LMFDB - Embedded newform 2475.4.a.bq.1.2

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 44x^{5} + 118x^{4} + 515x^{3} - 1279x^{2} - 892x + 1840 $ x^7 - 3x^6 - 44x^5 + 118x^4 + 515x^3 - 1279x^2 - 892x + 1840
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.00699$ 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)

$f(q)$	$=$	$q-4.00699 q^{2} +8.05595 q^{4} +26.2766 q^{7} -0.224208 q^{8} +O(q^{10})$	$q-4.00699 q^{2} +8.05595 q^{4} +26.2766 q^{7} -0.224208 q^{8} +11.0000 q^{11} -11.3069 q^{13} -105.290 q^{14} -63.5492 q^{16} -87.6018 q^{17} +148.978 q^{19} -44.0769 q^{22} +12.7783 q^{23} +45.3065 q^{26} +211.683 q^{28} +37.2177 q^{29} -30.4121 q^{31} +256.435 q^{32} +351.019 q^{34} +122.491 q^{37} -596.955 q^{38} -444.960 q^{41} +36.2510 q^{43} +88.6155 q^{44} -51.2025 q^{46} -78.5138 q^{47} +347.458 q^{49} -91.0876 q^{52} +342.345 q^{53} -5.89142 q^{56} -149.131 q^{58} -377.123 q^{59} -690.244 q^{61} +121.861 q^{62} -519.137 q^{64} -696.564 q^{67} -705.716 q^{68} -1014.66 q^{71} -889.198 q^{73} -490.818 q^{74} +1200.16 q^{76} +289.042 q^{77} -858.251 q^{79} +1782.95 q^{82} -115.843 q^{83} -145.257 q^{86} -2.46629 q^{88} -553.264 q^{89} -297.106 q^{91} +102.941 q^{92} +314.604 q^{94} +336.110 q^{97} -1392.26 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{2} + 41 q^{4} + 50 q^{7} - 21 q^{8}+O(q^{10}) $ 7 * q - 3 * q^2 + 41 * q^4 + 50 * q^7 - 21 * q^8	$ 7 q - 3 q^{2} + 41 q^{4} + 50 q^{7} - 21 q^{8} + 77 q^{11} + 24 q^{13} - 142 q^{14} + 181 q^{16} - 38 q^{17} + 26 q^{19} - 33 q^{22} - 228 q^{23} - 476 q^{26} + 840 q^{28} - 572 q^{29} - 140 q^{31} - 991 q^{32} - 806 q^{34} - 104 q^{37} - 498 q^{38} - 896 q^{41} + 614 q^{43} + 451 q^{44} - 344 q^{46} - 520 q^{47} + 295 q^{49} - 26 q^{52} - 380 q^{53} - 1522 q^{56} + 1600 q^{58} - 1316 q^{59} - 386 q^{61} - 440 q^{62} + 869 q^{64} + 348 q^{67} - 332 q^{68} - 804 q^{71} + 468 q^{73} + 748 q^{74} - 1698 q^{76} + 550 q^{77} - 374 q^{79} - 620 q^{82} - 3128 q^{83} + 2534 q^{86} - 231 q^{88} - 694 q^{89} - 3376 q^{91} + 1184 q^{92} - 2920 q^{94} - 8 q^{97} - 4211 q^{98}+O(q^{100}) $ 7 * q - 3 * q^2 + 41 * q^4 + 50 * q^7 - 21 * q^8 + 77 * q^11 + 24 * q^13 - 142 * q^14 + 181 * q^16 - 38 * q^17 + 26 * q^19 - 33 * q^22 - 228 * q^23 - 476 * q^26 + 840 * q^28 - 572 * q^29 - 140 * q^31 - 991 * q^32 - 806 * q^34 - 104 * q^37 - 498 * q^38 - 896 * q^41 + 614 * q^43 + 451 * q^44 - 344 * q^46 - 520 * q^47 + 295 * q^49 - 26 * q^52 - 380 * q^53 - 1522 * q^56 + 1600 * q^58 - 1316 * q^59 - 386 * q^61 - 440 * q^62 + 869 * q^64 + 348 * q^67 - 332 * q^68 - 804 * q^71 + 468 * q^73 + 748 * q^74 - 1698 * q^76 + 550 * q^77 - 374 * q^79 - 620 * q^82 - 3128 * q^83 + 2534 * q^86 - 231 * q^88 - 694 * q^89 - 3376 * q^91 + 1184 * q^92 - 2920 * q^94 - 8 * q^97 - 4211 * 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 $

$2$	−4.00699	−1.41668	−0.708342	−	0.705869i	$-0.750557\pi$
$2$	−4.00699	−1.41668	−0.708342	+	0.705869i	$0.750557\pi$
$3$	0	0
$3$	0	0
$4$	8.05595	1.00699
$5$	0	0
$5$	0	0
$6$	0	0
$7$	26.2766	1.41880	0.709401	−	0.704805i	$-0.248966\pi$
$7$	26.2766	1.41880	0.709401	+	0.704805i	$0.248966\pi$
$8$	−0.224208	−0.00990869
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−11.3069	−0.241228	−0.120614	−	0.992700i	$-0.538486\pi$
$13$	−11.3069	−0.241228	−0.120614	+	0.992700i	$0.538486\pi$
$14$	−105.290	−2.00999
$15$	0	0
$16$	−63.5492	−0.992957
$17$	−87.6018	−1.24980	−0.624899	−	0.780706i	$-0.714860\pi$
$17$	−87.6018	−1.24980	−0.624899	+	0.780706i	$0.714860\pi$
$18$	0	0
$19$	148.978	1.79884	0.899421	−	0.437083i	$-0.143988\pi$
$19$	148.978	1.79884	0.899421	+	0.437083i	$0.143988\pi$
$20$	0	0
$21$	0	0
$22$	−44.0769	−0.427146
$23$	12.7783	0.115846	0.0579230	−	0.998321i	$-0.481552\pi$
$23$	12.7783	0.115846	0.0579230	+	0.998321i	$0.481552\pi$
$24$	0	0
$25$	0	0
$26$	45.3065	0.341743
$27$	0	0
$28$	211.683	1.42872
$29$	37.2177	0.238315	0.119158	−	0.992875i	$-0.461981\pi$
$29$	37.2177	0.238315	0.119158	+	0.992875i	$0.461981\pi$
$30$	0	0
$31$	−30.4121	−0.176199	−0.0880995	−	0.996112i	$-0.528079\pi$
$31$	−30.4121	−0.176199	−0.0880995	+	0.996112i	$0.528079\pi$
$32$	256.435	1.41661
$33$	0	0
$34$	351.019	1.77057
$35$	0	0
$36$	0	0
$37$	122.491	0.544252	0.272126	−	0.962262i	$-0.412273\pi$
$37$	122.491	0.544252	0.272126	+	0.962262i	$0.412273\pi$
$38$	−596.955	−2.54839
$39$	0	0
$40$	0	0
$41$	−444.960	−1.69490	−0.847452	−	0.530871i	$-0.821865\pi$
$41$	−444.960	−1.69490	−0.847452	+	0.530871i	$0.821865\pi$
$42$	0	0
$43$	36.2510	0.128564	0.0642818	−	0.997932i	$-0.479524\pi$
$43$	36.2510	0.128564	0.0642818	+	0.997932i	$0.479524\pi$
$44$	88.6155	0.303620
$45$	0	0
$46$	−51.2025	−0.164117
$47$	−78.5138	−0.243668	−0.121834	−	0.992550i	$-0.538878\pi$
$47$	−78.5138	−0.243668	−0.121834	+	0.992550i	$0.538878\pi$
$48$	0	0
$49$	347.458	1.01300
$50$	0	0
$51$	0	0
$52$	−91.0876	−0.242915
$53$	342.345	0.887260	0.443630	−	0.896210i	$-0.353691\pi$
$53$	342.345	0.887260	0.443630	+	0.896210i	$0.353691\pi$
$54$	0	0
$55$	0	0
$56$	−5.89142	−0.0140585
$57$	0	0
$58$	−149.131	−0.337618
$59$	−377.123	−0.832156	−0.416078	−	0.909329i	$-0.636596\pi$
$59$	−377.123	−0.832156	−0.416078	+	0.909329i	$0.636596\pi$
$60$	0	0
$61$	−690.244	−1.44880	−0.724399	−	0.689380i	$-0.757883\pi$
$61$	−690.244	−1.44880	−0.724399	+	0.689380i	$0.757883\pi$
$62$	121.861	0.249618
$63$	0	0
$64$	−519.137	−1.01394
$65$	0	0
$66$	0	0
$67$	−696.564	−1.27013	−0.635066	−	0.772458i	$-0.719027\pi$
$67$	−696.564	−1.27013	−0.635066	+	0.772458i	$0.719027\pi$
$68$	−705.716	−1.25854
$69$	0	0
$70$	0	0
$71$	−1014.66	−1.69602	−0.848011	−	0.529978i	$-0.822200\pi$
$71$	−1014.66	−1.69602	−0.848011	+	0.529978i	$0.822200\pi$
$72$	0	0
$73$	−889.198	−1.42565	−0.712827	−	0.701339i	$-0.752586\pi$
$73$	−889.198	−1.42565	−0.712827	+	0.701339i	$0.752586\pi$
$74$	−490.818	−0.771033
$75$	0	0
$76$	1200.16	1.81142
$77$	289.042	0.427785
$78$	0	0
$79$	−858.251	−1.22229	−0.611144	−	0.791519i	$-0.709290\pi$
$79$	−858.251	−1.22229	−0.611144	+	0.791519i	$0.709290\pi$
$80$	0	0
$81$	0	0
$82$	1782.95	2.40115
$83$	−115.843	−0.153198	−0.0765989	−	0.997062i	$-0.524406\pi$
$83$	−115.843	−0.153198	−0.0765989	+	0.997062i	$0.524406\pi$
$84$	0	0
$85$	0	0
$86$	−145.257	−0.182134
$87$	0	0
$88$	−2.46629	−0.00298758
$89$	−553.264	−0.658943	−0.329471	−	0.944166i	$-0.606870\pi$
$89$	−553.264	−0.658943	−0.329471	+	0.944166i	$0.606870\pi$
$90$	0	0
$91$	−297.106	−0.342254
$92$	102.941	0.116656
$93$	0	0
$94$	314.604	0.345201
$95$	0	0
$96$	0	0
$97$	336.110	0.351823	0.175911	−	0.984406i	$-0.443713\pi$
$97$	336.110	0.351823	0.175911	+	0.984406i	$0.443713\pi$
$98$	−1392.26	−1.43510
$99$	0	0
$100$	0	0
$101$	−1277.11	−1.25819	−0.629097	−	0.777327i	$-0.716575\pi$
$101$	−1277.11	−1.25819	−0.629097	+	0.777327i	$0.716575\pi$
$102$	0	0
$103$	1075.63	1.02898	0.514490	−	0.857497i	$-0.327981\pi$
$103$	1075.63	1.02898	0.514490	+	0.857497i	$0.327981\pi$
$104$	2.53509	0.00239025
$105$	0	0
$106$	−1371.77	−1.25697
$107$	1903.79	1.72006	0.860030	−	0.510244i	$-0.170445\pi$
$107$	1903.79	1.72006	0.860030	+	0.510244i	$0.170445\pi$
$108$	0	0
$109$	−894.655	−0.786169	−0.393084	−	0.919502i	$-0.628592\pi$
$109$	−894.655	−0.786169	−0.393084	+	0.919502i	$0.628592\pi$
$110$	0	0
$111$	0	0
$112$	−1669.86	−1.40881
$113$	−0.252752	−0.000210415 0	−0.000105208	−	1.00000i	$-0.500033\pi$
$113$	−0.252752	−0.000210415 0	−0.000105208		1.00000i	$0.500033\pi$
$114$	0	0
$115$	0	0
$116$	299.824	0.239982
$117$	0	0
$118$	1511.13	1.17890
$119$	−2301.87	−1.77321
$120$	0	0
$121$	121.000	0.0909091
$122$	2765.80	2.05249
$123$	0	0
$124$	−244.998	−0.177431
$125$	0	0
$126$	0	0
$127$	2393.92	1.67264	0.836321	−	0.548240i	$-0.184702\pi$
$127$	2393.92	1.67264	0.836321	+	0.548240i	$0.184702\pi$
$128$	28.6979	0.0198169
$129$	0	0
$130$	0	0
$131$	2363.05	1.57604	0.788019	−	0.615651i	$-0.211107\pi$
$131$	2363.05	1.57604	0.788019	+	0.615651i	$0.211107\pi$
$132$	0	0
$133$	3914.64	2.55220
$134$	2791.12	1.79938
$135$	0	0
$136$	19.6410	0.0123839
$137$	208.264	0.129877	0.0649385	−	0.997889i	$-0.479315\pi$
$137$	208.264	0.129877	0.0649385	+	0.997889i	$0.479315\pi$
$138$	0	0
$139$	−612.112	−0.373515	−0.186758	−	0.982406i	$-0.559798\pi$
$139$	−612.112	−0.373515	−0.186758	+	0.982406i	$0.559798\pi$
$140$	0	0
$141$	0	0
$142$	4065.72	2.40273
$143$	−124.375	−0.0727329
$144$	0	0
$145$	0	0
$146$	3563.01	2.01970
$147$	0	0
$148$	986.778	0.548059
$149$	2116.42	1.16365	0.581826	−	0.813313i	$-0.302338\pi$
$149$	2116.42	1.16365	0.581826	+	0.813313i	$0.302338\pi$
$150$	0	0
$151$	−3405.63	−1.83540	−0.917701	−	0.397271i	$-0.869957\pi$
$151$	−3405.63	−1.83540	−0.917701	+	0.397271i	$0.869957\pi$
$152$	−33.4022	−0.0178242
$153$	0	0
$154$	−1158.19	−0.606036
$155$	0	0
$156$	0	0
$157$	294.346	0.149627	0.0748133	−	0.997198i	$-0.476164\pi$
$157$	294.346	0.149627	0.0748133	+	0.997198i	$0.476164\pi$
$158$	3439.00	1.73160
$159$	0	0
$160$	0	0
$161$	335.770	0.164362
$162$	0	0
$163$	−2877.76	−1.38284	−0.691421	−	0.722452i	$-0.743015\pi$
$163$	−2877.76	−1.38284	−0.691421	+	0.722452i	$0.743015\pi$
$164$	−3584.58	−1.70676
$165$	0	0
$166$	464.181	0.217033
$167$	−2506.59	−1.16147	−0.580735	−	0.814093i	$-0.697235\pi$
$167$	−2506.59	−1.16147	−0.580735	+	0.814093i	$0.697235\pi$
$168$	0	0
$169$	−2069.15	−0.941809
$170$	0	0
$171$	0	0
$172$	292.037	0.129463
$173$	−3092.28	−1.35897	−0.679485	−	0.733689i	$-0.737797\pi$
$173$	−3092.28	−1.35897	−0.679485	+	0.733689i	$0.737797\pi$
$174$	0	0
$175$	0	0
$176$	−699.042	−0.299388
$177$	0	0
$178$	2216.92	0.933514
$179$	3360.78	1.40333	0.701665	−	0.712507i	$-0.252440\pi$
$179$	3360.78	1.40333	0.701665	+	0.712507i	$0.252440\pi$
$180$	0	0
$181$	208.349	0.0855606	0.0427803	−	0.999085i	$-0.486378\pi$
$181$	208.349	0.0855606	0.0427803	+	0.999085i	$0.486378\pi$
$182$	1190.50	0.484866
$183$	0	0
$184$	−2.86499	−0.00114788
$185$	0	0
$186$	0	0
$187$	−963.620	−0.376828
$188$	−632.503	−0.245373
$189$	0	0
$190$	0	0
$191$	324.423	0.122903	0.0614514	−	0.998110i	$-0.480427\pi$
$191$	324.423	0.122903	0.0614514	+	0.998110i	$0.480427\pi$
$192$	0	0
$193$	−3274.46	−1.22125	−0.610624	−	0.791920i	$-0.709081\pi$
$193$	−3274.46	−1.22125	−0.610624	+	0.791920i	$0.709081\pi$
$194$	−1346.79	−0.498422
$195$	0	0
$196$	2799.11	1.02008
$197$	642.826	0.232484	0.116242	−	0.993221i	$-0.462915\pi$
$197$	642.826	0.232484	0.116242	+	0.993221i	$0.462915\pi$
$198$	0	0
$199$	−320.513	−0.114174	−0.0570868	−	0.998369i	$-0.518181\pi$
$199$	−320.513	−0.114174	−0.0570868	+	0.998369i	$0.518181\pi$
$200$	0	0
$201$	0	0
$202$	5117.38	1.78246
$203$	977.953	0.338122
$204$	0	0
$205$	0	0
$206$	−4310.03	−1.45774
$207$	0	0
$208$	718.542	0.239529
$209$	1638.76	0.542371
$210$	0	0
$211$	−5158.61	−1.68310	−0.841548	−	0.540182i	$-0.818356\pi$
$211$	−5158.61	−1.68310	−0.841548	+	0.540182i	$0.818356\pi$
$212$	2757.92	0.893465
$213$	0	0
$214$	−7628.47	−2.43678
$215$	0	0
$216$	0	0
$217$	−799.125	−0.249992
$218$	3584.87	1.11375
$219$	0	0
$220$	0	0
$221$	990.501	0.301486
$222$	0	0
$223$	2948.51	0.885413	0.442706	−	0.896667i	$-0.354018\pi$
$223$	2948.51	0.885413	0.442706	+	0.896667i	$0.354018\pi$
$224$	6738.22	2.00990
$225$	0	0
$226$	1.01277	0.000298092 0
$227$	−2281.12	−0.666974	−0.333487	−	0.942755i	$-0.608225\pi$
$227$	−2281.12	−0.666974	−0.333487	+	0.942755i	$0.608225\pi$
$228$	0	0
$229$	3616.61	1.04364	0.521818	−	0.853057i	$-0.325254\pi$
$229$	3616.61	1.04364	0.521818	+	0.853057i	$0.325254\pi$
$230$	0	0
$231$	0	0
$232$	−8.34450	−0.00236139
$233$	4571.20	1.28528	0.642638	−	0.766170i	$-0.277840\pi$
$233$	4571.20	1.28528	0.642638	+	0.766170i	$0.277840\pi$
$234$	0	0
$235$	0	0
$236$	−3038.08	−0.837976
$237$	0	0
$238$	9223.59	2.51208
$239$	−2515.86	−0.680911	−0.340455	−	0.940261i	$-0.610581\pi$
$239$	−2515.86	−0.680911	−0.340455	+	0.940261i	$0.610581\pi$
$240$	0	0
$241$	5325.35	1.42339	0.711693	−	0.702490i	$-0.247929\pi$
$241$	5325.35	1.42339	0.711693	+	0.702490i	$0.247929\pi$
$242$	−484.846	−0.128789
$243$	0	0
$244$	−5560.58	−1.45893
$245$	0	0
$246$	0	0
$247$	−1684.48	−0.433931
$248$	6.81863	0.00174590
$249$	0	0
$250$	0	0
$251$	4417.05	1.11076	0.555382	−	0.831595i	$-0.312572\pi$
$251$	4417.05	1.11076	0.555382	+	0.831595i	$0.312572\pi$
$252$	0	0
$253$	140.561	0.0349289
$254$	−9592.39	−2.36961
$255$	0	0
$256$	4038.10	0.985865
$257$	−1049.24	−0.254668	−0.127334	−	0.991860i	$-0.540642\pi$
$257$	−1049.24	−0.254668	−0.127334	+	0.991860i	$0.540642\pi$
$258$	0	0
$259$	3218.63	0.772185
$260$	0	0
$261$	0	0
$262$	−9468.73	−2.23275
$263$	2773.92	0.650370	0.325185	−	0.945650i	$-0.394573\pi$
$263$	2773.92	0.650370	0.325185	+	0.945650i	$0.394573\pi$
$264$	0	0
$265$	0	0
$266$	−15685.9	−3.61566
$267$	0	0
$268$	−5611.49	−1.27902
$269$	1645.90	0.373056	0.186528	−	0.982450i	$-0.440276\pi$
$269$	1645.90	0.373056	0.186528	+	0.982450i	$0.440276\pi$
$270$	0	0
$271$	−8599.58	−1.92763	−0.963814	−	0.266576i	$-0.914108\pi$
$271$	−8599.58	−1.92763	−0.963814	+	0.266576i	$0.914108\pi$
$272$	5567.03	1.24099
$273$	0	0
$274$	−834.510	−0.183995
$275$	0	0
$276$	0	0
$277$	3908.67	0.847832	0.423916	−	0.905701i	$-0.360655\pi$
$277$	3908.67	0.847832	0.423916	+	0.905701i	$0.360655\pi$
$278$	2452.72	0.529153
$279$	0	0
$280$	0	0
$281$	1987.21	0.421875	0.210938	−	0.977500i	$-0.432348\pi$
$281$	1987.21	0.421875	0.210938	+	0.977500i	$0.432348\pi$
$282$	0	0
$283$	1229.67	0.258291	0.129145	−	0.991626i	$-0.458777\pi$
$283$	1229.67	0.258291	0.129145	+	0.991626i	$0.458777\pi$
$284$	−8174.03	−1.70789
$285$	0	0
$286$	498.371	0.103040
$287$	−11692.0	−2.40473
$288$	0	0
$289$	2761.08	0.561994
$290$	0	0
$291$	0	0
$292$	−7163.34	−1.43563
$293$	5067.22	1.01034	0.505171	−	0.863019i	$-0.331429\pi$
$293$	5067.22	1.01034	0.505171	+	0.863019i	$0.331429\pi$
$294$	0	0
$295$	0	0
$296$	−27.4634	−0.00539282
$297$	0	0
$298$	−8480.49	−1.64853
$299$	−144.482	−0.0279453
$300$	0	0
$301$	952.553	0.182406
$302$	13646.3	2.60019
$303$	0	0
$304$	−9467.47	−1.78617
$305$	0	0
$306$	0	0
$307$	321.662	0.0597988	0.0298994	−	0.999553i	$-0.490481\pi$
$307$	321.662	0.0597988	0.0298994	+	0.999553i	$0.490481\pi$
$308$	2328.51	0.430777
$309$	0	0
$310$	0	0
$311$	5135.27	0.936316	0.468158	−	0.883645i	$-0.344918\pi$
$311$	5135.27	0.936316	0.468158	+	0.883645i	$0.344918\pi$
$312$	0	0
$313$	−8366.56	−1.51088	−0.755441	−	0.655217i	$-0.772577\pi$
$313$	−8366.56	−1.51088	−0.755441	+	0.655217i	$0.772577\pi$
$314$	−1179.44	−0.211974
$315$	0	0
$316$	−6914.03	−1.23084
$317$	7744.61	1.37218	0.686089	−	0.727517i	$-0.259326\pi$
$317$	7744.61	1.37218	0.686089	+	0.727517i	$0.259326\pi$
$318$	0	0
$319$	409.394	0.0718548
$320$	0	0
$321$	0	0
$322$	−1345.43	−0.232850
$323$	−13050.8	−2.24819
$324$	0	0
$325$	0	0
$326$	11531.1	1.95905
$327$	0	0
$328$	99.7636	0.0167943
$329$	−2063.07	−0.345717
$330$	0	0
$331$	201.426	0.0334483	0.0167242	−	0.999860i	$-0.494676\pi$
$331$	201.426	0.0334483	0.0167242	+	0.999860i	$0.494676\pi$
$332$	−933.225	−0.154269
$333$	0	0
$334$	10043.9	1.64544
$335$	0	0
$336$	0	0
$337$	1271.19	0.205478	0.102739	−	0.994708i	$-0.467239\pi$
$337$	1271.19	0.205478	0.102739	+	0.994708i	$0.467239\pi$
$338$	8291.08	1.33425
$339$	0	0
$340$	0	0
$341$	−334.533	−0.0531260
$342$	0	0
$343$	117.142	0.0184405
$344$	−8.12777	−0.00127390
$345$	0	0
$346$	12390.7	1.92523
$347$	−7274.99	−1.12548	−0.562740	−	0.826634i	$-0.690253\pi$
$347$	−7274.99	−1.12548	−0.562740	+	0.826634i	$0.690253\pi$
$348$	0	0
$349$	293.092	0.0449538	0.0224769	−	0.999747i	$-0.492845\pi$
$349$	293.092	0.0449538	0.0224769	+	0.999747i	$0.492845\pi$
$350$	0	0
$351$	0	0
$352$	2820.78	0.427125
$353$	−1120.68	−0.168974	−0.0844868	−	0.996425i	$-0.526925\pi$
$353$	−1120.68	−0.168974	−0.0844868	+	0.996425i	$0.526925\pi$
$354$	0	0
$355$	0	0
$356$	−4457.07	−0.663551
$357$	0	0
$358$	−13466.6	−1.98808
$359$	3543.35	0.520921	0.260461	−	0.965484i	$-0.416126\pi$
$359$	3543.35	0.520921	0.260461	+	0.965484i	$0.416126\pi$
$360$	0	0
$361$	15335.6	2.23583
$362$	−834.853	−0.121212
$363$	0	0
$364$	−2393.47	−0.344648
$365$	0	0
$366$	0	0
$367$	−5998.09	−0.853128	−0.426564	−	0.904457i	$-0.640276\pi$
$367$	−5998.09	−0.853128	−0.426564	+	0.904457i	$0.640276\pi$
$368$	−812.051	−0.115030
$369$	0	0
$370$	0	0
$371$	8995.66	1.25885
$372$	0	0
$373$	4701.91	0.652696	0.326348	−	0.945250i	$-0.394182\pi$
$373$	4701.91	0.652696	0.326348	+	0.945250i	$0.394182\pi$
$374$	3861.21	0.533846
$375$	0	0
$376$	17.6034	0.00241443
$377$	−420.815	−0.0574883
$378$	0	0
$379$	−10784.6	−1.46165	−0.730826	−	0.682563i	$-0.760865\pi$
$379$	−10784.6	−1.46165	−0.730826	+	0.682563i	$0.760865\pi$
$380$	0	0
$381$	0	0
$382$	−1299.96	−0.174115
$383$	−5317.16	−0.709385	−0.354692	−	0.934983i	$-0.615414\pi$
$383$	−5317.16	−0.709385	−0.354692	+	0.934983i	$0.615414\pi$
$384$	0	0
$385$	0	0
$386$	13120.7	1.73012
$387$	0	0
$388$	2707.69	0.354284
$389$	−10677.3	−1.39167	−0.695833	−	0.718203i	$-0.744965\pi$
$389$	−10677.3	−1.39167	−0.695833	+	0.718203i	$0.744965\pi$
$390$	0	0
$391$	−1119.40	−0.144784
$392$	−77.9029	−0.0100375
$393$	0	0
$394$	−2575.79	−0.329357
$395$	0	0
$396$	0	0
$397$	8922.89	1.12803	0.564014	−	0.825765i	$-0.309257\pi$
$397$	8922.89	1.12803	0.564014	+	0.825765i	$0.309257\pi$
$398$	1284.29	0.161748
$399$	0	0
$400$	0	0
$401$	3177.53	0.395707	0.197853	−	0.980232i	$-0.436603\pi$
$401$	3177.53	0.395707	0.197853	+	0.980232i	$0.436603\pi$
$402$	0	0
$403$	343.865	0.0425041
$404$	−10288.4	−1.26699
$405$	0	0
$406$	−3918.64	−0.479013
$407$	1347.40	0.164098
$408$	0	0
$409$	−2509.89	−0.303438	−0.151719	−	0.988424i	$-0.548481\pi$
$409$	−2509.89	−0.303438	−0.151719	+	0.988424i	$0.548481\pi$
$410$	0	0
$411$	0	0
$412$	8665.22	1.03618
$413$	−9909.49	−1.18066
$414$	0	0
$415$	0	0
$416$	−2899.47	−0.341727
$417$	0	0
$418$	−6566.51	−0.768369
$419$	−9452.10	−1.10207	−0.551033	−	0.834484i	$-0.685766\pi$
$419$	−9452.10	−1.10207	−0.551033	+	0.834484i	$0.685766\pi$
$420$	0	0
$421$	−1824.33	−0.211193	−0.105597	−	0.994409i	$-0.533675\pi$
$421$	−1824.33	−0.211193	−0.105597	+	0.994409i	$0.533675\pi$
$422$	20670.5	2.38442
$423$	0	0
$424$	−76.7566	−0.00879158
$425$	0	0
$426$	0	0
$427$	−18137.3	−2.05556
$428$	15336.9	1.73209
$429$	0	0
$430$	0	0
$431$	−15347.3	−1.71521	−0.857603	−	0.514312i	$-0.828047\pi$
$431$	−15347.3	−1.71521	−0.857603	+	0.514312i	$0.828047\pi$
$432$	0	0
$433$	5002.63	0.555222	0.277611	−	0.960694i	$-0.410457\pi$
$433$	5002.63	0.555222	0.277611	+	0.960694i	$0.410457\pi$
$434$	3202.09	0.354159
$435$	0	0
$436$	−7207.30	−0.791667
$437$	1903.69	0.208389
$438$	0	0
$439$	−2498.66	−0.271651	−0.135825	−	0.990733i	$-0.543369\pi$
$439$	−2498.66	−0.271651	−0.135825	+	0.990733i	$0.543369\pi$
$440$	0	0
$441$	0	0
$442$	−3968.93	−0.427110
$443$	3044.08	0.326476	0.163238	−	0.986587i	$-0.447806\pi$
$443$	3044.08	0.326476	0.163238	+	0.986587i	$0.447806\pi$
$444$	0	0
$445$	0	0
$446$	−11814.7	−1.25435
$447$	0	0
$448$	−13641.1	−1.43858
$449$	−13184.9	−1.38582	−0.692911	−	0.721023i	$-0.743672\pi$
$449$	−13184.9	−1.38582	−0.692911	+	0.721023i	$0.743672\pi$
$450$	0	0
$451$	−4894.56	−0.511033
$452$	−2.03616	−0.000211887 0
$453$	0	0
$454$	9140.42	0.944892
$455$	0	0
$456$	0	0
$457$	16938.6	1.73381	0.866907	−	0.498469i	$-0.166104\pi$
$457$	16938.6	1.73381	0.866907	+	0.498469i	$0.166104\pi$
$458$	−14491.7	−1.47850
$459$	0	0
$460$	0	0
$461$	−5188.91	−0.524234	−0.262117	−	0.965036i	$-0.584421\pi$
$461$	−5188.91	−0.524234	−0.262117	+	0.965036i	$0.584421\pi$
$462$	0	0
$463$	−15097.9	−1.51546	−0.757730	−	0.652569i	$-0.773691\pi$
$463$	−15097.9	−1.51546	−0.757730	+	0.652569i	$0.773691\pi$
$464$	−2365.15	−0.236637
$465$	0	0
$466$	−18316.7	−1.82083
$467$	−13597.2	−1.34733	−0.673665	−	0.739037i	$-0.735281\pi$
$467$	−13597.2	−1.34733	−0.673665	+	0.739037i	$0.735281\pi$
$468$	0	0
$469$	−18303.3	−1.80207
$470$	0	0
$471$	0	0
$472$	84.5539	0.00824557
$473$	398.761	0.0387634
$474$	0	0
$475$	0	0
$476$	−18543.8	−1.78562
$477$	0	0
$478$	10081.0	0.964636
$479$	6830.32	0.651535	0.325767	−	0.945450i	$-0.394377\pi$
$479$	6830.32	0.651535	0.325767	+	0.945450i	$0.394377\pi$
$480$	0	0
$481$	−1384.98	−0.131289
$482$	−21338.6	−2.01649
$483$	0	0
$484$	974.770	0.0915449
$485$	0	0
$486$	0	0
$487$	−762.315	−0.0709318	−0.0354659	−	0.999371i	$-0.511292\pi$
$487$	−762.315	−0.0709318	−0.0354659	+	0.999371i	$0.511292\pi$
$488$	154.758	0.0143557
$489$	0	0
$490$	0	0
$491$	−1478.23	−0.135868	−0.0679342	−	0.997690i	$-0.521641\pi$
$491$	−1478.23	−0.135868	−0.0679342	+	0.997690i	$0.521641\pi$
$492$	0	0
$493$	−3260.34	−0.297846
$494$	6749.69	0.614743
$495$	0	0
$496$	1932.66	0.174958
$497$	−26661.7	−2.40632
$498$	0	0
$499$	−11087.2	−0.994648	−0.497324	−	0.867565i	$-0.665684\pi$
$499$	−11087.2	−0.994648	−0.497324	+	0.867565i	$0.665684\pi$
$500$	0	0
$501$	0	0
$502$	−17699.1	−1.57360
$503$	7336.27	0.650314	0.325157	−	0.945660i	$-0.394583\pi$
$503$	7336.27	0.650314	0.325157	+	0.945660i	$0.394583\pi$
$504$	0	0
$505$	0	0
$506$	−563.227	−0.0494832
$507$	0	0
$508$	19285.3	1.68434
$509$	15419.2	1.34272	0.671360	−	0.741131i	$-0.265710\pi$
$509$	15419.2	1.34272	0.671360	+	0.741131i	$0.265710\pi$
$510$	0	0
$511$	−23365.1	−2.02272
$512$	−16410.2	−1.41648
$513$	0	0
$514$	4204.29	0.360785
$515$	0	0
$516$	0	0
$517$	−863.651	−0.0734688
$518$	−12897.0	−1.09394
$519$	0	0
$520$	0	0
$521$	−20182.9	−1.69717	−0.848587	−	0.529056i	$-0.822546\pi$
$521$	−20182.9	−1.69717	−0.848587	+	0.529056i	$0.822546\pi$
$522$	0	0
$523$	15479.7	1.29422	0.647112	−	0.762395i	$-0.275977\pi$
$523$	15479.7	1.29422	0.647112	+	0.762395i	$0.275977\pi$
$524$	19036.7	1.58706
$525$	0	0
$526$	−11115.1	−0.921369
$527$	2664.15	0.220213
$528$	0	0
$529$	−12003.7	−0.986580
$530$	0	0
$531$	0	0
$532$	31536.2	2.57005
$533$	5031.10	0.408858
$534$	0	0
$535$	0	0
$536$	156.175	0.0125853
$537$	0	0
$538$	−6595.09	−0.528503
$539$	3822.04	0.305430
$540$	0	0
$541$	−511.002	−0.0406094	−0.0203047	−	0.999794i	$-0.506464\pi$
$541$	−511.002	−0.0406094	−0.0203047	+	0.999794i	$0.506464\pi$
$542$	34458.4	2.73084
$543$	0	0
$544$	−22464.1	−1.77048
$545$	0	0
$546$	0	0
$547$	18912.8	1.47834	0.739170	−	0.673519i	$-0.235218\pi$
$547$	18912.8	1.47834	0.739170	+	0.673519i	$0.235218\pi$
$548$	1677.76	0.130785
$549$	0	0
$550$	0	0
$551$	5544.63	0.428692
$552$	0	0
$553$	−22551.9	−1.73418
$554$	−15662.0	−1.20111
$555$	0	0
$556$	−4931.14	−0.376128
$557$	7869.53	0.598640	0.299320	−	0.954153i	$-0.403240\pi$
$557$	7869.53	0.598640	0.299320	+	0.954153i	$0.403240\pi$
$558$	0	0
$559$	−409.885	−0.0310131
$560$	0	0
$561$	0	0
$562$	−7962.72	−0.597664
$563$	−12877.3	−0.963964	−0.481982	−	0.876181i	$-0.660083\pi$
$563$	−12877.3	−0.963964	−0.481982	+	0.876181i	$0.660083\pi$
$564$	0	0
$565$	0	0
$566$	−4927.28	−0.365917
$567$	0	0
$568$	227.494	0.0168054
$569$	7070.93	0.520965	0.260482	−	0.965479i	$-0.416118\pi$
$569$	7070.93	0.520965	0.260482	+	0.965479i	$0.416118\pi$
$570$	0	0
$571$	7455.30	0.546400	0.273200	−	0.961957i	$-0.411918\pi$
$571$	7455.30	0.546400	0.273200	+	0.961957i	$0.411918\pi$
$572$	−1001.96	−0.0732416
$573$	0	0
$574$	46849.8	3.40675
$575$	0	0
$576$	0	0
$577$	−13840.8	−0.998611	−0.499306	−	0.866426i	$-0.666412\pi$
$577$	−13840.8	−0.998611	−0.499306	+	0.866426i	$0.666412\pi$
$578$	−11063.6	−0.796168
$579$	0	0
$580$	0	0
$581$	−3043.95	−0.217357
$582$	0	0
$583$	3765.80	0.267519
$584$	199.365	0.0141264
$585$	0	0
$586$	−20304.3	−1.43134
$587$	−5382.51	−0.378466	−0.189233	−	0.981932i	$-0.560600\pi$
$587$	−5382.51	−0.378466	−0.189233	+	0.981932i	$0.560600\pi$
$588$	0	0
$589$	−4530.75	−0.316954
$590$	0	0
$591$	0	0
$592$	−7784.18	−0.540419
$593$	−17176.5	−1.18947	−0.594735	−	0.803922i	$-0.702743\pi$
$593$	−17176.5	−1.18947	−0.594735	+	0.803922i	$0.702743\pi$
$594$	0	0
$595$	0	0
$596$	17049.8	1.17179
$597$	0	0
$598$	578.939	0.0395896
$599$	−4846.64	−0.330598	−0.165299	−	0.986243i	$-0.552859\pi$
$599$	−4846.64	−0.330598	−0.165299	+	0.986243i	$0.552859\pi$
$600$	0	0
$601$	−5837.26	−0.396184	−0.198092	−	0.980183i	$-0.563475\pi$
$601$	−5837.26	−0.396184	−0.198092	+	0.980183i	$0.563475\pi$
$602$	−3816.87	−0.258412
$603$	0	0
$604$	−27435.6	−1.84824
$605$	0	0
$606$	0	0
$607$	−22838.2	−1.52714	−0.763572	−	0.645723i	$-0.776556\pi$
$607$	−22838.2	−1.52714	−0.763572	+	0.645723i	$0.776556\pi$
$608$	38203.3	2.54827
$609$	0	0
$610$	0	0
$611$	887.744	0.0587795
$612$	0	0
$613$	22566.4	1.48687	0.743433	−	0.668811i	$-0.233196\pi$
$613$	22566.4	1.48687	0.743433	+	0.668811i	$0.233196\pi$
$614$	−1288.90	−0.0847160
$615$	0	0
$616$	−64.8056	−0.00423878
$617$	22553.3	1.47158	0.735789	−	0.677211i	$-0.236812\pi$
$617$	22553.3	1.47158	0.735789	+	0.677211i	$0.236812\pi$
$618$	0	0
$619$	−4053.59	−0.263211	−0.131605	−	0.991302i	$-0.542013\pi$
$619$	−4053.59	−0.263211	−0.131605	+	0.991302i	$0.542013\pi$
$620$	0	0
$621$	0	0
$622$	−20576.9	−1.32646
$623$	−14537.9	−0.934909
$624$	0	0
$625$	0	0
$626$	33524.7	2.14044
$627$	0	0
$628$	2371.24	0.150673
$629$	−10730.4	−0.680205
$630$	0	0
$631$	18912.2	1.19316	0.596580	−	0.802554i	$-0.296526\pi$
$631$	18912.2	1.19316	0.596580	+	0.802554i	$0.296526\pi$
$632$	192.427	0.0121113
$633$	0	0
$634$	−31032.6	−1.94394
$635$	0	0
$636$	0	0
$637$	−3928.66	−0.244363
$638$	−1640.44	−0.101796
$639$	0	0
$640$	0	0
$641$	−11120.8	−0.685248	−0.342624	−	0.939473i	$-0.611316\pi$
$641$	−11120.8	−0.685248	−0.342624	+	0.939473i	$0.611316\pi$
$642$	0	0
$643$	−6066.39	−0.372061	−0.186030	−	0.982544i	$-0.559562\pi$
$643$	−6066.39	−0.372061	−0.186030	+	0.982544i	$0.559562\pi$
$644$	2704.94	0.165512
$645$	0	0
$646$	52294.3	3.18497
$647$	−20954.2	−1.27325	−0.636627	−	0.771172i	$-0.719671\pi$
$647$	−20954.2	−1.27325	−0.636627	+	0.771172i	$0.719671\pi$
$648$	0	0
$649$	−4148.35	−0.250904
$650$	0	0
$651$	0	0
$652$	−23183.1	−1.39251
$653$	−9497.10	−0.569142	−0.284571	−	0.958655i	$-0.591851\pi$
$653$	−9497.10	−0.569142	−0.284571	+	0.958655i	$0.591851\pi$
$654$	0	0
$655$	0	0
$656$	28276.9	1.68297
$657$	0	0
$658$	8266.71	0.489772
$659$	21562.5	1.27459	0.637295	−	0.770620i	$-0.280053\pi$
$659$	21562.5	1.27459	0.637295	+	0.770620i	$0.280053\pi$
$660$	0	0
$661$	−15127.3	−0.890141	−0.445070	−	0.895496i	$-0.646821\pi$
$661$	−15127.3	−0.890141	−0.445070	+	0.895496i	$0.646821\pi$
$662$	−807.114	−0.0473857
$663$	0	0
$664$	25.9729	0.00151799
$665$	0	0
$666$	0	0
$667$	475.578	0.0276079
$668$	−20192.9	−1.16959
$669$	0	0
$670$	0	0
$671$	−7592.69	−0.436829
$672$	0	0
$673$	2499.23	0.143148	0.0715738	−	0.997435i	$-0.477198\pi$
$673$	2499.23	0.143148	0.0715738	+	0.997435i	$0.477198\pi$
$674$	−5093.64	−0.291098
$675$	0	0
$676$	−16669.0	−0.948397
$677$	−32630.0	−1.85240	−0.926199	−	0.377036i	$-0.876943\pi$
$677$	−32630.0	−1.85240	−0.926199	+	0.377036i	$0.876943\pi$
$678$	0	0
$679$	8831.82	0.499167
$680$	0	0
$681$	0	0
$682$	1340.47	0.0752628
$683$	9737.59	0.545532	0.272766	−	0.962080i	$-0.412061\pi$
$683$	9737.59	0.545532	0.272766	+	0.962080i	$0.412061\pi$
$684$	0	0
$685$	0	0
$686$	−469.388	−0.0261244
$687$	0	0
$688$	−2303.73	−0.127658
$689$	−3870.85	−0.214032
$690$	0	0
$691$	−8508.21	−0.468404	−0.234202	−	0.972188i	$-0.575248\pi$
$691$	−8508.21	−0.468404	−0.234202	+	0.972188i	$0.575248\pi$
$692$	−24911.3	−1.36848
$693$	0	0
$694$	29150.8	1.59445
$695$	0	0
$696$	0	0
$697$	38979.3	2.11829
$698$	−1174.42	−0.0636853
$699$	0	0
$700$	0	0
$701$	22843.3	1.23078	0.615391	−	0.788222i	$-0.288998\pi$
$701$	22843.3	1.23078	0.615391	+	0.788222i	$0.288998\pi$
$702$	0	0
$703$	18248.5	0.979024
$704$	−5710.51	−0.305714
$705$	0	0
$706$	4490.54	0.239382
$707$	−33558.2	−1.78513
$708$	0	0
$709$	8510.36	0.450794	0.225397	−	0.974267i	$-0.427632\pi$
$709$	8510.36	0.450794	0.225397	+	0.974267i	$0.427632\pi$
$710$	0	0
$711$	0	0
$712$	124.046	0.00652926
$713$	−388.615	−0.0204120
$714$	0	0
$715$	0	0
$716$	27074.3	1.41315
$717$	0	0
$718$	−14198.2	−0.737981
$719$	−31326.2	−1.62486	−0.812428	−	0.583062i	$-0.801854\pi$
$719$	−31326.2	−1.62486	−0.812428	+	0.583062i	$0.801854\pi$
$720$	0	0
$721$	28263.8	1.45992
$722$	−61449.5	−3.16747
$723$	0	0
$724$	1678.45	0.0861591
$725$	0	0
$726$	0	0
$727$	13516.6	0.689549	0.344774	−	0.938686i	$-0.387955\pi$
$727$	13516.6	0.689549	0.344774	+	0.938686i	$0.387955\pi$
$728$	66.6134	0.00339129
$729$	0	0
$730$	0	0
$731$	−3175.66	−0.160678
$732$	0	0
$733$	9310.89	0.469175	0.234588	−	0.972095i	$-0.424626\pi$
$733$	9310.89	0.469175	0.234588	+	0.972095i	$0.424626\pi$
$734$	24034.3	1.20861
$735$	0	0
$736$	3276.80	0.164109
$737$	−7662.21	−0.382959
$738$	0	0
$739$	14829.0	0.738149	0.369074	−	0.929400i	$-0.379675\pi$
$739$	14829.0	0.738149	0.369074	+	0.929400i	$0.379675\pi$
$740$	0	0
$741$	0	0
$742$	−36045.5	−1.78339
$743$	−7376.85	−0.364240	−0.182120	−	0.983276i	$-0.558296\pi$
$743$	−7376.85	−0.364240	−0.182120	+	0.983276i	$0.558296\pi$
$744$	0	0
$745$	0	0
$746$	−18840.5	−0.924664
$747$	0	0
$748$	−7762.88	−0.379464
$749$	50025.1	2.44042
$750$	0	0
$751$	−2568.19	−0.124786	−0.0623932	−	0.998052i	$-0.519873\pi$
$751$	−2568.19	−0.124786	−0.0623932	+	0.998052i	$0.519873\pi$
$752$	4989.49	0.241952
$753$	0	0
$754$	1686.20	0.0814427
$755$	0	0
$756$	0	0
$757$	24908.1	1.19590	0.597952	−	0.801532i	$-0.295981\pi$
$757$	24908.1	1.19590	0.597952	+	0.801532i	$0.295981\pi$
$758$	43213.7	2.07070
$759$	0	0
$760$	0	0
$761$	−33333.8	−1.58784	−0.793922	−	0.608020i	$-0.791964\pi$
$761$	−33333.8	−1.58784	−0.793922	+	0.608020i	$0.791964\pi$
$762$	0	0
$763$	−23508.5	−1.11542
$764$	2613.54	0.123763
$765$	0	0
$766$	21305.8	1.00497
$767$	4264.07	0.200739
$768$	0	0
$769$	−27944.9	−1.31043	−0.655215	−	0.755443i	$-0.727422\pi$
$769$	−27944.9	−1.31043	−0.655215	+	0.755443i	$0.727422\pi$
$770$	0	0
$771$	0	0
$772$	−26378.9	−1.22979
$773$	−11831.3	−0.550510	−0.275255	−	0.961371i	$-0.588762\pi$
$773$	−11831.3	−0.550510	−0.275255	+	0.961371i	$0.588762\pi$
$774$	0	0
$775$	0	0
$776$	−75.3586	−0.00348610
$777$	0	0
$778$	42783.6	1.97155
$779$	−66289.5	−3.04887
$780$	0	0
$781$	−11161.2	−0.511370
$782$	4485.43	0.205113
$783$	0	0
$784$	−22080.7	−1.00586
$785$	0	0
$786$	0	0
$787$	−13469.7	−0.610092	−0.305046	−	0.952338i	$-0.598672\pi$
$787$	−13469.7	−0.610092	−0.305046	+	0.952338i	$0.598672\pi$
$788$	5178.57	0.234110
$789$	0	0
$790$	0	0
$791$	−6.64146	−0.000298537 0
$792$	0	0
$793$	7804.50	0.349490
$794$	−35753.9	−1.59806
$795$	0	0
$796$	−2582.04	−0.114972
$797$	897.262	0.0398778	0.0199389	−	0.999801i	$-0.493653\pi$
$797$	897.262	0.0398778	0.0199389	+	0.999801i	$0.493653\pi$
$798$	0	0
$799$	6877.95	0.304536
$800$	0	0
$801$	0	0
$802$	−12732.3	−0.560591
$803$	−9781.18	−0.429851
$804$	0	0
$805$	0	0
$806$	−1377.86	−0.0602149
$807$	0	0
$808$	286.339	0.0124671
$809$	5655.30	0.245772	0.122886	−	0.992421i	$-0.460785\pi$
$809$	5655.30	0.245772	0.122886	+	0.992421i	$0.460785\pi$
$810$	0	0
$811$	9401.84	0.407082	0.203541	−	0.979066i	$-0.434755\pi$
$811$	9401.84	0.407082	0.203541	+	0.979066i	$0.434755\pi$
$812$	7878.34	0.340487
$813$	0	0
$814$	−5399.00	−0.232475
$815$	0	0
$816$	0	0
$817$	5400.63	0.231266
$818$	10057.1	0.429875
$819$	0	0
$820$	0	0
$821$	−3692.36	−0.156960	−0.0784800	−	0.996916i	$-0.525007\pi$
$821$	−3692.36	−0.156960	−0.0784800	+	0.996916i	$0.525007\pi$
$822$	0	0
$823$	26803.8	1.13526	0.567632	−	0.823282i	$-0.307860\pi$
$823$	26803.8	1.13526	0.567632	+	0.823282i	$0.307860\pi$
$824$	−241.165	−0.0101958
$825$	0	0
$826$	39707.2	1.67263
$827$	11945.5	0.502279	0.251139	−	0.967951i	$-0.419195\pi$
$827$	11945.5	0.502279	0.251139	+	0.967951i	$0.419195\pi$
$828$	0	0
$829$	41215.0	1.72673	0.863364	−	0.504582i	$-0.168354\pi$
$829$	41215.0	1.72673	0.863364	+	0.504582i	$0.168354\pi$
$830$	0	0
$831$	0	0
$832$	5869.81	0.244590
$833$	−30438.0	−1.26604
$834$	0	0
$835$	0	0
$836$	13201.8	0.546165
$837$	0	0
$838$	37874.5	1.56128
$839$	−11797.4	−0.485447	−0.242724	−	0.970095i	$-0.578041\pi$
$839$	−11797.4	−0.485447	−0.242724	+	0.970095i	$0.578041\pi$
$840$	0	0
$841$	−23003.8	−0.943206
$842$	7310.06	0.299194
$843$	0	0
$844$	−41557.5	−1.69487
$845$	0	0
$846$	0	0
$847$	3179.46	0.128982
$848$	−21755.8	−0.881010
$849$	0	0
$850$	0	0
$851$	1565.22	0.0630494
$852$	0	0
$853$	−36817.4	−1.47785	−0.738923	−	0.673789i	$-0.764666\pi$
$853$	−36817.4	−1.47785	−0.738923	+	0.673789i	$0.764666\pi$
$854$	72675.8	2.91208
$855$	0	0
$856$	−426.845	−0.0170435
$857$	42922.8	1.71087	0.855435	−	0.517911i	$-0.173290\pi$
$857$	42922.8	1.71087	0.855435	+	0.517911i	$0.173290\pi$
$858$	0	0
$859$	11693.5	0.464468	0.232234	−	0.972660i	$-0.425396\pi$
$859$	11693.5	0.464468	0.232234	+	0.972660i	$0.425396\pi$
$860$	0	0
$861$	0	0
$862$	61496.5	2.42991
$863$	−28640.1	−1.12969	−0.564844	−	0.825198i	$-0.691064\pi$
$863$	−28640.1	−1.12969	−0.564844	+	0.825198i	$0.691064\pi$
$864$	0	0
$865$	0	0
$866$	−20045.5	−0.786574
$867$	0	0
$868$	−6437.72	−0.251740
$869$	−9440.76	−0.368534
$870$	0	0
$871$	7875.95	0.306391
$872$	200.589	0.00778990
$873$	0	0
$874$	−7628.07	−0.295221
$875$	0	0
$876$	0	0
$877$	−19344.1	−0.744815	−0.372407	−	0.928069i	$-0.621468\pi$
$877$	−19344.1	−0.744815	−0.372407	+	0.928069i	$0.621468\pi$
$878$	10012.1	0.384844
$879$	0	0
$880$	0	0
$881$	−24160.8	−0.923949	−0.461974	−	0.886893i	$-0.652859\pi$
$881$	−24160.8	−0.923949	−0.461974	+	0.886893i	$0.652859\pi$
$882$	0	0
$883$	−30965.1	−1.18014	−0.590068	−	0.807354i	$-0.700899\pi$
$883$	−30965.1	−1.18014	−0.590068	+	0.807354i	$0.700899\pi$
$884$	7979.43	0.303594
$885$	0	0
$886$	−12197.6	−0.462513
$887$	−40432.5	−1.53054	−0.765270	−	0.643709i	$-0.777395\pi$
$887$	−40432.5	−1.53054	−0.765270	+	0.643709i	$0.777395\pi$
$888$	0	0
$889$	62903.9	2.37315
$890$	0	0
$891$	0	0
$892$	23753.1	0.891606
$893$	−11696.9	−0.438321
$894$	0	0
$895$	0	0
$896$	754.083	0.0281162
$897$	0	0
$898$	52831.7	1.96327
$899$	−1131.87	−0.0419910
$900$	0	0
$901$	−29990.1	−1.10889
$902$	19612.5	0.723972
$903$	0	0
$904$	0.0566690	2.08494e−6 0
$905$	0	0
$906$	0	0
$907$	−13270.3	−0.485814	−0.242907	−	0.970050i	$-0.578101\pi$
$907$	−13270.3	−0.485814	−0.242907	+	0.970050i	$0.578101\pi$
$908$	−18376.6	−0.671639
$909$	0	0
$910$	0	0
$911$	−43574.4	−1.58472	−0.792362	−	0.610051i	$-0.791149\pi$
$911$	−43574.4	−1.58472	−0.792362	+	0.610051i	$0.791149\pi$
$912$	0	0
$913$	−1274.27	−0.0461909
$914$	−67872.7	−2.45627
$915$	0	0
$916$	29135.3	1.05093
$917$	62093.0	2.23609
$918$	0	0
$919$	41317.7	1.48307	0.741537	−	0.670912i	$-0.234097\pi$
$919$	41317.7	1.48307	0.741537	+	0.670912i	$0.234097\pi$
$920$	0	0
$921$	0	0
$922$	20791.9	0.742674
$923$	11472.6	0.409128
$924$	0	0
$925$	0	0
$926$	60497.0	2.14693
$927$	0	0
$928$	9543.90	0.337601
$929$	27886.3	0.984844	0.492422	−	0.870357i	$-0.336112\pi$
$929$	27886.3	0.984844	0.492422	+	0.870357i	$0.336112\pi$
$930$	0	0
$931$	51763.8	1.82222
$932$	36825.4	1.29427
$933$	0	0
$934$	54483.8	1.90874
$935$	0	0
$936$	0	0
$937$	16036.3	0.559105	0.279553	−	0.960130i	$-0.409814\pi$
$937$	16036.3	0.559105	0.279553	+	0.960130i	$0.409814\pi$
$938$	73341.2	2.55296
$939$	0	0
$940$	0	0
$941$	1506.93	0.0522046	0.0261023	−	0.999659i	$-0.491690\pi$
$941$	1506.93	0.0522046	0.0261023	+	0.999659i	$0.491690\pi$
$942$	0	0
$943$	−5685.83	−0.196348
$944$	23965.9	0.826294
$945$	0	0
$946$	−1597.83	−0.0549154
$947$	−7327.03	−0.251422	−0.125711	−	0.992067i	$-0.540121\pi$
$947$	−7327.03	−0.251422	−0.125711	+	0.992067i	$0.540121\pi$
$948$	0	0
$949$	10054.0	0.343907
$950$	0	0
$951$	0	0
$952$	516.099	0.0175702
$953$	−46874.7	−1.59331	−0.796653	−	0.604437i	$-0.793398\pi$
$953$	−46874.7	−1.59331	−0.796653	+	0.604437i	$0.793398\pi$
$954$	0	0
$955$	0	0
$956$	−20267.7	−0.685673
$957$	0	0
$958$	−27369.0	−0.923019
$959$	5472.45	0.184270
$960$	0	0
$961$	−28866.1	−0.968954
$962$	5549.61	0.185994
$963$	0	0
$964$	42900.8	1.43334
$965$	0	0
$966$	0	0
$967$	−2812.90	−0.0935437	−0.0467719	−	0.998906i	$-0.514893\pi$
$967$	−2812.90	−0.0935437	−0.0467719	+	0.998906i	$0.514893\pi$
$968$	−27.1292	−0.000900790 0
$969$	0	0
$970$	0	0
$971$	45216.4	1.49440	0.747200	−	0.664599i	$-0.231398\pi$
$971$	45216.4	1.49440	0.747200	+	0.664599i	$0.231398\pi$
$972$	0	0
$973$	−16084.2	−0.529944
$974$	3054.59	0.100488
$975$	0	0
$976$	43864.5	1.43859
$977$	−10647.0	−0.348647	−0.174323	−	0.984688i	$-0.555774\pi$
$977$	−10647.0	−0.348647	−0.174323	+	0.984688i	$0.555774\pi$
$978$	0	0
$979$	−6085.91	−0.198679
$980$	0	0
$981$	0	0
$982$	5923.23	0.192483
$983$	−1993.76	−0.0646909	−0.0323455	−	0.999477i	$-0.510298\pi$
$983$	−1993.76	−0.0646909	−0.0323455	+	0.999477i	$0.510298\pi$
$984$	0	0
$985$	0	0
$986$	13064.1	0.421954
$987$	0	0
$988$	−13570.1	−0.436966
$989$	463.226	0.0148936
$990$	0	0
$991$	−23143.6	−0.741858	−0.370929	−	0.928661i	$-0.620961\pi$
$991$	−23143.6	−0.741858	−0.370929	+	0.928661i	$0.620961\pi$
$992$	−7798.71	−0.249606
$993$	0	0
$994$	106833.	3.40899
$995$	0	0
$996$	0	0
$997$	3506.68	0.111392	0.0556959	−	0.998448i	$-0.482262\pi$
$997$	3506.68	0.111392	0.0556959	+	0.998448i	$0.482262\pi$
$998$	44426.1	1.40910
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bq.1.2		7
3.2	odd	2		825.4.a.bc.1.6		7
5.2	odd	4		495.4.c.c.199.4		14
5.3	odd	4		495.4.c.c.199.11		14
5.4	even	2		2475.4.a.br.1.6		7
15.2	even	4		165.4.c.a.34.11	yes	14
15.8	even	4		165.4.c.a.34.4	✓	14
15.14	odd	2		825.4.a.bb.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.c.a.34.4	✓	14	15.8	even	4
165.4.c.a.34.11	yes	14	15.2	even	4
495.4.c.c.199.4		14	5.2	odd	4
495.4.c.c.199.11		14	5.3	odd	4
825.4.a.bb.1.2		7	15.14	odd	2
825.4.a.bc.1.6		7	3.2	odd	2
2475.4.a.bq.1.2		7	1.1	even	1	trivial
2475.4.a.br.1.6		7	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)

\(f(q)\)	\(=\)	\(q-4.00699 q^{2} +8.05595 q^{4} +26.2766 q^{7} -0.224208 q^{8} +O(q^{10})\)	\(q-4.00699 q^{2} +8.05595 q^{4} +26.2766 q^{7} -0.224208 q^{8} +11.0000 q^{11} -11.3069 q^{13} -105.290 q^{14} -63.5492 q^{16} -87.6018 q^{17} +148.978 q^{19} -44.0769 q^{22} +12.7783 q^{23} +45.3065 q^{26} +211.683 q^{28} +37.2177 q^{29} -30.4121 q^{31} +256.435 q^{32} +351.019 q^{34} +122.491 q^{37} -596.955 q^{38} -444.960 q^{41} +36.2510 q^{43} +88.6155 q^{44} -51.2025 q^{46} -78.5138 q^{47} +347.458 q^{49} -91.0876 q^{52} +342.345 q^{53} -5.89142 q^{56} -149.131 q^{58} -377.123 q^{59} -690.244 q^{61} +121.861 q^{62} -519.137 q^{64} -696.564 q^{67} -705.716 q^{68} -1014.66 q^{71} -889.198 q^{73} -490.818 q^{74} +1200.16 q^{76} +289.042 q^{77} -858.251 q^{79} +1782.95 q^{82} -115.843 q^{83} -145.257 q^{86} -2.46629 q^{88} -553.264 q^{89} -297.106 q^{91} +102.941 q^{92} +314.604 q^{94} +336.110 q^{97} -1392.26 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{2} + 41 q^{4} + 50 q^{7} - 21 q^{8}+O(q^{10}) \) 7 * q - 3 * q^2 + 41 * q^4 + 50 * q^7 - 21 * q^8	\( 7 q - 3 q^{2} + 41 q^{4} + 50 q^{7} - 21 q^{8} + 77 q^{11} + 24 q^{13} - 142 q^{14} + 181 q^{16} - 38 q^{17} + 26 q^{19} - 33 q^{22} - 228 q^{23} - 476 q^{26} + 840 q^{28} - 572 q^{29} - 140 q^{31} - 991 q^{32} - 806 q^{34} - 104 q^{37} - 498 q^{38} - 896 q^{41} + 614 q^{43} + 451 q^{44} - 344 q^{46} - 520 q^{47} + 295 q^{49} - 26 q^{52} - 380 q^{53} - 1522 q^{56} + 1600 q^{58} - 1316 q^{59} - 386 q^{61} - 440 q^{62} + 869 q^{64} + 348 q^{67} - 332 q^{68} - 804 q^{71} + 468 q^{73} + 748 q^{74} - 1698 q^{76} + 550 q^{77} - 374 q^{79} - 620 q^{82} - 3128 q^{83} + 2534 q^{86} - 231 q^{88} - 694 q^{89} - 3376 q^{91} + 1184 q^{92} - 2920 q^{94} - 8 q^{97} - 4211 q^{98}+O(q^{100}) \) 7 * q - 3 * q^2 + 41 * q^4 + 50 * q^7 - 21 * q^8 + 77 * q^11 + 24 * q^13 - 142 * q^14 + 181 * q^16 - 38 * q^17 + 26 * q^19 - 33 * q^22 - 228 * q^23 - 476 * q^26 + 840 * q^28 - 572 * q^29 - 140 * q^31 - 991 * q^32 - 806 * q^34 - 104 * q^37 - 498 * q^38 - 896 * q^41 + 614 * q^43 + 451 * q^44 - 344 * q^46 - 520 * q^47 + 295 * q^49 - 26 * q^52 - 380 * q^53 - 1522 * q^56 + 1600 * q^58 - 1316 * q^59 - 386 * q^61 - 440 * q^62 + 869 * q^64 + 348 * q^67 - 332 * q^68 - 804 * q^71 + 468 * q^73 + 748 * q^74 - 1698 * q^76 + 550 * q^77 - 374 * q^79 - 620 * q^82 - 3128 * q^83 + 2534 * q^86 - 231 * q^88 - 694 * q^89 - 3376 * q^91 + 1184 * q^92 - 2920 * q^94 - 8 * q^97 - 4211 * q^98

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