$439$	464.472	0.0504967	0.0252483	−	0.999681i	$-0.491962\pi$
$439$	464.472	0.0504967	0.0252483	+	0.999681i	$0.491962\pi$
$440$	0	0
$441$	0	0
$442$	5388.37	0.579861
$443$	5796.17	0.621635	0.310818	−	0.950470i	$-0.399397\pi$
$443$	5796.17	0.621635	0.310818	+	0.950470i	$0.399397\pi$
$444$	0	0
$445$	0	0
$446$	8967.58	0.952079
$447$	0	0
$448$	4878.20	0.514449
$449$	8955.57	0.941291	0.470645	−	0.882323i	$-0.344021\pi$
$449$	8955.57	0.941291	0.470645	+	0.882323i	$0.344021\pi$
$450$	0	0
$451$	1972.87	0.205984
$452$	5661.42	0.589139
$453$	0	0
$454$	−1113.64	−0.115122
$455$	0	0
$456$	0	0
$457$	10067.8	1.03053	0.515267	−	0.857030i	$-0.327693\pi$
$457$	10067.8	1.03053	0.515267	+	0.857030i	$0.327693\pi$
$458$	6930.31	0.707057
$459$	0	0
$460$	0	0
$461$	8293.06	0.837844	0.418922	−	0.908022i	$-0.362408\pi$
$461$	8293.06	0.837844	0.418922	+	0.908022i	$0.362408\pi$
$462$	0	0
$463$	−9901.59	−0.993879	−0.496939	−	0.867785i	$-0.665543\pi$
$463$	−9901.59	−0.993879	−0.496939	+	0.867785i	$0.665543\pi$
$464$	−2248.79	−0.224994
$465$	0	0
$466$	959.073	0.0953395
$467$	−10915.0	−1.08156	−0.540779	−	0.841165i	$-0.681870\pi$
$467$	−10915.0	−1.08156	−0.540779	+	0.841165i	$0.681870\pi$
$468$	0	0
$469$	15245.2	1.50097
$470$	0	0
$471$	0	0
$472$	−10412.7	−1.01543
$473$	−995.336	−0.0967560
$474$	0	0
$475$	0	0
$476$	−5528.60	−0.532359
$477$	0	0
$478$	−8106.02	−0.775650
$479$	−5725.15	−0.546114	−0.273057	−	0.961998i	$-0.588035\pi$
$479$	−5725.15	−0.546114	−0.273057	+	0.961998i	$0.588035\pi$
$480$	0	0
$481$	−21351.9	−2.02403
$482$	−552.334	−0.0521953
$483$	0	0
$484$	−672.948	−0.0631995
$485$	0	0
$486$	0	0
$487$	16620.3	1.54648	0.773242	−	0.634111i	$-0.218634\pi$
$487$	16620.3	1.54648	0.773242	+	0.634111i	$0.218634\pi$
$488$	−2689.65	−0.249498
$489$	0	0
$490$	0	0
$491$	9769.75	0.897969	0.448985	−	0.893540i	$-0.351786\pi$
$491$	9769.75	0.897969	0.448985	+	0.893540i	$0.351786\pi$
$492$	0	0
$493$	−8064.17	−0.736698
$494$	15901.4	1.44825
$495$	0	0
$496$	1726.32	0.156278
$497$	−7522.20	−0.678907
$498$	0	0
$499$	−1886.64	−0.169254	−0.0846270	−	0.996413i	$-0.526970\pi$
$499$	−1886.64	−0.169254	−0.0846270	+	0.996413i	$0.526970\pi$
$500$	0	0
$501$	0	0
$502$	−4080.20	−0.362765
$503$	2920.56	0.258889	0.129445	−	0.991587i	$-0.458681\pi$
$503$	2920.56	0.258889	0.129445	+	0.991587i	$0.458681\pi$
$504$	0	0
$505$	0	0
$506$	170.771	0.0150034
$507$	0	0
$508$	6572.96	0.574070
$509$	−7956.30	−0.692842	−0.346421	−	0.938079i	$-0.612603\pi$
$509$	−7956.30	−0.692842	−0.346421	+	0.938079i	$0.612603\pi$
$510$	0	0
$511$	−27782.1	−2.40510
$512$	4074.36	0.351686
$513$	0	0
$514$	−3801.41	−0.326212
$515$	0	0
$516$	0	0
$517$	5317.78	0.452371
$518$	−9605.30	−0.814735
$519$	0	0
$520$	0	0
$521$	−20260.6	−1.70371	−0.851853	−	0.523780i	$-0.824521\pi$
$521$	−20260.6	−1.70371	−0.851853	+	0.523780i	$0.824521\pi$
$522$	0	0
$523$	3457.70	0.289091	0.144546	−	0.989498i	$-0.453828\pi$
$523$	3457.70	0.289091	0.144546	+	0.989498i	$0.453828\pi$
$524$	−14544.9	−1.21259
$525$	0	0
$526$	11250.1	0.932559
$527$	6190.60	0.511702
$528$	0	0
$529$	−12068.2	−0.991876
$530$	0	0
$531$	0	0
$532$	−16315.2	−1.32961
$533$	15107.8	1.22775
$534$	0	0
$535$	0	0
$536$	−13304.0	−1.07210
$537$	0	0
$538$	−8170.55	−0.654753
$539$	2704.76	0.216145
$540$	0	0
$541$	−3020.10	−0.240008	−0.120004	−	0.992773i	$-0.538291\pi$
$541$	−3020.10	−0.240008	−0.120004	+	0.992773i	$0.538291\pi$
$542$	9730.24	0.771125
$543$	0	0
$544$	7670.71	0.604557
$545$	0	0
$546$	0	0
$547$	17068.8	1.33421	0.667103	−	0.744965i	$-0.267534\pi$
$547$	17068.8	1.33421	0.667103	+	0.744965i	$0.267534\pi$
$548$	−2588.17	−0.201754
$549$	0	0
$550$	0	0
$551$	−23797.9	−1.83997
$552$	0	0
$553$	−2122.44	−0.163210
$554$	−7714.42	−0.591614
$555$	0	0
$556$	−15625.3	−1.19184
$557$	4965.05	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$557$	4965.05	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$558$	0	0
$559$	−7622.10	−0.576709
$560$	0	0
$561$	0	0
$562$	2759.21	0.207100
$563$	−18839.1	−1.41025	−0.705126	−	0.709082i	$-0.749110\pi$
$563$	−18839.1	−1.41025	−0.705126	+	0.709082i	$0.749110\pi$
$564$	0	0
$565$	0	0
$566$	−8282.92	−0.615119
$567$	0	0
$568$	6564.40	0.484922
$569$	−25785.9	−1.89983	−0.949914	−	0.312513i	$-0.898829\pi$
$569$	−25785.9	−1.89983	−0.949914	+	0.312513i	$0.898829\pi$
$570$	0	0
$571$	1732.89	0.127004	0.0635018	−	0.997982i	$-0.479773\pi$
$571$	1732.89	0.127004	0.0635018	+	0.997982i	$0.479773\pi$
$572$	−5153.31	−0.376697
$573$	0	0
$574$	6796.38	0.494208
$575$	0	0
$576$	0	0
$577$	−8872.77	−0.640170	−0.320085	−	0.947389i	$-0.603711\pi$
$577$	−8872.77	−0.640170	−0.320085	+	0.947389i	$0.603711\pi$
$578$	−5051.54	−0.363523
$579$	0	0
$580$	0	0
$581$	−25240.8	−1.80235
$582$	0	0
$583$	−6240.83	−0.443343
$584$	24244.6	1.71789
$585$	0	0
$586$	−13048.9	−0.919875
$587$	−11144.0	−0.783582	−0.391791	−	0.920054i	$-0.628144\pi$
$587$	−11144.0	−0.783582	−0.391791	+	0.920054i	$0.628144\pi$
$588$	0	0
$589$	18268.8	1.27802
$590$	0	0
$591$	0	0
$592$	−2895.54	−0.201023
$593$	6643.23	0.460042	0.230021	−	0.973186i	$-0.426121\pi$
$593$	6643.23	0.460042	0.230021	+	0.973186i	$0.426121\pi$
$594$	0	0
$595$	0	0
$596$	−3245.14	−0.223031
$597$	0	0
$598$	1307.74	0.0894270
$599$	−26138.0	−1.78292	−0.891462	−	0.453096i	$-0.850320\pi$
$599$	−26138.0	−1.78292	−0.891462	+	0.453096i	$0.850320\pi$
$600$	0	0
$601$	8628.54	0.585633	0.292817	−	0.956169i	$-0.405407\pi$
$601$	8628.54	0.585633	0.292817	+	0.956169i	$0.405407\pi$
$602$	−3428.86	−0.232143
$603$	0	0
$604$	−6920.40	−0.466204
$605$	0	0
$606$	0	0
$607$	−4043.74	−0.270396	−0.135198	−	0.990819i	$-0.543167\pi$
$607$	−4043.74	−0.270396	−0.135198	+	0.990819i	$0.543167\pi$
$608$	22636.7	1.50994
$609$	0	0
$610$	0	0
$611$	40722.6	2.69633
$612$	0	0
$613$	−5443.62	−0.358671	−0.179336	−	0.983788i	$-0.557395\pi$
$613$	−5443.62	−0.358671	−0.179336	+	0.983788i	$0.557395\pi$
$614$	13364.2	0.878397
$615$	0	0
$616$	−5652.95	−0.369746
$617$	234.868	0.0153248	0.00766241	−	0.999971i	$-0.497561\pi$
$617$	234.868	0.0153248	0.00766241	+	0.999971i	$0.497561\pi$
$618$	0	0
$619$	−4517.44	−0.293330	−0.146665	−	0.989186i	$-0.546854\pi$
$619$	−4517.44	−0.293330	−0.146665	+	0.989186i	$0.546854\pi$
$620$	0	0
$621$	0	0
$622$	−9580.42	−0.617588
$623$	−9481.68	−0.609752
$624$	0	0
$625$	0	0
$626$	−6384.59	−0.407635
$627$	0	0
$628$	1690.66	0.107428
$629$	−10383.4	−0.658211
$630$	0	0
$631$	26533.3	1.67397	0.836983	−	0.547229i	$-0.184317\pi$
$631$	26533.3	1.67397	0.836983	+	0.547229i	$0.184317\pi$
$632$	1852.19	0.116576
$633$	0	0
$634$	12425.9	0.778386
$635$	0	0
$636$	0	0
$637$	20712.6	1.28832
$638$	−3381.47	−0.209834
$639$	0	0
$640$	0	0
$641$	8233.77	0.507355	0.253677	−	0.967289i	$-0.418360\pi$
$641$	8233.77	0.507355	0.253677	+	0.967289i	$0.418360\pi$
$642$	0	0
$643$	−4844.30	−0.297108	−0.148554	−	0.988904i	$-0.547462\pi$
$643$	−4844.30	−0.297108	−0.148554	+	0.988904i	$0.547462\pi$
$644$	−1341.77	−0.0821012
$645$	0	0
$646$	7732.87	0.470969
$647$	29856.8	1.81421	0.907104	−	0.420907i	$-0.138288\pi$
$647$	29856.8	1.81421	0.907104	+	0.420907i	$0.138288\pi$
$648$	0	0
$649$	5408.66	0.327132
$650$	0	0
$651$	0	0
$652$	−1445.90	−0.0868497
$653$	4212.65	0.252456	0.126228	−	0.992001i	$-0.459713\pi$
$653$	4212.65	0.252456	0.126228	+	0.992001i	$0.459713\pi$
$654$	0	0
$655$	0	0
$656$	2048.78	0.121938
$657$	0	0
$658$	18319.4	1.08535
$659$	−14549.4	−0.860039	−0.430020	−	0.902820i	$-0.641493\pi$
$659$	−14549.4	−0.860039	−0.430020	+	0.902820i	$0.641493\pi$
$660$	0	0
$661$	2494.86	0.146806	0.0734029	−	0.997302i	$-0.476614\pi$
$661$	2494.86	0.146806	0.0734029	+	0.997302i	$0.476614\pi$
$662$	−4862.63	−0.285485
$663$	0	0
$664$	22026.9	1.28736
$665$	0	0
$666$	0	0
$667$	−1957.14	−0.113615
$668$	−19289.5	−1.11726
$669$	0	0
$670$	0	0
$671$	1397.09	0.0803784
$672$	0	0
$673$	−8223.98	−0.471042	−0.235521	−	0.971869i	$-0.575680\pi$
$673$	−8223.98	−0.471042	−0.235521	+	0.971869i	$0.575680\pi$
$674$	12909.6	0.737773
$675$	0	0
$676$	−27244.4	−1.55009
$677$	9369.49	0.531904	0.265952	−	0.963986i	$-0.414314\pi$
$677$	9369.49	0.531904	0.265952	+	0.963986i	$0.414314\pi$
$678$	0	0
$679$	4017.34	0.227057
$680$	0	0
$681$	0	0
$682$	2595.85	0.145748
$683$	561.238	0.0314424	0.0157212	−	0.999876i	$-0.494996\pi$
$683$	561.238	0.0314424	0.0157212	+	0.999876i	$0.494996\pi$
$684$	0	0
$685$	0	0
$686$	−3680.01	−0.204815
$687$	0	0
$688$	−1033.64	−0.0572777
$689$	−47791.1	−2.64252
$690$	0	0
$691$	11456.5	0.630720	0.315360	−	0.948972i	$-0.397875\pi$
$691$	11456.5	0.630720	0.315360	+	0.948972i	$0.397875\pi$
$692$	−21491.7	−1.18063
$693$	0	0
$694$	13543.4	0.740779
$695$	0	0
$696$	0	0
$697$	7346.96	0.399263
$698$	5233.09	0.283775
$699$	0	0
$700$	0	0
$701$	2782.39	0.149914	0.0749569	−	0.997187i	$-0.476118\pi$
$701$	2782.39	0.149914	0.0749569	+	0.997187i	$0.476118\pi$
$702$	0	0
$703$	−30642.1	−1.64394
$704$	2211.24	0.118380
$705$	0	0
$706$	57.1404	0.00304604
$707$	−36651.8	−1.94969
$708$	0	0
$709$	4824.70	0.255565	0.127783	−	0.991802i	$-0.459214\pi$
$709$	4824.70	0.255565	0.127783	+	0.991802i	$0.459214\pi$
$710$	0	0
$711$	0	0
$712$	8274.37	0.435527
$713$	1502.43	0.0789153
$714$	0	0
$715$	0	0
$716$	911.371	0.0475692
$717$	0	0
$718$	19971.0	1.03804
$719$	−20717.6	−1.07460	−0.537298	−	0.843392i	$-0.680555\pi$
$719$	−20717.6	−1.07460	−0.537298	+	0.843392i	$0.680555\pi$
$720$	0	0
$721$	30231.4	1.56155
$722$	12109.5	0.624195
$723$	0	0
$724$	17801.3	0.913783
$725$	0	0
$726$	0	0
$727$	−22731.2	−1.15963	−0.579816	−	0.814747i	$-0.696876\pi$
$727$	−22731.2	−1.15963	−0.579816	+	0.814747i	$0.696876\pi$
$728$	−43289.2	−2.20385
$729$	0	0
$730$	0	0
$731$	−3706.64	−0.187544
$732$	0	0
$733$	24311.8	1.22507	0.612536	−	0.790443i	$-0.290149\pi$
$733$	24311.8	1.22507	0.612536	+	0.790443i	$0.290149\pi$
$734$	−2020.11	−0.101586
$735$	0	0
$736$	1861.65	0.0932357
$737$	6910.49	0.345388
$738$	0	0
$739$	16033.9	0.798126	0.399063	−	0.916924i	$-0.369335\pi$
$739$	16033.9	0.798126	0.399063	+	0.916924i	$0.369335\pi$
$740$	0	0
$741$	0	0
$742$	−21499.2	−1.06369
$743$	−19512.2	−0.963435	−0.481718	−	0.876326i	$-0.659987\pi$
$743$	−19512.2	−0.963435	−0.481718	+	0.876326i	$0.659987\pi$
$744$	0	0
$745$	0	0
$746$	13489.7	0.662053
$747$	0	0
$748$	−2506.06	−0.122501
$749$	13360.6	0.651785
$750$	0	0
$751$	8393.65	0.407841	0.203920	−	0.978987i	$-0.434632\pi$
$751$	8393.65	0.407841	0.203920	+	0.978987i	$0.434632\pi$
$752$	5522.41	0.267795
$753$	0	0
$754$	−25894.7	−1.25070
$755$	0	0
$756$	0	0
$757$	−20208.6	−0.970270	−0.485135	−	0.874439i	$-0.661229\pi$
$757$	−20208.6	−0.970270	−0.485135	+	0.874439i	$0.661229\pi$
$758$	2894.38	0.138692
$759$	0	0
$760$	0	0
$761$	−3529.19	−0.168112	−0.0840558	−	0.996461i	$-0.526787\pi$
$761$	−3529.19	−0.168112	−0.0840558	+	0.996461i	$0.526787\pi$
$762$	0	0
$763$	33495.8	1.58929
$764$	13464.7	0.637612
$765$	0	0
$766$	6072.01	0.286411
$767$	41418.5	1.94985
$768$	0	0
$769$	−26692.6	−1.25171	−0.625853	−	0.779941i	$-0.715249\pi$
$769$	−26692.6	−1.25171	−0.625853	+	0.779941i	$0.715249\pi$
$770$	0	0
$771$	0	0
$772$	12655.0	0.589977
$773$	32009.0	1.48937	0.744684	−	0.667417i	$-0.232600\pi$
$773$	32009.0	1.48937	0.744684	+	0.667417i	$0.232600\pi$
$774$	0	0
$775$	0	0
$776$	−3505.81	−0.162180
$777$	0	0
$778$	16850.9	0.776523
$779$	21681.3	0.997194
$780$	0	0
$781$	−3409.74	−0.156223
$782$	635.954	0.0290814
$783$	0	0
$784$	2808.84	0.127954
$785$	0	0
$786$	0	0
$787$	−8180.18	−0.370511	−0.185255	−	0.982690i	$-0.559311\pi$
$787$	−8180.18	−0.370511	−0.185255	+	0.982690i	$0.559311\pi$
$788$	10889.2	0.492272
$789$	0	0
$790$	0	0
$791$	−24702.8	−1.11040
$792$	0	0
$793$	10698.6	0.479091
$794$	4490.91	0.200726
$795$	0	0
$796$	10986.1	0.489184
$797$	−25471.0	−1.13203	−0.566015	−	0.824395i	$-0.691516\pi$
$797$	−25471.0	−1.13203	−0.566015	+	0.824395i	$0.691516\pi$
$798$	0	0
$799$	19803.4	0.876840
$800$	0	0
$801$	0	0
$802$	−11892.6	−0.523619
$803$	−12593.4	−0.553437
$804$	0	0
$805$	0	0
$806$	19878.5	0.868723
$807$	0	0
$808$	31984.9	1.39261
$809$	−12381.1	−0.538067	−0.269034	−	0.963131i	$-0.586704\pi$
$809$	−12381.1	−0.538067	−0.269034	+	0.963131i	$0.586704\pi$
$810$	0	0
$811$	29129.5	1.26125	0.630627	−	0.776086i	$-0.282798\pi$
$811$	29129.5	1.26125	0.630627	+	0.776086i	$0.282798\pi$
$812$	26568.6	1.14825
$813$	0	0
$814$	−4353.99	−0.187478
$815$	0	0
$816$	0	0
$817$	−10938.5	−0.468409
$818$	−5920.64	−0.253069
$819$	0	0
$820$	0	0
$821$	5279.00	0.224407	0.112204	−	0.993685i	$-0.464209\pi$
$821$	5279.00	0.224407	0.112204	+	0.993685i	$0.464209\pi$
$822$	0	0
$823$	−19991.7	−0.846738	−0.423369	−	0.905957i	$-0.639153\pi$
$823$	−19991.7	−0.846738	−0.423369	+	0.905957i	$0.639153\pi$
$824$	−26382.0	−1.11537
$825$	0	0
$826$	18632.4	0.784874
$827$	9918.83	0.417063	0.208532	−	0.978016i	$-0.433132\pi$
$827$	9918.83	0.417063	0.208532	+	0.978016i	$0.433132\pi$
$828$	0	0
$829$	−5671.76	−0.237622	−0.118811	−	0.992917i	$-0.537908\pi$
$829$	−5671.76	−0.237622	−0.118811	+	0.992917i	$0.537908\pi$
$830$	0	0
$831$	0	0
$832$	16933.3	0.705595
$833$	10072.5	0.418959
$834$	0	0
$835$	0	0
$836$	−7395.54	−0.305957
$837$	0	0
$838$	−13752.1	−0.566894
$839$	29160.1	1.19990	0.599951	−	0.800036i	$-0.295187\pi$
$839$	29160.1	1.19990	0.599951	+	0.800036i	$0.295187\pi$
$840$	0	0
$841$	14364.7	0.588984
$842$	−12071.9	−0.494092
$843$	0	0
$844$	−6562.21	−0.267631
$845$	0	0
$846$	0	0
$847$	2936.31	0.119118
$848$	−6480.98	−0.262450
$849$	0	0
$850$	0	0
$851$	−2520.02	−0.101510
$852$	0	0
$853$	−18361.2	−0.737017	−0.368508	−	0.929624i	$-0.620131\pi$
$853$	−18361.2	−0.737017	−0.368508	+	0.929624i	$0.620131\pi$
$854$	4812.86	0.192849
$855$	0	0
$856$	−11659.4	−0.465550
$857$	1755.28	0.0699642	0.0349821	−	0.999388i	$-0.488863\pi$
$857$	1755.28	0.0699642	0.0349821	+	0.999388i	$0.488863\pi$
$858$	0	0
$859$	−31317.3	−1.24392	−0.621962	−	0.783047i	$-0.713664\pi$
$859$	−31317.3	−1.24392	−0.621962	+	0.783047i	$0.713664\pi$
$860$	0	0
$861$	0	0
$862$	−14082.0	−0.556422
$863$	−42862.3	−1.69067	−0.845336	−	0.534235i	$-0.820600\pi$
$863$	−42862.3	−1.69067	−0.845336	+	0.534235i	$0.820600\pi$
$864$	0	0
$865$	0	0
$866$	21548.0	0.845530
$867$	0	0
$868$	−20395.9	−0.797558
$869$	−962.082	−0.0375563
$870$	0	0
$871$	52919.3	2.05867
$872$	−29230.8	−1.13518
$873$	0	0
$874$	1876.74	0.0726335
$875$	0	0
$876$	0	0
$877$	−38306.6	−1.47494	−0.737471	−	0.675379i	$-0.763980\pi$
$877$	−38306.6	−1.47494	−0.737471	+	0.675379i	$0.763980\pi$
$878$	725.298	0.0278788
$879$	0	0
$880$	0	0
$881$	12994.5	0.496931	0.248466	−	0.968641i	$-0.420074\pi$
$881$	12994.5	0.496931	0.248466	+	0.968641i	$0.420074\pi$
$882$	0	0
$883$	4256.48	0.162222	0.0811110	−	0.996705i	$-0.474153\pi$
$883$	4256.48	0.162222	0.0811110	+	0.996705i	$0.474153\pi$
$884$	−19191.0	−0.730160
$885$	0	0
$886$	9051.03	0.343200
$887$	38188.8	1.44561	0.722804	−	0.691053i	$-0.242853\pi$
$887$	38188.8	1.44561	0.722804	+	0.691053i	$0.242853\pi$
$888$	0	0
$889$	−28680.1	−1.08200
$890$	0	0
$891$	0	0
$892$	−31938.5	−1.19886
$893$	58441.1	2.18999
$894$	0	0
$895$	0	0
$896$	−28735.3	−1.07141
$897$	0	0
$898$	13984.6	0.519679
$899$	−29750.0	−1.10369
$900$	0	0
$901$	−23240.9	−0.859341
$902$	3080.73	0.113722
$903$	0	0
$904$	21557.3	0.793127
$905$	0	0
$906$	0	0
$907$	31257.8	1.14432	0.572161	−	0.820142i	$-0.306105\pi$
$907$	31257.8	1.14432	0.572161	+	0.820142i	$0.306105\pi$
$908$	3966.28	0.144962
$909$	0	0
$910$	0	0
$911$	35781.4	1.30131	0.650653	−	0.759375i	$-0.274495\pi$
$911$	35781.4	1.30131	0.650653	+	0.759375i	$0.274495\pi$
$912$	0	0
$913$	−11441.4	−0.414737
$914$	15721.5	0.568950
$915$	0	0
$916$	−24682.7	−0.890326
$917$	63464.6	2.28548
$918$	0	0
$919$	31489.1	1.13028	0.565140	−	0.824995i	$-0.308822\pi$
$919$	31489.1	1.13028	0.565140	+	0.824995i	$0.308822\pi$
$920$	0	0
$921$	0	0
$922$	12950.0	0.462567
$923$	−26111.2	−0.931159
$924$	0	0
$925$	0	0
$926$	−15461.9	−0.548713
$927$	0	0
$928$	−36862.9	−1.30397
$929$	40466.1	1.42912	0.714559	−	0.699575i	$-0.246627\pi$
$929$	40466.1	1.42912	0.714559	+	0.699575i	$0.246627\pi$
$930$	0	0
$931$	29724.7	1.04639
$932$	−3415.79	−0.120051
$933$	0	0
$934$	−17044.4	−0.597120
$935$	0	0
$936$	0	0
$937$	−39041.1	−1.36117	−0.680586	−	0.732668i	$-0.738275\pi$
$937$	−39041.1	−1.36117	−0.680586	+	0.732668i	$0.738275\pi$
$938$	23806.1	0.828676
$939$	0	0
$940$	0	0
$941$	−10842.0	−0.375598	−0.187799	−	0.982207i	$-0.560135\pi$
$941$	−10842.0	−0.375598	−0.187799	+	0.982207i	$0.560135\pi$
$942$	0	0
$943$	1783.08	0.0615748
$944$	5616.79	0.193656
$945$	0	0
$946$	−1554.27	−0.0534182
$947$	−17810.1	−0.611142	−0.305571	−	0.952169i	$-0.598847\pi$
$947$	−17810.1	−0.611142	−0.305571	+	0.952169i	$0.598847\pi$
$948$	0	0
$949$	−96437.6	−3.29873
$950$	0	0
$951$	0	0
$952$	−21051.6	−0.716687
$953$	−45621.7	−1.55071	−0.775357	−	0.631523i	$-0.782430\pi$
$953$	−45621.7	−1.55071	−0.775357	+	0.631523i	$0.782430\pi$
$954$	0	0
$955$	0	0
$956$	28870.0	0.976697
$957$	0	0
$958$	−8940.12	−0.301506
$959$	11293.1	0.380264
$960$	0	0
$961$	−6952.93	−0.233390
$962$	−33342.0	−1.11745
$963$	0	0
$964$	1967.17	0.0657242
$965$	0	0
$966$	0	0
$967$	41111.6	1.36718	0.683588	−	0.729868i	$-0.260419\pi$
$967$	41111.6	1.36718	0.683588	+	0.729868i	$0.260419\pi$
$968$	−2562.43	−0.0850821
$969$	0	0
$970$	0	0
$971$	−22030.1	−0.728093	−0.364046	−	0.931381i	$-0.618605\pi$
$971$	−22030.1	−0.728093	−0.364046	+	0.931381i	$0.618605\pi$
$972$	0	0
$973$	68178.7	2.24636
$974$	25953.5	0.853802
$975$	0	0
$976$	1450.85	0.0475825
$977$	−34593.0	−1.13278	−0.566391	−	0.824137i	$-0.691661\pi$
$977$	−34593.0	−1.13278	−0.566391	+	0.824137i	$0.691661\pi$
$978$	0	0
$979$	−4297.95	−0.140310
$980$	0	0
$981$	0	0
$982$	15256.0	0.495762
$983$	−20638.9	−0.669663	−0.334831	−	0.942278i	$-0.608679\pi$
$983$	−20638.9	−0.669663	−0.334831	+	0.942278i	$0.608679\pi$
$984$	0	0
$985$	0	0
$986$	−12592.6	−0.406725
$987$	0	0
$988$	−56633.7	−1.82364
$989$	−899.586	−0.0289233
$990$	0	0
$991$	−35229.7	−1.12927	−0.564636	−	0.825340i	$-0.690983\pi$
$991$	−35229.7	−1.12927	−0.564636	+	0.825340i	$0.690983\pi$
$992$	28298.4	0.905722
$993$	0	0
$994$	−11746.3	−0.374819
$995$	0	0
$996$	0	0
$997$	−29366.2	−0.932834	−0.466417	−	0.884565i	$-0.654455\pi$
$997$	−29366.2	−0.932834	−0.466417	+	0.884565i	$0.654455\pi$
$998$	−2946.09	−0.0934439
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bb.1.4		4
3.2	odd	2		825.4.a.v.1.2	yes	4
5.4	even	2		2475.4.a.bd.1.1		4
15.2	even	4		825.4.c.q.199.4		8
15.8	even	4		825.4.c.q.199.5		8
15.14	odd	2		825.4.a.u.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.u.1.3	✓	4	15.14	odd	2
825.4.a.v.1.2	yes	4	3.2	odd	2
825.4.c.q.199.4		8	15.2	even	4
825.4.c.q.199.5		8	15.8	even	4
2475.4.a.bb.1.4		4	1.1	even	1	trivial
2475.4.a.bd.1.1		4	5.4	even	2

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{2} -5.56155 q^{4} +24.2670 q^{7} -21.1771 q^{8} +O(q^{10})\)	\(q+1.56155 q^{2} -5.56155 q^{4} +24.2670 q^{7} -21.1771 q^{8} +11.0000 q^{11} +84.2360 q^{13} +37.8942 q^{14} +11.4233 q^{16} +40.9641 q^{17} +120.887 q^{19} +17.1771 q^{22} +9.94182 q^{23} +131.539 q^{26} -134.962 q^{28} -196.860 q^{29} +151.123 q^{31} +187.255 q^{32} +63.9675 q^{34} -253.477 q^{37} +188.772 q^{38} +179.351 q^{41} -90.4851 q^{43} -61.1771 q^{44} +15.5247 q^{46} +483.434 q^{47} +245.887 q^{49} -468.483 q^{52} -567.348 q^{53} -513.904 q^{56} -307.407 q^{58} +491.696 q^{59} +127.008 q^{61} +235.986 q^{62} +201.022 q^{64} +628.226 q^{67} -227.824 q^{68} -309.976 q^{71} -1144.85 q^{73} -395.817 q^{74} -672.322 q^{76} +266.937 q^{77} -87.4620 q^{79} +280.067 q^{82} -1040.13 q^{83} -141.297 q^{86} -232.948 q^{88} -390.723 q^{89} +2044.15 q^{91} -55.2920 q^{92} +754.908 q^{94} +165.548 q^{97} +383.966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 14 q^{4} + 11 q^{7} + 6 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 14 * q^4 + 11 * q^7 + 6 * q^8	\( 4 q - 2 q^{2} - 14 q^{4} + 11 q^{7} + 6 q^{8} + 44 q^{11} - 25 q^{13} + 3 q^{14} - 78 q^{16} + 85 q^{17} + 118 q^{19} - 22 q^{22} - 10 q^{23} + 157 q^{26} - 47 q^{28} - 251 q^{29} - 135 q^{31} + 246 q^{32} + 289 q^{34} - 419 q^{37} - 76 q^{38} + 103 q^{41} - 15 q^{43} - 154 q^{44} - 420 q^{46} + 665 q^{47} + 1003 q^{49} - 57 q^{52} - 116 q^{53} - 77 q^{56} - 189 q^{58} - 951 q^{59} - 350 q^{61} - 213 q^{62} + 1538 q^{64} - 266 q^{67} - 629 q^{68} - 1526 q^{71} + 566 q^{73} - 1091 q^{74} - 396 q^{76} + 121 q^{77} + 2567 q^{79} - 9 q^{82} - 961 q^{83} - 1837 q^{86} + 66 q^{88} - 1053 q^{89} + 2150 q^{91} + 460 q^{92} + 2209 q^{94} + 172 q^{97} - 1819 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 14 * q^4 + 11 * q^7 + 6 * q^8 + 44 * q^11 - 25 * q^13 + 3 * q^14 - 78 * q^16 + 85 * q^17 + 118 * q^19 - 22 * q^22 - 10 * q^23 + 157 * q^26 - 47 * q^28 - 251 * q^29 - 135 * q^31 + 246 * q^32 + 289 * q^34 - 419 * q^37 - 76 * q^38 + 103 * q^41 - 15 * q^43 - 154 * q^44 - 420 * q^46 + 665 * q^47 + 1003 * q^49 - 57 * q^52 - 116 * q^53 - 77 * q^56 - 189 * q^58 - 951 * q^59 - 350 * q^61 - 213 * q^62 + 1538 * q^64 - 266 * q^67 - 629 * q^68 - 1526 * q^71 + 566 * q^73 - 1091 * q^74 - 396 * q^76 + 121 * q^77 + 2567 * q^79 - 9 * q^82 - 961 * q^83 - 1837 * q^86 + 66 * q^88 - 1053 * q^89 + 2150 * q^91 + 460 * q^92 + 2209 * q^94 + 172 * q^97 - 1819 * q^98

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