LMFDB - Embedded newform 3675.2.a.bm.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 2x + 1 $ x^4 - 2x^3 - 3x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.77462$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.77462 q^{2} -1.00000 q^{3} +1.14929 q^{4} -1.77462 q^{6} -1.50970 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.77462 q^{2} -1.00000 q^{3} +1.14929 q^{4} -1.77462 q^{6} -1.50970 q^{8} +1.00000 q^{9} -1.21112 q^{11} -1.14929 q^{12} +1.62534 q^{13} -4.97771 q^{16} +5.33812 q^{17} +1.77462 q^{18} +6.22248 q^{19} -2.14929 q^{22} -1.31873 q^{23} +1.50970 q^{24} +2.88436 q^{26} -1.00000 q^{27} -3.80827 q^{29} -2.85585 q^{31} -5.81417 q^{32} +1.21112 q^{33} +9.47316 q^{34} +1.14929 q^{36} +3.37767 q^{37} +11.0426 q^{38} -1.62534 q^{39} +2.98061 q^{41} -11.4610 q^{43} -1.39193 q^{44} -2.34025 q^{46} -6.87601 q^{47} +4.97771 q^{48} -5.33812 q^{51} +1.86798 q^{52} +12.5686 q^{53} -1.77462 q^{54} -6.22248 q^{57} -6.75824 q^{58} +12.4925 q^{59} +5.17157 q^{61} -5.06806 q^{62} -0.362537 q^{64} +2.14929 q^{66} +3.03878 q^{67} +6.13503 q^{68} +1.31873 q^{69} +12.4845 q^{71} -1.50970 q^{72} +12.3290 q^{73} +5.99410 q^{74} +7.15141 q^{76} -2.88436 q^{78} +13.8387 q^{79} +1.00000 q^{81} +5.28946 q^{82} +8.74824 q^{83} -20.3390 q^{86} +3.80827 q^{87} +1.82843 q^{88} +12.5329 q^{89} -1.51560 q^{92} +2.85585 q^{93} -12.2023 q^{94} +5.81417 q^{96} -4.81707 q^{97} -1.21112 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^9	$ 4 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9} - 4 q^{11} - 2 q^{12} - 6 q^{16} + 4 q^{17} - 2 q^{18} + 8 q^{19} - 6 q^{22} + 12 q^{26} - 4 q^{27} - 4 q^{29} + 8 q^{31} - 2 q^{32} + 4 q^{33} + 8 q^{34} + 2 q^{36} - 16 q^{37} - 4 q^{38} + 24 q^{41} - 20 q^{43} + 14 q^{44} - 6 q^{46} - 8 q^{47} + 6 q^{48} - 4 q^{51} - 16 q^{52} + 20 q^{53} + 2 q^{54} - 8 q^{57} + 6 q^{58} + 8 q^{59} + 32 q^{61} - 28 q^{62} - 12 q^{64} + 6 q^{66} - 12 q^{67} + 12 q^{68} + 4 q^{71} + 34 q^{74} + 40 q^{76} - 12 q^{78} + 4 q^{81} - 16 q^{82} + 20 q^{83} - 14 q^{86} + 4 q^{87} - 4 q^{88} + 8 q^{89} + 10 q^{92} - 8 q^{93} - 32 q^{94} + 2 q^{96} - 24 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^9 - 4 * q^11 - 2 * q^12 - 6 * q^16 + 4 * q^17 - 2 * q^18 + 8 * q^19 - 6 * q^22 + 12 * q^26 - 4 * q^27 - 4 * q^29 + 8 * q^31 - 2 * q^32 + 4 * q^33 + 8 * q^34 + 2 * q^36 - 16 * q^37 - 4 * q^38 + 24 * q^41 - 20 * q^43 + 14 * q^44 - 6 * q^46 - 8 * q^47 + 6 * q^48 - 4 * q^51 - 16 * q^52 + 20 * q^53 + 2 * q^54 - 8 * q^57 + 6 * q^58 + 8 * q^59 + 32 * q^61 - 28 * q^62 - 12 * q^64 + 6 * q^66 - 12 * q^67 + 12 * q^68 + 4 * q^71 + 34 * q^74 + 40 * q^76 - 12 * q^78 + 4 * q^81 - 16 * q^82 + 20 * q^83 - 14 * q^86 + 4 * q^87 - 4 * q^88 + 8 * q^89 + 10 * q^92 - 8 * q^93 - 32 * q^94 + 2 * q^96 - 24 * q^97 - 4 * q^99

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 $

$2$	1.77462	1.25485	0.627424	−	0.778678i	$-0.284109\pi$
$2$	1.77462	1.25485	0.627424	+	0.778678i	$0.284109\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.14929	0.574643
$5$	0	0
$5$	0	0
$6$	−1.77462	−0.724487
$7$	0	0
$7$	0	0
$8$	−1.50970	−0.533758
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.21112	−0.365167	−0.182584	−	0.983190i	$-0.558446\pi$
$11$	−1.21112	−0.365167	−0.182584	+	0.983190i	$0.558446\pi$
$12$	−1.14929	−0.331770
$13$	1.62534	0.450787	0.225394	−	0.974268i	$-0.427633\pi$
$13$	1.62534	0.450787	0.225394	+	0.974268i	$0.427633\pi$
$14$	0	0
$15$	0	0
$16$	−4.97771	−1.24443
$17$	5.33812	1.29468	0.647342	−	0.762199i	$-0.275880\pi$
$17$	5.33812	1.29468	0.647342	+	0.762199i	$0.275880\pi$
$18$	1.77462	0.418283
$19$	6.22248	1.42754	0.713768	−	0.700383i	$-0.246987\pi$
$19$	6.22248	1.42754	0.713768	+	0.700383i	$0.246987\pi$
$20$	0	0
$21$	0	0
$22$	−2.14929	−0.458229
$23$	−1.31873	−0.274974	−0.137487	−	0.990504i	$-0.543903\pi$
$23$	−1.31873	−0.274974	−0.137487	+	0.990504i	$0.543903\pi$
$24$	1.50970	0.308165
$25$	0	0
$26$	2.88436	0.565669
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.80827	−0.707178	−0.353589	−	0.935401i	$-0.615039\pi$
$29$	−3.80827	−0.707178	−0.353589	+	0.935401i	$0.615039\pi$
$30$	0	0
$31$	−2.85585	−0.512926	−0.256463	−	0.966554i	$-0.582557\pi$
$31$	−2.85585	−0.512926	−0.256463	+	0.966554i	$0.582557\pi$
$32$	−5.81417	−1.02781
$33$	1.21112	0.210829
$34$	9.47316	1.62463
$35$	0	0
$36$	1.14929	0.191548
$37$	3.37767	0.555286	0.277643	−	0.960684i	$-0.410447\pi$
$37$	3.37767	0.555286	0.277643	+	0.960684i	$0.410447\pi$
$38$	11.0426	1.79134
$39$	−1.62534	−0.260262
$40$	0	0
$41$	2.98061	0.465493	0.232746	−	0.972537i	$-0.425229\pi$
$41$	2.98061	0.465493	0.232746	+	0.972537i	$0.425229\pi$
$42$	0	0
$43$	−11.4610	−1.74779	−0.873895	−	0.486114i	$-0.838414\pi$
$43$	−11.4610	−1.74779	−0.873895	+	0.486114i	$0.838414\pi$
$44$	−1.39193	−0.209841
$45$	0	0
$46$	−2.34025	−0.345051
$47$	−6.87601	−1.00297	−0.501485	−	0.865167i	$-0.667213\pi$
$47$	−6.87601	−1.00297	−0.501485	+	0.865167i	$0.667213\pi$
$48$	4.97771	0.718471
$49$	0	0
$50$	0	0
$51$	−5.33812	−0.747487
$52$	1.86798	0.259042
$53$	12.5686	1.72644	0.863218	−	0.504832i	$-0.168446\pi$
$53$	12.5686	1.72644	0.863218	+	0.504832i	$0.168446\pi$
$54$	−1.77462	−0.241496
$55$	0	0
$56$	0	0
$57$	−6.22248	−0.824188
$58$	−6.75824	−0.887400
$59$	12.4925	1.62639	0.813196	−	0.581991i	$-0.197726\pi$
$59$	12.4925	1.62639	0.813196	+	0.581991i	$0.197726\pi$
$60$	0	0
$61$	5.17157	0.662152	0.331076	−	0.943604i	$-0.392588\pi$
$61$	5.17157	0.662152	0.331076	+	0.943604i	$0.392588\pi$
$62$	−5.06806	−0.643644
$63$	0	0
$64$	−0.362537	−0.0453172
$65$	0	0
$66$	2.14929	0.264559
$67$	3.03878	0.371246	0.185623	−	0.982621i	$-0.440570\pi$
$67$	3.03878	0.371246	0.185623	+	0.982621i	$0.440570\pi$
$68$	6.13503	0.743982
$69$	1.31873	0.158757
$70$	0	0
$71$	12.4845	1.48164	0.740820	−	0.671704i	$-0.234437\pi$
$71$	12.4845	1.48164	0.740820	+	0.671704i	$0.234437\pi$
$72$	−1.50970	−0.177919
$73$	12.3290	1.44300	0.721501	−	0.692414i	$-0.243453\pi$
$73$	12.3290	1.44300	0.721501	+	0.692414i	$0.243453\pi$
$74$	5.99410	0.696799
$75$	0	0
$76$	7.15141	0.820323
$77$	0	0
$78$	−2.88436	−0.326589
$79$	13.8387	1.55698	0.778488	−	0.627660i	$-0.215987\pi$
$79$	13.8387	1.55698	0.778488	+	0.627660i	$0.215987\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.28946	0.584123
$83$	8.74824	0.960244	0.480122	−	0.877202i	$-0.340592\pi$
$83$	8.74824	0.960244	0.480122	+	0.877202i	$0.340592\pi$
$84$	0	0
$85$	0	0
$86$	−20.3390	−2.19321
$87$	3.80827	0.408289
$88$	1.82843	0.194911
$89$	12.5329	1.32848	0.664240	−	0.747519i	$-0.268755\pi$
$89$	12.5329	1.32848	0.664240	+	0.747519i	$0.268755\pi$
$90$	0	0
$91$	0	0
$92$	−1.51560	−0.158012
$93$	2.85585	0.296138
$94$	−12.2023	−1.25857
$95$	0	0
$96$	5.81417	0.593407
$97$	−4.81707	−0.489099	−0.244550	−	0.969637i	$-0.578640\pi$
$97$	−4.81707	−0.489099	−0.244550	+	0.969637i	$0.578640\pi$
$98$	0	0
$99$	−1.21112	−0.121722
$100$	0	0
$101$	−0.166550	−0.0165723	−0.00828617	−	0.999966i	$-0.502638\pi$
$101$	−0.166550	−0.0165723	−0.00828617	+	0.999966i	$0.502638\pi$
$102$	−9.47316	−0.937982
$103$	0.605945	0.0597055	0.0298528	−	0.999554i	$-0.490496\pi$
$103$	0.605945	0.0597055	0.0298528	+	0.999554i	$0.490496\pi$
$104$	−2.45376	−0.240611
$105$	0	0
$106$	22.3046	2.16641
$107$	3.26312	0.315457	0.157729	−	0.987482i	$-0.449583\pi$
$107$	3.26312	0.315457	0.157729	+	0.987482i	$0.449583\pi$
$108$	−1.14929	−0.110590
$109$	12.4762	1.19500	0.597500	−	0.801869i	$-0.296161\pi$
$109$	12.4762	1.19500	0.597500	+	0.801869i	$0.296161\pi$
$110$	0	0
$111$	−3.37767	−0.320595
$112$	0	0
$113$	3.95619	0.372167	0.186084	−	0.982534i	$-0.440420\pi$
$113$	3.95619	0.372167	0.186084	+	0.982534i	$0.440420\pi$
$114$	−11.0426	−1.03423
$115$	0	0
$116$	−4.37679	−0.406375
$117$	1.62534	0.150262
$118$	22.1696	2.04087
$119$	0	0
$120$	0	0
$121$	−9.53318	−0.866653
$122$	9.17759	0.830900
$123$	−2.98061	−0.268752
$124$	−3.28219	−0.294749
$125$	0	0
$126$	0	0
$127$	−15.2540	−1.35357	−0.676787	−	0.736179i	$-0.736628\pi$
$127$	−15.2540	−1.35357	−0.676787	+	0.736179i	$0.736628\pi$
$128$	10.9850	0.970944
$129$	11.4610	1.00909
$130$	0	0
$131$	−16.2844	−1.42278	−0.711389	−	0.702799i	$-0.751933\pi$
$131$	−16.2844	−1.42278	−0.711389	+	0.702799i	$0.751933\pi$
$132$	1.39193	0.121152
$133$	0	0
$134$	5.39269	0.465858
$135$	0	0
$136$	−8.05894	−0.691049
$137$	18.9909	1.62250	0.811250	−	0.584699i	$-0.198787\pi$
$137$	18.9909	1.62250	0.811250	+	0.584699i	$0.198787\pi$
$138$	2.34025	0.199215
$139$	15.3096	1.29854	0.649272	−	0.760556i	$-0.275074\pi$
$139$	15.3096	1.29854	0.649272	+	0.760556i	$0.275074\pi$
$140$	0	0
$141$	6.87601	0.579064
$142$	22.1553	1.85923
$143$	−1.96848	−0.164613
$144$	−4.97771	−0.414809
$145$	0	0
$146$	21.8793	1.81075
$147$	0	0
$148$	3.88191	0.319091
$149$	−10.4172	−0.853412	−0.426706	−	0.904390i	$-0.640326\pi$
$149$	−10.4172	−0.853412	−0.426706	+	0.904390i	$0.640326\pi$
$150$	0	0
$151$	17.3331	1.41055	0.705274	−	0.708935i	$-0.250824\pi$
$151$	17.3331	1.41055	0.705274	+	0.708935i	$0.250824\pi$
$152$	−9.39406	−0.761958
$153$	5.33812	0.431562
$154$	0	0
$155$	0	0
$156$	−1.86798	−0.149558
$157$	−19.9657	−1.59344	−0.796718	−	0.604351i	$-0.793433\pi$
$157$	−19.9657	−1.59344	−0.796718	+	0.604351i	$0.793433\pi$
$158$	24.5585	1.95377
$159$	−12.5686	−0.996758
$160$	0	0
$161$	0	0
$162$	1.77462	0.139428
$163$	−25.3079	−1.98227	−0.991135	−	0.132862i	$-0.957583\pi$
$163$	−25.3079	−1.98227	−0.991135	+	0.132862i	$0.957583\pi$
$164$	3.42557	0.267492
$165$	0	0
$166$	15.5248	1.20496
$167$	−4.97181	−0.384730	−0.192365	−	0.981323i	$-0.561616\pi$
$167$	−4.97181	−0.384730	−0.192365	+	0.981323i	$0.561616\pi$
$168$	0	0
$169$	−10.3583	−0.796791
$170$	0	0
$171$	6.22248	0.475845
$172$	−13.1720	−1.00436
$173$	12.0155	0.913518	0.456759	−	0.889590i	$-0.349010\pi$
$173$	12.0155	0.913518	0.456759	+	0.889590i	$0.349010\pi$
$174$	6.75824	0.512341
$175$	0	0
$176$	6.02862	0.454425
$177$	−12.4925	−0.938997
$178$	22.2411	1.66704
$179$	4.57017	0.341591	0.170795	−	0.985307i	$-0.445366\pi$
$179$	4.57017	0.341591	0.170795	+	0.985307i	$0.445366\pi$
$180$	0	0
$181$	8.36330	0.621640	0.310820	−	0.950469i	$-0.399396\pi$
$181$	8.36330	0.621640	0.310820	+	0.950469i	$0.399396\pi$
$182$	0	0
$183$	−5.17157	−0.382294
$184$	1.99088	0.146770
$185$	0	0
$186$	5.06806	0.371608
$187$	−6.46512	−0.472777
$188$	−7.90250	−0.576349
$189$	0	0
$190$	0	0
$191$	−9.91085	−0.717124	−0.358562	−	0.933506i	$-0.616733\pi$
$191$	−9.91085	−0.717124	−0.358562	+	0.933506i	$0.616733\pi$
$192$	0.362537	0.0261639
$193$	−15.5804	−1.12151	−0.560753	−	0.827983i	$-0.689488\pi$
$193$	−15.5804	−1.12151	−0.560753	+	0.827983i	$0.689488\pi$
$194$	−8.54848	−0.613745
$195$	0	0
$196$	0	0
$197$	−9.96647	−0.710081	−0.355041	−	0.934851i	$-0.615533\pi$
$197$	−9.96647	−0.710081	−0.355041	+	0.934851i	$0.615533\pi$
$198$	−2.14929	−0.152743
$199$	−8.82689	−0.625722	−0.312861	−	0.949799i	$-0.601287\pi$
$199$	−8.82689	−0.625722	−0.312861	+	0.949799i	$0.601287\pi$
$200$	0	0
$201$	−3.03878	−0.214339
$202$	−0.295563	−0.0207958
$203$	0	0
$204$	−6.13503	−0.429538
$205$	0	0
$206$	1.07532	0.0749214
$207$	−1.31873	−0.0916582
$208$	−8.09046	−0.560972
$209$	−7.53619	−0.521289
$210$	0	0
$211$	−14.5401	−1.00098	−0.500492	−	0.865741i	$-0.666847\pi$
$211$	−14.5401	−1.00098	−0.500492	+	0.865741i	$0.666847\pi$
$212$	14.4450	0.992084
$213$	−12.4845	−0.855425
$214$	5.79080	0.395851
$215$	0	0
$216$	1.50970	0.102722
$217$	0	0
$218$	22.1405	1.49954
$219$	−12.3290	−0.833117
$220$	0	0
$221$	8.67625	0.583627
$222$	−5.99410	−0.402297
$223$	−4.26597	−0.285671	−0.142835	−	0.989746i	$-0.545622\pi$
$223$	−4.26597	−0.285671	−0.142835	+	0.989746i	$0.545622\pi$
$224$	0	0
$225$	0	0
$226$	7.02075	0.467014
$227$	−2.51680	−0.167046	−0.0835229	−	0.996506i	$-0.526617\pi$
$227$	−2.51680	−0.167046	−0.0835229	+	0.996506i	$0.526617\pi$
$228$	−7.15141	−0.473614
$229$	17.7391	1.17223	0.586117	−	0.810226i	$-0.300656\pi$
$229$	17.7391	1.17223	0.586117	+	0.810226i	$0.300656\pi$
$230$	0	0
$231$	0	0
$232$	5.74933	0.377462
$233$	−0.366631	−0.0240188	−0.0120094	−	0.999928i	$-0.503823\pi$
$233$	−0.366631	−0.0240188	−0.0120094	+	0.999928i	$0.503823\pi$
$234$	2.88436	0.188556
$235$	0	0
$236$	14.3575	0.934595
$237$	−13.8387	−0.898920
$238$	0	0
$239$	−1.69717	−0.109781	−0.0548904	−	0.998492i	$-0.517481\pi$
$239$	−1.69717	−0.109781	−0.0548904	+	0.998492i	$0.517481\pi$
$240$	0	0
$241$	−2.18217	−0.140566	−0.0702828	−	0.997527i	$-0.522390\pi$
$241$	−2.18217	−0.140566	−0.0702828	+	0.997527i	$0.522390\pi$
$242$	−16.9178	−1.08752
$243$	−1.00000	−0.0641500
$244$	5.94362	0.380501
$245$	0	0
$246$	−5.28946	−0.337243
$247$	10.1136	0.643515
$248$	4.31147	0.273778
$249$	−8.74824	−0.554397
$250$	0	0
$251$	17.8637	1.12755	0.563774	−	0.825929i	$-0.309349\pi$
$251$	17.8637	1.12755	0.563774	+	0.825929i	$0.309349\pi$
$252$	0	0
$253$	1.59715	0.100412
$254$	−27.0701	−1.69853
$255$	0	0
$256$	20.2193	1.26370
$257$	−2.34206	−0.146094	−0.0730469	−	0.997329i	$-0.523272\pi$
$257$	−2.34206	−0.146094	−0.0730469	+	0.997329i	$0.523272\pi$
$258$	20.3390	1.26625
$259$	0	0
$260$	0	0
$261$	−3.80827	−0.235726
$262$	−28.8987	−1.78537
$263$	3.09324	0.190737	0.0953687	−	0.995442i	$-0.469597\pi$
$263$	3.09324	0.190737	0.0953687	+	0.995442i	$0.469597\pi$
$264$	−1.82843	−0.112532
$265$	0	0
$266$	0	0
$267$	−12.5329	−0.766999
$268$	3.49243	0.213334
$269$	23.7919	1.45062	0.725308	−	0.688424i	$-0.241697\pi$
$269$	23.7919	1.45062	0.725308	+	0.688424i	$0.241697\pi$
$270$	0	0
$271$	13.8551	0.841636	0.420818	−	0.907145i	$-0.361743\pi$
$271$	13.8551	0.841636	0.420818	+	0.907145i	$0.361743\pi$
$272$	−26.5716	−1.61114
$273$	0	0
$274$	33.7017	2.03599
$275$	0	0
$276$	1.51560	0.0912284
$277$	30.4510	1.82962	0.914811	−	0.403882i	$-0.132339\pi$
$277$	30.4510	1.82962	0.914811	+	0.403882i	$0.132339\pi$
$278$	27.1688	1.62948
$279$	−2.85585	−0.170975
$280$	0	0
$281$	−15.8185	−0.943655	−0.471828	−	0.881691i	$-0.656406\pi$
$281$	−15.8185	−0.943655	−0.471828	+	0.881691i	$0.656406\pi$
$282$	12.2023	0.726638
$283$	−29.9055	−1.77770	−0.888849	−	0.458200i	$-0.848494\pi$
$283$	−29.9055	−1.77770	−0.888849	+	0.458200i	$0.848494\pi$
$284$	14.3483	0.851414
$285$	0	0
$286$	−3.49331	−0.206564
$287$	0	0
$288$	−5.81417	−0.342603
$289$	11.4956	0.676209
$290$	0	0
$291$	4.81707	0.282382
$292$	14.1696	0.829211
$293$	−26.8499	−1.56859	−0.784294	−	0.620389i	$-0.786975\pi$
$293$	−26.8499	−1.56859	−0.784294	+	0.620389i	$0.786975\pi$
$294$	0	0
$295$	0	0
$296$	−5.09926	−0.296388
$297$	1.21112	0.0702765
$298$	−18.4866	−1.07090
$299$	−2.14338	−0.123955
$300$	0	0
$301$	0	0
$302$	30.7597	1.77002
$303$	0.166550	0.00956805
$304$	−30.9737	−1.77647
$305$	0	0
$306$	9.47316	0.541544
$307$	24.0817	1.37441	0.687206	−	0.726462i	$-0.258837\pi$
$307$	24.0817	1.37441	0.687206	+	0.726462i	$0.258837\pi$
$308$	0	0
$309$	−0.605945	−0.0344710
$310$	0	0
$311$	−7.36238	−0.417482	−0.208741	−	0.977971i	$-0.566937\pi$
$311$	−7.36238	−0.417482	−0.208741	+	0.977971i	$0.566937\pi$
$312$	2.45376	0.138917
$313$	−3.87934	−0.219273	−0.109637	−	0.993972i	$-0.534969\pi$
$313$	−3.87934	−0.219273	−0.109637	+	0.993972i	$0.534969\pi$
$314$	−35.4316	−1.99952
$315$	0	0
$316$	15.9046	0.894705
$317$	3.10505	0.174397	0.0871984	−	0.996191i	$-0.472209\pi$
$317$	3.10505	0.174397	0.0871984	+	0.996191i	$0.472209\pi$
$318$	−22.3046	−1.25078
$319$	4.61228	0.258238
$320$	0	0
$321$	−3.26312	−0.182129
$322$	0	0
$323$	33.2164	1.84821
$324$	1.14929	0.0638492
$325$	0	0
$326$	−44.9120	−2.48745
$327$	−12.4762	−0.689933
$328$	−4.49981	−0.248461
$329$	0	0
$330$	0	0
$331$	−3.37767	−0.185654	−0.0928268	−	0.995682i	$-0.529590\pi$
$331$	−3.37767	−0.185654	−0.0928268	+	0.995682i	$0.529590\pi$
$332$	10.0542	0.551798
$333$	3.37767	0.185095
$334$	−8.82309	−0.482778
$335$	0	0
$336$	0	0
$337$	4.83870	0.263581	0.131790	−	0.991278i	$-0.457927\pi$
$337$	4.83870	0.263581	0.131790	+	0.991278i	$0.457927\pi$
$338$	−18.3820	−0.999851
$339$	−3.95619	−0.214871
$340$	0	0
$341$	3.45879	0.187304
$342$	11.0426	0.597113
$343$	0	0
$344$	17.3027	0.932897
$345$	0	0
$346$	21.3229	1.14633
$347$	−0.947077	−0.0508418	−0.0254209	−	0.999677i	$-0.508093\pi$
$347$	−0.947077	−0.0508418	−0.0254209	+	0.999677i	$0.508093\pi$
$348$	4.37679	0.234621
$349$	20.0315	1.07226	0.536131	−	0.844135i	$-0.319885\pi$
$349$	20.0315	1.07226	0.536131	+	0.844135i	$0.319885\pi$
$350$	0	0
$351$	−1.62534	−0.0867540
$352$	7.04168	0.375323
$353$	−19.5424	−1.04014	−0.520068	−	0.854125i	$-0.674093\pi$
$353$	−19.5424	−1.04014	−0.520068	+	0.854125i	$0.674093\pi$
$354$	−22.1696	−1.17830
$355$	0	0
$356$	14.4039	0.763403
$357$	0	0
$358$	8.11033	0.428644
$359$	−3.27841	−0.173028	−0.0865140	−	0.996251i	$-0.527573\pi$
$359$	−3.27841	−0.173028	−0.0865140	+	0.996251i	$0.527573\pi$
$360$	0	0
$361$	19.7193	1.03786
$362$	14.8417	0.780063
$363$	9.53318	0.500362
$364$	0	0
$365$	0	0
$366$	−9.17759	−0.479720
$367$	−0.387948	−0.0202507	−0.0101254	−	0.999949i	$-0.503223\pi$
$367$	−0.387948	−0.0202507	−0.0101254	+	0.999949i	$0.503223\pi$
$368$	6.56427	0.342186
$369$	2.98061	0.155164
$370$	0	0
$371$	0	0
$372$	3.28219	0.170174
$373$	−23.1845	−1.20045	−0.600225	−	0.799831i	$-0.704922\pi$
$373$	−23.1845	−1.20045	−0.600225	+	0.799831i	$0.704922\pi$
$374$	−11.4732	−0.593263
$375$	0	0
$376$	10.3807	0.535343
$377$	−6.18972	−0.318787
$378$	0	0
$379$	0.445893	0.0229040	0.0114520	−	0.999934i	$-0.496355\pi$
$379$	0.445893	0.0229040	0.0114520	+	0.999934i	$0.496355\pi$
$380$	0	0
$381$	15.2540	0.781486
$382$	−17.5880	−0.899882
$383$	−6.56563	−0.335488	−0.167744	−	0.985831i	$-0.553648\pi$
$383$	−6.56563	−0.335488	−0.167744	+	0.985831i	$0.553648\pi$
$384$	−10.9850	−0.560575
$385$	0	0
$386$	−27.6494	−1.40732
$387$	−11.4610	−0.582597
$388$	−5.53619	−0.281058
$389$	−33.9079	−1.71920	−0.859600	−	0.510968i	$-0.829287\pi$
$389$	−33.9079	−1.71920	−0.859600	+	0.510968i	$0.829287\pi$
$390$	0	0
$391$	−7.03955	−0.356005
$392$	0	0
$393$	16.2844	0.821441
$394$	−17.6867	−0.891044
$395$	0	0
$396$	−1.39193	−0.0699470
$397$	−26.1125	−1.31055	−0.655275	−	0.755390i	$-0.727447\pi$
$397$	−26.1125	−1.31055	−0.655275	+	0.755390i	$0.727447\pi$
$398$	−15.6644	−0.785186
$399$	0	0
$400$	0	0
$401$	19.1220	0.954906	0.477453	−	0.878657i	$-0.341560\pi$
$401$	19.1220	0.954906	0.477453	+	0.878657i	$0.341560\pi$
$402$	−5.39269	−0.268963
$403$	−4.64172	−0.231220
$404$	−0.191414	−0.00952319
$405$	0	0
$406$	0	0
$407$	−4.09078	−0.202772
$408$	8.05894	0.398977
$409$	−23.9402	−1.18377	−0.591883	−	0.806024i	$-0.701615\pi$
$409$	−23.9402	−1.18377	−0.591883	+	0.806024i	$0.701615\pi$
$410$	0	0
$411$	−18.9909	−0.936751
$412$	0.696404	0.0343094
$413$	0	0
$414$	−2.34025	−0.115017
$415$	0	0
$416$	−9.44999	−0.463324
$417$	−15.3096	−0.749715
$418$	−13.3739	−0.654139
$419$	−11.4327	−0.558523	−0.279261	−	0.960215i	$-0.590090\pi$
$419$	−11.4327	−0.558523	−0.279261	+	0.960215i	$0.590090\pi$
$420$	0	0
$421$	−7.79413	−0.379863	−0.189931	−	0.981797i	$-0.560827\pi$
$421$	−7.79413	−0.379863	−0.189931	+	0.981797i	$0.560827\pi$
$422$	−25.8032	−1.25608
$423$	−6.87601	−0.334323
$424$	−18.9748	−0.921499
$425$	0	0
$426$	−22.1553	−1.07343
$427$	0	0
$428$	3.75026	0.181275
$429$	1.96848	0.0950392
$430$	0	0
$431$	28.5571	1.37555	0.687773	−	0.725926i	$-0.258589\pi$
$431$	28.5571	1.37555	0.687773	+	0.725926i	$0.258589\pi$
$432$	4.97771	0.239490
$433$	−22.5789	−1.08507	−0.542537	−	0.840032i	$-0.682536\pi$
$433$	−22.5789	−1.08507	−0.542537	+	0.840032i	$0.682536\pi$
$434$	0	0
$435$	0	0
$436$	14.3387	0.686699
$437$	−8.20578	−0.392536
$438$	−21.8793	−1.04544
$439$	37.3497	1.78260	0.891302	−	0.453410i	$-0.149793\pi$
$439$	37.3497	1.78260	0.891302	+	0.453410i	$0.149793\pi$
$440$	0	0
$441$	0	0
$442$	15.3971	0.732364
$443$	−0.174900	−0.00830975	−0.00415487	−	0.999991i	$-0.501323\pi$
$443$	−0.174900	−0.00830975	−0.00415487	+	0.999991i	$0.501323\pi$
$444$	−3.88191	−0.184227
$445$	0	0
$446$	−7.57049	−0.358473
$447$	10.4172	0.492718
$448$	0	0
$449$	13.4569	0.635072	0.317536	−	0.948246i	$-0.397145\pi$
$449$	13.4569	0.635072	0.317536	+	0.948246i	$0.397145\pi$
$450$	0	0
$451$	−3.60988	−0.169983
$452$	4.54680	0.213863
$453$	−17.3331	−0.814380
$454$	−4.46637	−0.209617
$455$	0	0
$456$	9.39406	0.439917
$457$	−5.42225	−0.253642	−0.126821	−	0.991926i	$-0.540477\pi$
$457$	−5.42225	−0.253642	−0.126821	+	0.991926i	$0.540477\pi$
$458$	31.4803	1.47098
$459$	−5.33812	−0.249162
$460$	0	0
$461$	33.9796	1.58259	0.791294	−	0.611435i	$-0.209408\pi$
$461$	33.9796	1.58259	0.791294	+	0.611435i	$0.209408\pi$
$462$	0	0
$463$	8.59136	0.399274	0.199637	−	0.979870i	$-0.436024\pi$
$463$	8.59136	0.399274	0.199637	+	0.979870i	$0.436024\pi$
$464$	18.9565	0.880032
$465$	0	0
$466$	−0.650632	−0.0301400
$467$	8.67068	0.401231	0.200616	−	0.979670i	$-0.435706\pi$
$467$	8.67068	0.401231	0.200616	+	0.979670i	$0.435706\pi$
$468$	1.86798	0.0863473
$469$	0	0
$470$	0	0
$471$	19.9657	0.919971
$472$	−18.8599	−0.868099
$473$	13.8807	0.638236
$474$	−24.5585	−1.12801
$475$	0	0
$476$	0	0
$477$	12.5686	0.575478
$478$	−3.01184	−0.137758
$479$	32.3486	1.47804	0.739022	−	0.673682i	$-0.235288\pi$
$479$	32.3486	1.47804	0.739022	+	0.673682i	$0.235288\pi$
$480$	0	0
$481$	5.48985	0.250316
$482$	−3.87252	−0.176388
$483$	0	0
$484$	−10.9564	−0.498016
$485$	0	0
$486$	−1.77462	−0.0804985
$487$	−15.3389	−0.695071	−0.347536	−	0.937667i	$-0.612981\pi$
$487$	−15.3389	−0.695071	−0.347536	+	0.937667i	$0.612981\pi$
$488$	−7.80750	−0.353429
$489$	25.3079	1.14446
$490$	0	0
$491$	−30.7282	−1.38675	−0.693373	−	0.720579i	$-0.743876\pi$
$491$	−30.7282	−1.38675	−0.693373	+	0.720579i	$0.743876\pi$
$492$	−3.42557	−0.154437
$493$	−20.3290	−0.915572
$494$	17.9479	0.807513
$495$	0	0
$496$	14.2156	0.638300
$497$	0	0
$498$	−15.5248	−0.695684
$499$	23.3137	1.04366	0.521832	−	0.853048i	$-0.325249\pi$
$499$	23.3137	1.04366	0.521832	+	0.853048i	$0.325249\pi$
$500$	0	0
$501$	4.97181	0.222124
$502$	31.7014	1.41490
$503$	−43.5102	−1.94003	−0.970013	−	0.243053i	$-0.921851\pi$
$503$	−43.5102	−1.94003	−0.970013	+	0.243053i	$0.921851\pi$
$504$	0	0
$505$	0	0
$506$	2.83433	0.126001
$507$	10.3583	0.460027
$508$	−17.5312	−0.777822
$509$	−9.85445	−0.436791	−0.218395	−	0.975860i	$-0.570082\pi$
$509$	−9.85445	−0.436791	−0.218395	+	0.975860i	$0.570082\pi$
$510$	0	0
$511$	0	0
$512$	13.9116	0.614813
$513$	−6.22248	−0.274729
$514$	−4.15628	−0.183325
$515$	0	0
$516$	13.1720	0.579865
$517$	8.32769	0.366251
$518$	0	0
$519$	−12.0155	−0.527420
$520$	0	0
$521$	34.0604	1.49221	0.746107	−	0.665826i	$-0.231921\pi$
$521$	34.0604	1.49221	0.746107	+	0.665826i	$0.231921\pi$
$522$	−6.75824	−0.295800
$523$	−18.2146	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$523$	−18.2146	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$524$	−18.7155	−0.817589
$525$	0	0
$526$	5.48933	0.239346
$527$	−15.2449	−0.664078
$528$	−6.02862	−0.262362
$529$	−21.2609	−0.924389
$530$	0	0
$531$	12.4925	0.542130
$532$	0	0
$533$	4.84449	0.209838
$534$	−22.2411	−0.962467
$535$	0	0
$536$	−4.58764	−0.198156
$537$	−4.57017	−0.197217
$538$	42.2216	1.82030
$539$	0	0
$540$	0	0
$541$	21.4246	0.921117	0.460559	−	0.887629i	$-0.347649\pi$
$541$	21.4246	0.921117	0.460559	+	0.887629i	$0.347649\pi$
$542$	24.5875	1.05613
$543$	−8.36330	−0.358904
$544$	−31.0368	−1.33069
$545$	0	0
$546$	0	0
$547$	11.5526	0.493952	0.246976	−	0.969022i	$-0.420563\pi$
$547$	11.5526	0.493952	0.246976	+	0.969022i	$0.420563\pi$
$548$	21.8260	0.932359
$549$	5.17157	0.220717
$550$	0	0
$551$	−23.6969	−1.00952
$552$	−1.99088	−0.0847376
$553$	0	0
$554$	54.0390	2.29590
$555$	0	0
$556$	17.5951	0.746200
$557$	26.1311	1.10721	0.553605	−	0.832779i	$-0.313252\pi$
$557$	26.1311	1.10721	0.553605	+	0.832779i	$0.313252\pi$
$558$	−5.06806	−0.214548
$559$	−18.6280	−0.787882
$560$	0	0
$561$	6.46512	0.272958
$562$	−28.0719	−1.18414
$563$	−22.9795	−0.968471	−0.484236	−	0.874938i	$-0.660902\pi$
$563$	−22.9795	−0.968471	−0.484236	+	0.874938i	$0.660902\pi$
$564$	7.90250	0.332755
$565$	0	0
$566$	−53.0710	−2.23074
$567$	0	0
$568$	−18.8478	−0.790837
$569$	26.9655	1.13045	0.565227	−	0.824935i	$-0.308789\pi$
$569$	26.9655	1.13045	0.565227	+	0.824935i	$0.308789\pi$
$570$	0	0
$571$	−23.3761	−0.978261	−0.489130	−	0.872211i	$-0.662686\pi$
$571$	−23.3761	−0.978261	−0.489130	+	0.872211i	$0.662686\pi$
$572$	−2.26235	−0.0945936
$573$	9.91085	0.414032
$574$	0	0
$575$	0	0
$576$	−0.362537	−0.0151057
$577$	38.4437	1.60043	0.800216	−	0.599711i	$-0.204718\pi$
$577$	38.4437	1.60043	0.800216	+	0.599711i	$0.204718\pi$
$578$	20.4003	0.848540
$579$	15.5804	0.647501
$580$	0	0
$581$	0	0
$582$	8.54848	0.354346
$583$	−15.2222	−0.630438
$584$	−18.6130	−0.770213
$585$	0	0
$586$	−47.6484	−1.96834
$587$	−19.2796	−0.795753	−0.397877	−	0.917439i	$-0.630253\pi$
$587$	−19.2796	−0.795753	−0.397877	+	0.917439i	$0.630253\pi$
$588$	0	0
$589$	−17.7705	−0.732220
$590$	0	0
$591$	9.96647	0.409966
$592$	−16.8131	−0.691014
$593$	−3.91937	−0.160949	−0.0804745	−	0.996757i	$-0.525644\pi$
$593$	−3.91937	−0.160949	−0.0804745	+	0.996757i	$0.525644\pi$
$594$	2.14929	0.0881863
$595$	0	0
$596$	−11.9724	−0.490408
$597$	8.82689	0.361261
$598$	−3.80370	−0.155545
$599$	29.1220	1.18989	0.594946	−	0.803766i	$-0.297173\pi$
$599$	29.1220	1.18989	0.594946	+	0.803766i	$0.297173\pi$
$600$	0	0
$601$	−31.4886	−1.28445	−0.642224	−	0.766517i	$-0.721988\pi$
$601$	−31.4886	−1.28445	−0.642224	+	0.766517i	$0.721988\pi$
$602$	0	0
$603$	3.03878	0.123749
$604$	19.9207	0.810562
$605$	0	0
$606$	0.295563	0.0120064
$607$	6.58988	0.267475	0.133737	−	0.991017i	$-0.457302\pi$
$607$	6.58988	0.267475	0.133737	+	0.991017i	$0.457302\pi$
$608$	−36.1786	−1.46724
$609$	0	0
$610$	0	0
$611$	−11.1758	−0.452126
$612$	6.13503	0.247994
$613$	−33.9260	−1.37026	−0.685129	−	0.728422i	$-0.740254\pi$
$613$	−33.9260	−1.37026	−0.685129	+	0.728422i	$0.740254\pi$
$614$	42.7359	1.72468
$615$	0	0
$616$	0	0
$617$	−38.5251	−1.55096	−0.775480	−	0.631372i	$-0.782492\pi$
$617$	−38.5251	−1.55096	−0.775480	+	0.631372i	$0.782492\pi$
$618$	−1.07532	−0.0432559
$619$	−2.17365	−0.0873665	−0.0436833	−	0.999045i	$-0.513909\pi$
$619$	−2.17365	−0.0873665	−0.0436833	+	0.999045i	$0.513909\pi$
$620$	0	0
$621$	1.31873	0.0529189
$622$	−13.0654	−0.523876
$623$	0	0
$624$	8.09046	0.323878
$625$	0	0
$626$	−6.88436	−0.275154
$627$	7.53619	0.300966
$628$	−22.9463	−0.915657
$629$	18.0304	0.718920
$630$	0	0
$631$	−27.0633	−1.07737	−0.538686	−	0.842507i	$-0.681079\pi$
$631$	−27.0633	−1.07737	−0.538686	+	0.842507i	$0.681079\pi$
$632$	−20.8922	−0.831048
$633$	14.5401	0.577918
$634$	5.51029	0.218842
$635$	0	0
$636$	−14.4450	−0.572780
$637$	0	0
$638$	8.18506	0.324050
$639$	12.4845	0.493880
$640$	0	0
$641$	−45.6521	−1.80315	−0.901574	−	0.432625i	$-0.857587\pi$
$641$	−45.6521	−1.80315	−0.901574	+	0.432625i	$0.857587\pi$
$642$	−5.79080	−0.228545
$643$	−10.8224	−0.426794	−0.213397	−	0.976966i	$-0.568453\pi$
$643$	−10.8224	−0.426794	−0.213397	+	0.976966i	$0.568453\pi$
$644$	0	0
$645$	0	0
$646$	58.9465	2.31922
$647$	−38.6409	−1.51913	−0.759565	−	0.650432i	$-0.774588\pi$
$647$	−38.6409	−1.51913	−0.759565	+	0.650432i	$0.774588\pi$
$648$	−1.50970	−0.0593065
$649$	−15.1300	−0.593905
$650$	0	0
$651$	0	0
$652$	−29.0860	−1.13910
$653$	34.5989	1.35396	0.676980	−	0.736001i	$-0.263288\pi$
$653$	34.5989	1.35396	0.676980	+	0.736001i	$0.263288\pi$
$654$	−22.1405	−0.865762
$655$	0	0
$656$	−14.8366	−0.579272
$657$	12.3290	0.481000
$658$	0	0
$659$	−30.6765	−1.19499	−0.597493	−	0.801874i	$-0.703836\pi$
$659$	−30.6765	−1.19499	−0.597493	+	0.801874i	$0.703836\pi$
$660$	0	0
$661$	−4.10908	−0.159825	−0.0799124	−	0.996802i	$-0.525464\pi$
$661$	−4.10908	−0.159825	−0.0799124	+	0.996802i	$0.525464\pi$
$662$	−5.99410	−0.232967
$663$	−8.67625	−0.336957
$664$	−13.2072	−0.512538
$665$	0	0
$666$	5.99410	0.232266
$667$	5.02208	0.194456
$668$	−5.71403	−0.221083
$669$	4.26597	0.164932
$670$	0	0
$671$	−6.26341	−0.241796
$672$	0	0
$673$	10.6778	0.411598	0.205799	−	0.978594i	$-0.434021\pi$
$673$	10.6778	0.411598	0.205799	+	0.978594i	$0.434021\pi$
$674$	8.58687	0.330754
$675$	0	0
$676$	−11.9046	−0.457870
$677$	45.0874	1.73285	0.866424	−	0.499309i	$-0.166413\pi$
$677$	45.0874	1.73285	0.866424	+	0.499309i	$0.166413\pi$
$678$	−7.02075	−0.269630
$679$	0	0
$680$	0	0
$681$	2.51680	0.0964440
$682$	6.13804	0.235038
$683$	30.0053	1.14812	0.574060	−	0.818814i	$-0.305368\pi$
$683$	30.0053	1.14812	0.574060	+	0.818814i	$0.305368\pi$
$684$	7.15141	0.273441
$685$	0	0
$686$	0	0
$687$	−17.7391	−0.676790
$688$	57.0497	2.17500
$689$	20.4283	0.778255
$690$	0	0
$691$	−31.2018	−1.18697	−0.593487	−	0.804844i	$-0.702249\pi$
$691$	−31.2018	−1.18697	−0.593487	+	0.804844i	$0.702249\pi$
$692$	13.8092	0.524947
$693$	0	0
$694$	−1.68070	−0.0637987
$695$	0	0
$696$	−5.74933	−0.217928
$697$	15.9109	0.602667
$698$	35.5484	1.34553
$699$	0.366631	0.0138673
$700$	0	0
$701$	17.5698	0.663602	0.331801	−	0.943349i	$-0.392344\pi$
$701$	17.5698	0.663602	0.331801	+	0.943349i	$0.392344\pi$
$702$	−2.88436	−0.108863
$703$	21.0175	0.792690
$704$	0.439077	0.0165483
$705$	0	0
$706$	−34.6803	−1.30521
$707$	0	0
$708$	−14.3575	−0.539588
$709$	−12.8263	−0.481700	−0.240850	−	0.970562i	$-0.577426\pi$
$709$	−12.8263	−0.481700	−0.240850	+	0.970562i	$0.577426\pi$
$710$	0	0
$711$	13.8387	0.518992
$712$	−18.9208	−0.709087
$713$	3.76610	0.141042
$714$	0	0
$715$	0	0
$716$	5.25244	0.196293
$717$	1.69717	0.0633820
$718$	−5.81795	−0.217124
$719$	−22.7166	−0.847185	−0.423592	−	0.905853i	$-0.639231\pi$
$719$	−22.7166	−0.847185	−0.423592	+	0.905853i	$0.639231\pi$
$720$	0	0
$721$	0	0
$722$	34.9943	1.30235
$723$	2.18217	0.0811556
$724$	9.61183	0.357221
$725$	0	0
$726$	16.9178	0.627879
$727$	25.0638	0.929565	0.464782	−	0.885425i	$-0.346133\pi$
$727$	25.0638	0.929565	0.464782	+	0.885425i	$0.346133\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−61.1804	−2.26284
$732$	−5.94362	−0.219682
$733$	33.7893	1.24804	0.624018	−	0.781410i	$-0.285499\pi$
$733$	33.7893	1.24804	0.624018	+	0.781410i	$0.285499\pi$
$734$	−0.688461	−0.0254115
$735$	0	0
$736$	7.66733	0.282622
$737$	−3.68034	−0.135567
$738$	5.28946	0.194708
$739$	−16.5704	−0.609553	−0.304777	−	0.952424i	$-0.598582\pi$
$739$	−16.5704	−0.609553	−0.304777	+	0.952424i	$0.598582\pi$
$740$	0	0
$741$	−10.1136	−0.371533
$742$	0	0
$743$	8.25553	0.302866	0.151433	−	0.988468i	$-0.451611\pi$
$743$	8.25553	0.302866	0.151433	+	0.988468i	$0.451611\pi$
$744$	−4.31147	−0.158066
$745$	0	0
$746$	−41.1438	−1.50638
$747$	8.74824	0.320081
$748$	−7.43028	−0.271678
$749$	0	0
$750$	0	0
$751$	−37.1248	−1.35470	−0.677352	−	0.735659i	$-0.736873\pi$
$751$	−37.1248	−1.35470	−0.677352	+	0.735659i	$0.736873\pi$
$752$	34.2268	1.24812
$753$	−17.8637	−0.650990
$754$	−10.9844	−0.400029
$755$	0	0
$756$	0	0
$757$	−1.67625	−0.0609242	−0.0304621	−	0.999536i	$-0.509698\pi$
$757$	−1.67625	−0.0609242	−0.0304621	+	0.999536i	$0.509698\pi$
$758$	0.791292	0.0287410
$759$	−1.59715	−0.0579727
$760$	0	0
$761$	3.45987	0.125420	0.0627101	−	0.998032i	$-0.480026\pi$
$761$	3.45987	0.125420	0.0627101	+	0.998032i	$0.480026\pi$
$762$	27.0701	0.980646
$763$	0	0
$764$	−11.3904	−0.412091
$765$	0	0
$766$	−11.6515	−0.420986
$767$	20.3046	0.733156
$768$	−20.2193	−0.729600
$769$	31.3828	1.13169	0.565846	−	0.824511i	$-0.308550\pi$
$769$	31.3828	1.13169	0.565846	+	0.824511i	$0.308550\pi$
$770$	0	0
$771$	2.34206	0.0843473
$772$	−17.9064	−0.644465
$773$	37.6039	1.35252	0.676259	−	0.736664i	$-0.263600\pi$
$773$	37.6039	1.35252	0.676259	+	0.736664i	$0.263600\pi$
$774$	−20.3390	−0.731070
$775$	0	0
$776$	7.27231	0.261061
$777$	0	0
$778$	−60.1738	−2.15733
$779$	18.5468	0.664507
$780$	0	0
$781$	−15.1203	−0.541046
$782$	−12.4925	−0.446733
$783$	3.80827	0.136096
$784$	0	0
$785$	0	0
$786$	28.8987	1.03078
$787$	33.8482	1.20656	0.603279	−	0.797530i	$-0.293860\pi$
$787$	33.8482	1.20656	0.603279	+	0.797530i	$0.293860\pi$
$788$	−11.4543	−0.408044
$789$	−3.09324	−0.110122
$790$	0	0
$791$	0	0
$792$	1.82843	0.0649703
$793$	8.40555	0.298490
$794$	−46.3399	−1.64454
$795$	0	0
$796$	−10.1446	−0.359567
$797$	23.7421	0.840990	0.420495	−	0.907295i	$-0.361857\pi$
$797$	23.7421	0.840990	0.420495	+	0.907295i	$0.361857\pi$
$798$	0	0
$799$	−36.7050	−1.29853
$800$	0	0
$801$	12.5329	0.442827
$802$	33.9343	1.19826
$803$	−14.9319	−0.526937
$804$	−3.49243	−0.123169
$805$	0	0
$806$	−8.23730	−0.290147
$807$	−23.7919	−0.837514
$808$	0.251440	0.00884562
$809$	45.8224	1.61103	0.805515	−	0.592576i	$-0.201889\pi$
$809$	45.8224	1.61103	0.805515	+	0.592576i	$0.201889\pi$
$810$	0	0
$811$	−32.6473	−1.14640	−0.573201	−	0.819415i	$-0.694298\pi$
$811$	−32.6473	−1.14640	−0.573201	+	0.819415i	$0.694298\pi$
$812$	0	0
$813$	−13.8551	−0.485919
$814$	−7.25959	−0.254448
$815$	0	0
$816$	26.5716	0.930194
$817$	−71.3160	−2.49503
$818$	−42.4848	−1.48545
$819$	0	0
$820$	0	0
$821$	42.8023	1.49381	0.746906	−	0.664930i	$-0.231539\pi$
$821$	42.8023	1.49381	0.746906	+	0.664930i	$0.231539\pi$
$822$	−33.7017	−1.17548
$823$	27.8329	0.970194	0.485097	−	0.874460i	$-0.338784\pi$
$823$	27.8329	0.970194	0.485097	+	0.874460i	$0.338784\pi$
$824$	−0.914792	−0.0318683
$825$	0	0
$826$	0	0
$827$	−12.9741	−0.451152	−0.225576	−	0.974226i	$-0.572426\pi$
$827$	−12.9741	−0.451152	−0.225576	+	0.974226i	$0.572426\pi$
$828$	−1.51560	−0.0526707
$829$	49.0822	1.70469	0.852347	−	0.522976i	$-0.175178\pi$
$829$	49.0822	1.70469	0.852347	+	0.522976i	$0.175178\pi$
$830$	0	0
$831$	−30.4510	−1.05633
$832$	−0.589245	−0.0204284
$833$	0	0
$834$	−27.1688	−0.940778
$835$	0	0
$836$	−8.66124	−0.299555
$837$	2.85585	0.0987126
$838$	−20.2887	−0.700861
$839$	3.84798	0.132847	0.0664235	−	0.997792i	$-0.478841\pi$
$839$	3.84798	0.132847	0.0664235	+	0.997792i	$0.478841\pi$
$840$	0	0
$841$	−14.4971	−0.499900
$842$	−13.8316	−0.476670
$843$	15.8185	0.544820
$844$	−16.7108	−0.575209
$845$	0	0
$846$	−12.2023	−0.419525
$847$	0	0
$848$	−62.5631	−2.14842
$849$	29.9055	1.02635
$850$	0	0
$851$	−4.45424	−0.152689
$852$	−14.3483	−0.491564
$853$	19.7420	0.675952	0.337976	−	0.941155i	$-0.390258\pi$
$853$	19.7420	0.675952	0.337976	+	0.941155i	$0.390258\pi$
$854$	0	0
$855$	0	0
$856$	−4.92631	−0.168378
$857$	2.12793	0.0726887	0.0363443	−	0.999339i	$-0.488429\pi$
$857$	2.12793	0.0726887	0.0363443	+	0.999339i	$0.488429\pi$
$858$	3.49331	0.119260
$859$	−19.9610	−0.681060	−0.340530	−	0.940234i	$-0.610607\pi$
$859$	−19.9610	−0.681060	−0.340530	+	0.940234i	$0.610607\pi$
$860$	0	0
$861$	0	0
$862$	50.6780	1.72610
$863$	6.63068	0.225711	0.112855	−	0.993611i	$-0.464000\pi$
$863$	6.63068	0.225711	0.112855	+	0.993611i	$0.464000\pi$
$864$	5.81417	0.197802
$865$	0	0
$866$	−40.0691	−1.36160
$867$	−11.4956	−0.390410
$868$	0	0
$869$	−16.7604	−0.568557
$870$	0	0
$871$	4.93904	0.167353
$872$	−18.8352	−0.637841
$873$	−4.81707	−0.163033
$874$	−14.5622	−0.492573
$875$	0	0
$876$	−14.1696	−0.478745
$877$	−28.2776	−0.954867	−0.477434	−	0.878668i	$-0.658433\pi$
$877$	−28.2776	−0.954867	−0.477434	+	0.878668i	$0.658433\pi$
$878$	66.2817	2.23690
$879$	26.8499	0.905625
$880$	0	0
$881$	−44.4417	−1.49728	−0.748640	−	0.662977i	$-0.769293\pi$
$881$	−44.4417	−1.49728	−0.748640	+	0.662977i	$0.769293\pi$
$882$	0	0
$883$	−45.6589	−1.53654	−0.768272	−	0.640124i	$-0.778883\pi$
$883$	−45.6589	−1.53654	−0.768272	+	0.640124i	$0.778883\pi$
$884$	9.97149	0.335378
$885$	0	0
$886$	−0.310381	−0.0104275
$887$	20.5971	0.691584	0.345792	−	0.938311i	$-0.387610\pi$
$887$	20.5971	0.691584	0.345792	+	0.938311i	$0.387610\pi$
$888$	5.09926	0.171120
$889$	0	0
$890$	0	0
$891$	−1.21112	−0.0405741
$892$	−4.90282	−0.164159
$893$	−42.7858	−1.43177
$894$	18.4866	0.618286
$895$	0	0
$896$	0	0
$897$	2.14338	0.0715654
$898$	23.8810	0.796919
$899$	10.8758	0.362730
$900$	0	0
$901$	67.0929	2.23519
$902$	−6.40618	−0.213303
$903$	0	0
$904$	−5.97265	−0.198647
$905$	0	0
$906$	−30.7597	−1.02192
$907$	−8.64837	−0.287165	−0.143582	−	0.989638i	$-0.545862\pi$
$907$	−8.64837	−0.287165	−0.143582	+	0.989638i	$0.545862\pi$
$908$	−2.89252	−0.0959918
$909$	−0.166550	−0.00552412
$910$	0	0
$911$	−37.0446	−1.22734	−0.613672	−	0.789561i	$-0.710308\pi$
$911$	−37.0446	−1.22734	−0.613672	+	0.789561i	$0.710308\pi$
$912$	30.9737	1.02564
$913$	−10.5952	−0.350650
$914$	−9.62244	−0.318282
$915$	0	0
$916$	20.3873	0.673617
$917$	0	0
$918$	−9.47316	−0.312661
$919$	−50.4076	−1.66279	−0.831396	−	0.555680i	$-0.812458\pi$
$919$	−50.4076	−1.66279	−0.831396	+	0.555680i	$0.812458\pi$
$920$	0	0
$921$	−24.0817	−0.793518
$922$	60.3010	1.98591
$923$	20.2915	0.667904
$924$	0	0
$925$	0	0
$926$	15.2464	0.501028
$927$	0.605945	0.0199018
$928$	22.1419	0.726845
$929$	−18.9401	−0.621405	−0.310703	−	0.950507i	$-0.600564\pi$
$929$	−18.9401	−0.621405	−0.310703	+	0.950507i	$0.600564\pi$
$930$	0	0
$931$	0	0
$932$	−0.421364	−0.0138022
$933$	7.36238	0.241033
$934$	15.3872	0.503484
$935$	0	0
$936$	−2.45376	−0.0802038
$937$	−9.23461	−0.301682	−0.150841	−	0.988558i	$-0.548198\pi$
$937$	−9.23461	−0.301682	−0.150841	+	0.988558i	$0.548198\pi$
$938$	0	0
$939$	3.87934	0.126597
$940$	0	0
$941$	−4.94676	−0.161260	−0.0806299	−	0.996744i	$-0.525693\pi$
$941$	−4.94676	−0.161260	−0.0806299	+	0.996744i	$0.525693\pi$
$942$	35.4316	1.15442
$943$	−3.93062	−0.127999
$944$	−62.1843	−2.02393
$945$	0	0
$946$	24.6330	0.800889
$947$	49.1088	1.59582	0.797910	−	0.602776i	$-0.205939\pi$
$947$	49.1088	1.59582	0.797910	+	0.602776i	$0.205939\pi$
$948$	−15.9046	−0.516558
$949$	20.0388	0.650486
$950$	0	0
$951$	−3.10505	−0.100688
$952$	0	0
$953$	−6.74856	−0.218607	−0.109304	−	0.994008i	$-0.534862\pi$
$953$	−6.74856	−0.218607	−0.109304	+	0.994008i	$0.534862\pi$
$954$	22.3046	0.722138
$955$	0	0
$956$	−1.95054	−0.0630848
$957$	−4.61228	−0.149094
$958$	57.4065	1.85472
$959$	0	0
$960$	0	0
$961$	−22.8441	−0.736907
$962$	9.74242	0.314108
$963$	3.26312	0.105152
$964$	−2.50793	−0.0807751
$965$	0	0
$966$	0	0
$967$	−22.5196	−0.724181	−0.362090	−	0.932143i	$-0.617937\pi$
$967$	−22.5196	−0.724181	−0.362090	+	0.932143i	$0.617937\pi$
$968$	14.3922	0.462583
$969$	−33.2164	−1.06706
$970$	0	0
$971$	36.1599	1.16043	0.580213	−	0.814465i	$-0.302969\pi$
$971$	36.1599	1.16043	0.580213	+	0.814465i	$0.302969\pi$
$972$	−1.14929	−0.0368634
$973$	0	0
$974$	−27.2207	−0.872209
$975$	0	0
$976$	−25.7426	−0.824001
$977$	40.8982	1.30845	0.654225	−	0.756300i	$-0.272995\pi$
$977$	40.8982	1.30845	0.654225	+	0.756300i	$0.272995\pi$
$978$	44.9120	1.43613
$979$	−15.1788	−0.485118
$980$	0	0
$981$	12.4762	0.398333
$982$	−54.5310	−1.74016
$983$	30.2589	0.965110	0.482555	−	0.875866i	$-0.339709\pi$
$983$	30.2589	0.965110	0.482555	+	0.875866i	$0.339709\pi$
$984$	4.49981	0.143449
$985$	0	0
$986$	−36.0763	−1.14890
$987$	0	0
$988$	11.6235	0.369791
$989$	15.1140	0.480598
$990$	0	0
$991$	2.66504	0.0846579	0.0423289	−	0.999104i	$-0.486522\pi$
$991$	2.66504	0.0846579	0.0423289	+	0.999104i	$0.486522\pi$
$992$	16.6044	0.527191
$993$	3.37767	0.107187
$994$	0	0
$995$	0	0
$996$	−10.0542	−0.318581
$997$	34.9092	1.10559	0.552793	−	0.833318i	$-0.313562\pi$
$997$	34.9092	1.10559	0.552793	+	0.833318i	$0.313562\pi$
$998$	41.3730	1.30964
$999$	−3.37767	−0.106865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bm.1.4	✓	4
5.4	even	2		3675.2.a.ca.1.1	yes	4
7.6	odd	2		3675.2.a.bo.1.4	yes	4
35.34	odd	2		3675.2.a.by.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3675.2.a.bm.1.4	✓	4	1.1	even	1	trivial
3675.2.a.bo.1.4	yes	4	7.6	odd	2
3675.2.a.by.1.1	yes	4	35.34	odd	2
3675.2.a.ca.1.1	yes	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:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+1.77462 q^{2} -1.00000 q^{3} +1.14929 q^{4} -1.77462 q^{6} -1.50970 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.77462 q^{2} -1.00000 q^{3} +1.14929 q^{4} -1.77462 q^{6} -1.50970 q^{8} +1.00000 q^{9} -1.21112 q^{11} -1.14929 q^{12} +1.62534 q^{13} -4.97771 q^{16} +5.33812 q^{17} +1.77462 q^{18} +6.22248 q^{19} -2.14929 q^{22} -1.31873 q^{23} +1.50970 q^{24} +2.88436 q^{26} -1.00000 q^{27} -3.80827 q^{29} -2.85585 q^{31} -5.81417 q^{32} +1.21112 q^{33} +9.47316 q^{34} +1.14929 q^{36} +3.37767 q^{37} +11.0426 q^{38} -1.62534 q^{39} +2.98061 q^{41} -11.4610 q^{43} -1.39193 q^{44} -2.34025 q^{46} -6.87601 q^{47} +4.97771 q^{48} -5.33812 q^{51} +1.86798 q^{52} +12.5686 q^{53} -1.77462 q^{54} -6.22248 q^{57} -6.75824 q^{58} +12.4925 q^{59} +5.17157 q^{61} -5.06806 q^{62} -0.362537 q^{64} +2.14929 q^{66} +3.03878 q^{67} +6.13503 q^{68} +1.31873 q^{69} +12.4845 q^{71} -1.50970 q^{72} +12.3290 q^{73} +5.99410 q^{74} +7.15141 q^{76} -2.88436 q^{78} +13.8387 q^{79} +1.00000 q^{81} +5.28946 q^{82} +8.74824 q^{83} -20.3390 q^{86} +3.80827 q^{87} +1.82843 q^{88} +12.5329 q^{89} -1.51560 q^{92} +2.85585 q^{93} -12.2023 q^{94} +5.81417 q^{96} -4.81707 q^{97} -1.21112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^9	\( 4 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9} - 4 q^{11} - 2 q^{12} - 6 q^{16} + 4 q^{17} - 2 q^{18} + 8 q^{19} - 6 q^{22} + 12 q^{26} - 4 q^{27} - 4 q^{29} + 8 q^{31} - 2 q^{32} + 4 q^{33} + 8 q^{34} + 2 q^{36} - 16 q^{37} - 4 q^{38} + 24 q^{41} - 20 q^{43} + 14 q^{44} - 6 q^{46} - 8 q^{47} + 6 q^{48} - 4 q^{51} - 16 q^{52} + 20 q^{53} + 2 q^{54} - 8 q^{57} + 6 q^{58} + 8 q^{59} + 32 q^{61} - 28 q^{62} - 12 q^{64} + 6 q^{66} - 12 q^{67} + 12 q^{68} + 4 q^{71} + 34 q^{74} + 40 q^{76} - 12 q^{78} + 4 q^{81} - 16 q^{82} + 20 q^{83} - 14 q^{86} + 4 q^{87} - 4 q^{88} + 8 q^{89} + 10 q^{92} - 8 q^{93} - 32 q^{94} + 2 q^{96} - 24 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^9 - 4 * q^11 - 2 * q^12 - 6 * q^16 + 4 * q^17 - 2 * q^18 + 8 * q^19 - 6 * q^22 + 12 * q^26 - 4 * q^27 - 4 * q^29 + 8 * q^31 - 2 * q^32 + 4 * q^33 + 8 * q^34 + 2 * q^36 - 16 * q^37 - 4 * q^38 + 24 * q^41 - 20 * q^43 + 14 * q^44 - 6 * q^46 - 8 * q^47 + 6 * q^48 - 4 * q^51 - 16 * q^52 + 20 * q^53 + 2 * q^54 - 8 * q^57 + 6 * q^58 + 8 * q^59 + 32 * q^61 - 28 * q^62 - 12 * q^64 + 6 * q^66 - 12 * q^67 + 12 * q^68 + 4 * q^71 + 34 * q^74 + 40 * q^76 - 12 * q^78 + 4 * q^81 - 16 * q^82 + 20 * q^83 - 14 * q^86 + 4 * q^87 - 4 * q^88 + 8 * q^89 + 10 * q^92 - 8 * q^93 - 32 * q^94 + 2 * q^96 - 24 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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