LMFDB - Embedded newform 525.4.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, 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("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.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:	$30.9760027530$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} - 3x + 46 $ x^4 - 24x^2 - 3x + 46
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			$-4.60171$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.60171 q^{2} -3.00000 q^{3} +13.1758 q^{4} -13.8051 q^{6} -7.00000 q^{7} +23.8174 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.60171 q^{2} -3.00000 q^{3} +13.1758 q^{4} -13.8051 q^{6} -7.00000 q^{7} +23.8174 q^{8} +9.00000 q^{9} -52.9465 q^{11} -39.5273 q^{12} -19.6024 q^{13} -32.2120 q^{14} +4.19486 q^{16} -61.5076 q^{17} +41.4154 q^{18} +27.0928 q^{19} +21.0000 q^{21} -243.645 q^{22} -19.2163 q^{23} -71.4523 q^{24} -90.2046 q^{26} -27.0000 q^{27} -92.2304 q^{28} -167.409 q^{29} +225.638 q^{31} -171.236 q^{32} +158.839 q^{33} -283.040 q^{34} +118.582 q^{36} -311.705 q^{37} +124.673 q^{38} +58.8071 q^{39} +12.8071 q^{41} +96.6360 q^{42} +114.311 q^{43} -697.611 q^{44} -88.4281 q^{46} +207.919 q^{47} -12.5846 q^{48} +49.0000 q^{49} +184.523 q^{51} -258.277 q^{52} -227.402 q^{53} -124.246 q^{54} -166.722 q^{56} -81.2784 q^{57} -770.369 q^{58} -605.531 q^{59} -315.981 q^{61} +1038.32 q^{62} -63.0000 q^{63} -821.538 q^{64} +730.934 q^{66} -720.254 q^{67} -810.410 q^{68} +57.6490 q^{69} -56.2766 q^{71} +214.357 q^{72} +1154.36 q^{73} -1434.38 q^{74} +356.969 q^{76} +370.625 q^{77} +270.614 q^{78} +1152.80 q^{79} +81.0000 q^{81} +58.9347 q^{82} -692.581 q^{83} +276.691 q^{84} +526.029 q^{86} +502.227 q^{87} -1261.05 q^{88} +1417.70 q^{89} +137.217 q^{91} -253.190 q^{92} -676.914 q^{93} +956.783 q^{94} +513.708 q^{96} -661.229 q^{97} +225.484 q^{98} -476.518 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 16 * q^4 - 28 * q^7 - 9 * q^8 + 36 * q^9	$ 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9} + 21 q^{11} - 48 q^{12} - 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 84 q^{21} - 221 q^{22} - 102 q^{23} + 27 q^{24} + 129 q^{26} - 108 q^{27} - 112 q^{28} - 240 q^{29} + 351 q^{31} - 72 q^{32} - 63 q^{33} - 285 q^{34} + 144 q^{36} - 399 q^{37} - 324 q^{38} + 15 q^{39} + 381 q^{41} - 460 q^{43} - 975 q^{44} + 550 q^{46} - 60 q^{47} - 216 q^{48} + 196 q^{49} + 297 q^{51} - 223 q^{52} - 873 q^{53} + 63 q^{56} - 216 q^{57} - 1408 q^{58} - 855 q^{59} + 687 q^{61} + 477 q^{62} - 252 q^{63} - 1285 q^{64} + 663 q^{66} + 503 q^{67} - 1725 q^{68} + 306 q^{69} - 681 q^{71} - 81 q^{72} - 1228 q^{73} - 3369 q^{74} + 1910 q^{76} - 147 q^{77} - 387 q^{78} + 345 q^{79} + 324 q^{81} + 495 q^{82} - 1509 q^{83} + 336 q^{84} - 540 q^{86} + 720 q^{87} - 2266 q^{88} - 198 q^{89} + 35 q^{91} - 2994 q^{92} - 1053 q^{93} - 1732 q^{94} + 216 q^{96} + 372 q^{97} + 189 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 16 * q^4 - 28 * q^7 - 9 * q^8 + 36 * q^9 + 21 * q^11 - 48 * q^12 - 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 84 * q^21 - 221 * q^22 - 102 * q^23 + 27 * q^24 + 129 * q^26 - 108 * q^27 - 112 * q^28 - 240 * q^29 + 351 * q^31 - 72 * q^32 - 63 * q^33 - 285 * q^34 + 144 * q^36 - 399 * q^37 - 324 * q^38 + 15 * q^39 + 381 * q^41 - 460 * q^43 - 975 * q^44 + 550 * q^46 - 60 * q^47 - 216 * q^48 + 196 * q^49 + 297 * q^51 - 223 * q^52 - 873 * q^53 + 63 * q^56 - 216 * q^57 - 1408 * q^58 - 855 * q^59 + 687 * q^61 + 477 * q^62 - 252 * q^63 - 1285 * q^64 + 663 * q^66 + 503 * q^67 - 1725 * q^68 + 306 * q^69 - 681 * q^71 - 81 * q^72 - 1228 * q^73 - 3369 * q^74 + 1910 * q^76 - 147 * q^77 - 387 * q^78 + 345 * q^79 + 324 * q^81 + 495 * q^82 - 1509 * q^83 + 336 * q^84 - 540 * q^86 + 720 * q^87 - 2266 * q^88 - 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1053 * q^93 - 1732 * q^94 + 216 * q^96 + 372 * q^97 + 189 * 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 $

$2$	4.60171	1.62695	0.813476	−	0.581599i	$-0.197573\pi$
$2$	4.60171	1.62695	0.813476	+	0.581599i	$0.197573\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	13.1758	1.64697
$5$	0	0
$5$	0	0
$6$	−13.8051	−0.939321
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	23.8174	1.05259
$9$	9.00000	0.333333
$10$	0	0
$11$	−52.9465	−1.45127	−0.725635	−	0.688080i	$-0.758454\pi$
$11$	−52.9465	−1.45127	−0.725635	+	0.688080i	$0.758454\pi$
$12$	−39.5273	−0.950880
$13$	−19.6024	−0.418209	−0.209105	−	0.977893i	$-0.567055\pi$
$13$	−19.6024	−0.418209	−0.209105	+	0.977893i	$0.567055\pi$
$14$	−32.2120	−0.614930
$15$	0	0
$16$	4.19486	0.0655446
$17$	−61.5076	−0.877516	−0.438758	−	0.898605i	$-0.644582\pi$
$17$	−61.5076	−0.877516	−0.438758	+	0.898605i	$0.644582\pi$
$18$	41.4154	0.542317
$19$	27.0928	0.327133	0.163566	−	0.986532i	$-0.447700\pi$
$19$	27.0928	0.327133	0.163566	+	0.986532i	$0.447700\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	−243.645	−2.36115
$23$	−19.2163	−0.174212	−0.0871062	−	0.996199i	$-0.527762\pi$
$23$	−19.2163	−0.174212	−0.0871062	+	0.996199i	$0.527762\pi$
$24$	−71.4523	−0.607714
$25$	0	0
$26$	−90.2046	−0.680407
$27$	−27.0000	−0.192450
$28$	−92.2304	−0.622497
$29$	−167.409	−1.07197	−0.535984	−	0.844228i	$-0.680059\pi$
$29$	−167.409	−1.07197	−0.535984	+	0.844228i	$0.680059\pi$
$30$	0	0
$31$	225.638	1.30728	0.653642	−	0.756804i	$-0.273240\pi$
$31$	225.638	1.30728	0.653642	+	0.756804i	$0.273240\pi$
$32$	−171.236	−0.945954
$33$	158.839	0.837891
$34$	−283.040	−1.42768
$35$	0	0
$36$	118.582	0.548991
$37$	−311.705	−1.38497	−0.692485	−	0.721432i	$-0.743484\pi$
$37$	−311.705	−1.38497	−0.692485	+	0.721432i	$0.743484\pi$
$38$	124.673	0.532229
$39$	58.8071	0.241453
$40$	0	0
$41$	12.8071	0.0487838	0.0243919	−	0.999702i	$-0.492235\pi$
$41$	12.8071	0.0487838	0.0243919	+	0.999702i	$0.492235\pi$
$42$	96.6360	0.355030
$43$	114.311	0.405403	0.202702	−	0.979241i	$-0.435028\pi$
$43$	114.311	0.405403	0.202702	+	0.979241i	$0.435028\pi$
$44$	−697.611	−2.39020
$45$	0	0
$46$	−88.4281	−0.283435
$47$	207.919	0.645278	0.322639	−	0.946522i	$-0.395430\pi$
$47$	207.919	0.645278	0.322639	+	0.946522i	$0.395430\pi$
$48$	−12.5846	−0.0378422
$49$	49.0000	0.142857
$50$	0	0
$51$	184.523	0.506634
$52$	−258.277	−0.688779
$53$	−227.402	−0.589360	−0.294680	−	0.955596i	$-0.595213\pi$
$53$	−227.402	−0.589360	−0.294680	+	0.955596i	$0.595213\pi$
$54$	−124.246	−0.313107
$55$	0	0
$56$	−166.722	−0.397842
$57$	−81.2784	−0.188870
$58$	−770.369	−1.74404
$59$	−605.531	−1.33616	−0.668080	−	0.744090i	$-0.732884\pi$
$59$	−605.531	−1.33616	−0.668080	+	0.744090i	$0.732884\pi$
$60$	0	0
$61$	−315.981	−0.663232	−0.331616	−	0.943414i	$-0.607594\pi$
$61$	−315.981	−0.663232	−0.331616	+	0.943414i	$0.607594\pi$
$62$	1038.32	2.12689
$63$	−63.0000	−0.125988
$64$	−821.538	−1.60457
$65$	0	0
$66$	730.934	1.36321
$67$	−720.254	−1.31333	−0.656664	−	0.754183i	$-0.728033\pi$
$67$	−720.254	−1.31333	−0.656664	+	0.754183i	$0.728033\pi$
$68$	−810.410	−1.44524
$69$	57.6490	0.100582
$70$	0	0
$71$	−56.2766	−0.0940676	−0.0470338	−	0.998893i	$-0.514977\pi$
$71$	−56.2766	−0.0940676	−0.0470338	+	0.998893i	$0.514977\pi$
$72$	214.357	0.350864
$73$	1154.36	1.85079	0.925393	−	0.379010i	$-0.123735\pi$
$73$	1154.36	1.85079	0.925393	+	0.379010i	$0.123735\pi$
$74$	−1434.38	−2.25328
$75$	0	0
$76$	356.969	0.538778
$77$	370.625	0.548528
$78$	270.614	0.392833
$79$	1152.80	1.64177	0.820885	−	0.571093i	$-0.193480\pi$
$79$	1152.80	1.64177	0.820885	+	0.571093i	$0.193480\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	58.9347	0.0793689
$83$	−692.581	−0.915911	−0.457956	−	0.888975i	$-0.651418\pi$
$83$	−692.581	−0.915911	−0.457956	+	0.888975i	$0.651418\pi$
$84$	276.691	0.359399
$85$	0	0
$86$	526.029	0.659571
$87$	502.227	0.618901
$88$	−1261.05	−1.52759
$89$	1417.70	1.68849	0.844246	−	0.535956i	$-0.180049\pi$
$89$	1417.70	1.68849	0.844246	+	0.535956i	$0.180049\pi$
$90$	0	0
$91$	137.217	0.158068
$92$	−253.190	−0.286923
$93$	−676.914	−0.754760
$94$	956.783	1.04984
$95$	0	0
$96$	513.708	0.546147
$97$	−661.229	−0.692140	−0.346070	−	0.938209i	$-0.612484\pi$
$97$	−661.229	−0.692140	−0.346070	+	0.938209i	$0.612484\pi$
$98$	225.484	0.232422
$99$	−476.518	−0.483757
$100$	0	0
$101$	187.231	0.184458	0.0922289	−	0.995738i	$-0.470601\pi$
$101$	187.231	0.184458	0.0922289	+	0.995738i	$0.470601\pi$
$102$	849.121	0.824270
$103$	−1076.28	−1.02961	−0.514803	−	0.857308i	$-0.672135\pi$
$103$	−1076.28	−1.02961	−0.514803	+	0.857308i	$0.672135\pi$
$104$	−466.879	−0.440204
$105$	0	0
$106$	−1046.44	−0.958861
$107$	−1591.65	−1.43805	−0.719024	−	0.694986i	$-0.755411\pi$
$107$	−1591.65	−1.43805	−0.719024	+	0.694986i	$0.755411\pi$
$108$	−355.746	−0.316960
$109$	1706.29	1.49938	0.749692	−	0.661786i	$-0.230201\pi$
$109$	1706.29	1.49938	0.749692	+	0.661786i	$0.230201\pi$
$110$	0	0
$111$	935.114	0.799613
$112$	−29.3640	−0.0247735
$113$	560.000	0.466198	0.233099	−	0.972453i	$-0.425113\pi$
$113$	560.000	0.466198	0.233099	+	0.972453i	$0.425113\pi$
$114$	−374.020	−0.307282
$115$	0	0
$116$	−2205.74	−1.76550
$117$	−176.421	−0.139403
$118$	−2786.48	−2.17387
$119$	430.553	0.331670
$120$	0	0
$121$	1472.33	1.10618
$122$	−1454.05	−1.07905
$123$	−38.4214	−0.0281653
$124$	2972.96	2.15306
$125$	0	0
$126$	−289.908	−0.204977
$127$	−2098.96	−1.46655	−0.733276	−	0.679931i	$-0.762010\pi$
$127$	−2098.96	−1.46655	−0.733276	+	0.679931i	$0.762010\pi$
$128$	−2410.60	−1.66460
$129$	−342.934	−0.234060
$130$	0	0
$131$	1675.28	1.11733	0.558664	−	0.829394i	$-0.311314\pi$
$131$	1675.28	1.11733	0.558664	+	0.829394i	$0.311314\pi$
$132$	2092.83	1.37998
$133$	−189.650	−0.123644
$134$	−3314.40	−2.13672
$135$	0	0
$136$	−1464.95	−0.923667
$137$	2678.78	1.67054	0.835269	−	0.549841i	$-0.185312\pi$
$137$	2678.78	1.67054	0.835269	+	0.549841i	$0.185312\pi$
$138$	265.284	0.163641
$139$	1032.11	0.629800	0.314900	−	0.949125i	$-0.398029\pi$
$139$	1032.11	0.629800	0.314900	+	0.949125i	$0.398029\pi$
$140$	0	0
$141$	−623.756	−0.372552
$142$	−258.969	−0.153043
$143$	1037.88	0.606935
$144$	37.7537	0.0218482
$145$	0	0
$146$	5312.02	3.01114
$147$	−147.000	−0.0824786
$148$	−4106.95	−2.28101
$149$	1187.33	0.652815	0.326408	−	0.945229i	$-0.394162\pi$
$149$	1187.33	0.652815	0.326408	+	0.945229i	$0.394162\pi$
$150$	0	0
$151$	−138.484	−0.0746338	−0.0373169	−	0.999303i	$-0.511881\pi$
$151$	−138.484	−0.0746338	−0.0373169	+	0.999303i	$0.511881\pi$
$152$	645.282	0.344337
$153$	−553.568	−0.292505
$154$	1705.51	0.892429
$155$	0	0
$156$	774.830	0.397667
$157$	−441.575	−0.224469	−0.112234	−	0.993682i	$-0.535801\pi$
$157$	−441.575	−0.224469	−0.112234	+	0.993682i	$0.535801\pi$
$158$	5304.85	2.67108
$159$	682.207	0.340267
$160$	0	0
$161$	134.514	0.0658461
$162$	372.739	0.180772
$163$	−2116.62	−1.01710	−0.508548	−	0.861034i	$-0.669817\pi$
$163$	−2116.62	−1.01710	−0.508548	+	0.861034i	$0.669817\pi$
$164$	168.744	0.0803455
$165$	0	0
$166$	−3187.06	−1.49014
$167$	−843.566	−0.390881	−0.195440	−	0.980716i	$-0.562614\pi$
$167$	−843.566	−0.390881	−0.195440	+	0.980716i	$0.562614\pi$
$168$	500.166	0.229694
$169$	−1812.75	−0.825101
$170$	0	0
$171$	243.835	0.109044
$172$	1506.14	0.667687
$173$	4319.01	1.89808	0.949042	−	0.315150i	$-0.102055\pi$
$173$	4319.01	1.89808	0.949042	+	0.315150i	$0.102055\pi$
$174$	2311.11	1.00692
$175$	0	0
$176$	−222.103	−0.0951229
$177$	1816.59	0.771432
$178$	6523.84	2.74709
$179$	−421.744	−0.176104	−0.0880520	−	0.996116i	$-0.528064\pi$
$179$	−421.744	−0.176104	−0.0880520	+	0.996116i	$0.528064\pi$
$180$	0	0
$181$	791.205	0.324916	0.162458	−	0.986715i	$-0.448058\pi$
$181$	791.205	0.324916	0.162458	+	0.986715i	$0.448058\pi$
$182$	631.432	0.257170
$183$	947.942	0.382917
$184$	−457.684	−0.183375
$185$	0	0
$186$	−3114.96	−1.22796
$187$	3256.61	1.27351
$188$	2739.49	1.06276
$189$	189.000	0.0727393
$190$	0	0
$191$	1790.22	0.678199	0.339100	−	0.940750i	$-0.389878\pi$
$191$	1790.22	0.678199	0.339100	+	0.940750i	$0.389878\pi$
$192$	2464.61	0.926397
$193$	2301.10	0.858221	0.429111	−	0.903252i	$-0.358827\pi$
$193$	2301.10	0.858221	0.429111	+	0.903252i	$0.358827\pi$
$194$	−3042.79	−1.12608
$195$	0	0
$196$	645.613	0.235282
$197$	−3490.59	−1.26241	−0.631203	−	0.775617i	$-0.717439\pi$
$197$	−3490.59	−1.26241	−0.631203	+	0.775617i	$0.717439\pi$
$198$	−2192.80	−0.787048
$199$	822.357	0.292942	0.146471	−	0.989215i	$-0.453209\pi$
$199$	822.357	0.292942	0.146471	+	0.989215i	$0.453209\pi$
$200$	0	0
$201$	2160.76	0.758251
$202$	861.586	0.300104
$203$	1171.86	0.405166
$204$	2431.23	0.834413
$205$	0	0
$206$	−4952.75	−1.67512
$207$	−172.947	−0.0580708
$208$	−82.2292	−0.0274114
$209$	−1434.47	−0.474757
$210$	0	0
$211$	−2323.79	−0.758181	−0.379091	−	0.925360i	$-0.623763\pi$
$211$	−2323.79	−0.758181	−0.379091	+	0.925360i	$0.623763\pi$
$212$	−2996.20	−0.970660
$213$	168.830	0.0543099
$214$	−7324.34	−2.33963
$215$	0	0
$216$	−643.071	−0.202571
$217$	−1579.47	−0.494107
$218$	7851.86	2.43943
$219$	−3463.07	−1.06855
$220$	0	0
$221$	1205.70	0.366986
$222$	4303.13	1.30093
$223$	2526.13	0.758574	0.379287	−	0.925279i	$-0.376169\pi$
$223$	2526.13	0.758574	0.379287	+	0.925279i	$0.376169\pi$
$224$	1198.65	0.357537
$225$	0	0
$226$	2576.96	0.758482
$227$	−3924.06	−1.14735	−0.573676	−	0.819082i	$-0.694483\pi$
$227$	−3924.06	−1.14735	−0.573676	+	0.819082i	$0.694483\pi$
$228$	−1070.91	−0.311064
$229$	3591.55	1.03640	0.518201	−	0.855259i	$-0.326602\pi$
$229$	3591.55	1.03640	0.518201	+	0.855259i	$0.326602\pi$
$230$	0	0
$231$	−1111.88	−0.316693
$232$	−3987.26	−1.12835
$233$	−4957.57	−1.39391	−0.696955	−	0.717115i	$-0.745462\pi$
$233$	−4957.57	−1.39391	−0.696955	+	0.717115i	$0.745462\pi$
$234$	−811.841	−0.226802
$235$	0	0
$236$	−7978.34	−2.20062
$237$	−3458.39	−0.947877
$238$	1981.28	0.539611
$239$	−5663.57	−1.53283	−0.766414	−	0.642347i	$-0.777961\pi$
$239$	−5663.57	−1.53283	−0.766414	+	0.642347i	$0.777961\pi$
$240$	0	0
$241$	−2246.64	−0.600493	−0.300246	−	0.953862i	$-0.597069\pi$
$241$	−2246.64	−0.600493	−0.300246	+	0.953862i	$0.597069\pi$
$242$	6775.24	1.79971
$243$	−243.000	−0.0641500
$244$	−4163.29	−1.09232
$245$	0	0
$246$	−176.804	−0.0458236
$247$	−531.084	−0.136810
$248$	5374.12	1.37604
$249$	2077.74	0.528802
$250$	0	0
$251$	−3382.51	−0.850606	−0.425303	−	0.905051i	$-0.639833\pi$
$251$	−3382.51	−0.850606	−0.425303	+	0.905051i	$0.639833\pi$
$252$	−830.074	−0.207499
$253$	1017.44	0.252829
$254$	−9658.79	−2.38601
$255$	0	0
$256$	−4520.57	−1.10365
$257$	−8116.97	−1.97013	−0.985064	−	0.172190i	$-0.944916\pi$
$257$	−8116.97	−1.97013	−0.985064	+	0.172190i	$0.944916\pi$
$258$	−1578.09	−0.380804
$259$	2181.93	0.523470
$260$	0	0
$261$	−1506.68	−0.357323
$262$	7709.17	1.81784
$263$	−3643.53	−0.854257	−0.427129	−	0.904191i	$-0.640475\pi$
$263$	−3643.53	−0.854257	−0.427129	+	0.904191i	$0.640475\pi$
$264$	3783.15	0.881957
$265$	0	0
$266$	−872.714	−0.201164
$267$	−4253.10	−0.974851
$268$	−9489.90	−2.16302
$269$	7509.24	1.70203	0.851015	−	0.525141i	$-0.175987\pi$
$269$	7509.24	1.70203	0.851015	+	0.525141i	$0.175987\pi$
$270$	0	0
$271$	−8350.12	−1.87171	−0.935855	−	0.352385i	$-0.885371\pi$
$271$	−8350.12	−1.87171	−0.935855	+	0.352385i	$0.885371\pi$
$272$	−258.015	−0.0575165
$273$	−411.650	−0.0912608
$274$	12327.0	2.71788
$275$	0	0
$276$	759.571	0.165655
$277$	−3359.70	−0.728753	−0.364377	−	0.931252i	$-0.618718\pi$
$277$	−3359.70	−0.728753	−0.364377	+	0.931252i	$0.618718\pi$
$278$	4749.47	1.02465
$279$	2030.74	0.435761
$280$	0	0
$281$	6403.33	1.35940	0.679699	−	0.733491i	$-0.262110\pi$
$281$	6403.33	1.35940	0.679699	+	0.733491i	$0.262110\pi$
$282$	−2870.35	−0.606123
$283$	5862.59	1.23143	0.615715	−	0.787969i	$-0.288867\pi$
$283$	5862.59	1.23143	0.615715	+	0.787969i	$0.288867\pi$
$284$	−741.487	−0.154927
$285$	0	0
$286$	4776.01	0.987453
$287$	−89.6498	−0.0184385
$288$	−1541.12	−0.315318
$289$	−1129.82	−0.229965
$290$	0	0
$291$	1983.69	0.399607
$292$	15209.6	3.04819
$293$	−3866.04	−0.770840	−0.385420	−	0.922741i	$-0.625944\pi$
$293$	−3866.04	−0.770840	−0.385420	+	0.922741i	$0.625944\pi$
$294$	−676.452	−0.134189
$295$	0	0
$296$	−7424.01	−1.45781
$297$	1429.56	0.279297
$298$	5463.73	1.06210
$299$	376.686	0.0728573
$300$	0	0
$301$	−800.180	−0.153228
$302$	−637.266	−0.121426
$303$	−561.694	−0.106497
$304$	113.650	0.0214418
$305$	0	0
$306$	−2547.36	−0.475892
$307$	−7084.81	−1.31711	−0.658553	−	0.752535i	$-0.728831\pi$
$307$	−7084.81	−1.31711	−0.658553	+	0.752535i	$0.728831\pi$
$308$	4883.28	0.903411
$309$	3228.85	0.594443
$310$	0	0
$311$	−10616.6	−1.93572	−0.967862	−	0.251483i	$-0.919082\pi$
$311$	−10616.6	−1.93572	−0.967862	+	0.251483i	$0.919082\pi$
$312$	1400.64	0.254152
$313$	7247.78	1.30885	0.654423	−	0.756128i	$-0.272911\pi$
$313$	7247.78	1.30885	0.654423	+	0.756128i	$0.272911\pi$
$314$	−2032.00	−0.365199
$315$	0	0
$316$	15189.0	2.70395
$317$	−6308.24	−1.11768	−0.558842	−	0.829274i	$-0.688754\pi$
$317$	−6308.24	−1.11768	−0.558842	+	0.829274i	$0.688754\pi$
$318$	3139.32	0.553599
$319$	8863.72	1.55572
$320$	0	0
$321$	4774.96	0.830257
$322$	618.997	0.107128
$323$	−1666.41	−0.287064
$324$	1067.24	0.182997
$325$	0	0
$326$	−9740.09	−1.65477
$327$	−5118.87	−0.865670
$328$	305.033	0.0513494
$329$	−1455.43	−0.243892
$330$	0	0
$331$	6656.25	1.10532	0.552660	−	0.833407i	$-0.313613\pi$
$331$	6656.25	1.10532	0.552660	+	0.833407i	$0.313613\pi$
$332$	−9125.29	−1.50848
$333$	−2805.34	−0.461657
$334$	−3881.85	−0.635944
$335$	0	0
$336$	88.0920	0.0143030
$337$	11941.8	1.93030	0.965152	−	0.261691i	$-0.0842803\pi$
$337$	11941.8	1.93030	0.965152	+	0.261691i	$0.0842803\pi$
$338$	−8341.74	−1.34240
$339$	−1680.00	−0.269160
$340$	0	0
$341$	−11946.7	−1.89722
$342$	1122.06	0.177410
$343$	−343.000	−0.0539949
$344$	2722.61	0.426724
$345$	0	0
$346$	19874.9	3.08809
$347$	6390.65	0.988669	0.494335	−	0.869272i	$-0.335412\pi$
$347$	6390.65	0.988669	0.494335	+	0.869272i	$0.335412\pi$
$348$	6617.23	1.01931
$349$	−7760.54	−1.19029	−0.595147	−	0.803617i	$-0.702906\pi$
$349$	−7760.54	−1.19029	−0.595147	+	0.803617i	$0.702906\pi$
$350$	0	0
$351$	529.264	0.0804844
$352$	9066.34	1.37283
$353$	−5416.46	−0.816684	−0.408342	−	0.912829i	$-0.633893\pi$
$353$	−5416.46	−0.816684	−0.408342	+	0.912829i	$0.633893\pi$
$354$	8359.44	1.25508
$355$	0	0
$356$	18679.3	2.78090
$357$	−1291.66	−0.191490
$358$	−1940.74	−0.286513
$359$	−2563.26	−0.376834	−0.188417	−	0.982089i	$-0.560336\pi$
$359$	−2563.26	−0.376834	−0.188417	+	0.982089i	$0.560336\pi$
$360$	0	0
$361$	−6124.98	−0.892984
$362$	3640.90	0.528623
$363$	−4416.99	−0.638655
$364$	1807.94	0.260334
$365$	0	0
$366$	4362.16	0.622988
$367$	10118.8	1.43923	0.719614	−	0.694374i	$-0.244319\pi$
$367$	10118.8	1.43923	0.719614	+	0.694374i	$0.244319\pi$
$368$	−80.6098	−0.0114187
$369$	115.264	0.0162613
$370$	0	0
$371$	1591.82	0.222757
$372$	−8918.87	−1.24307
$373$	−1870.64	−0.259673	−0.129837	−	0.991535i	$-0.541445\pi$
$373$	−1870.64	−0.259673	−0.129837	+	0.991535i	$0.541445\pi$
$374$	14986.0	2.07194
$375$	0	0
$376$	4952.09	0.679215
$377$	3281.62	0.448307
$378$	869.724	0.118343
$379$	3804.87	0.515681	0.257841	−	0.966187i	$-0.416989\pi$
$379$	3804.87	0.515681	0.257841	+	0.966187i	$0.416989\pi$
$380$	0	0
$381$	6296.87	0.846715
$382$	8238.10	1.10340
$383$	−11709.3	−1.56219	−0.781094	−	0.624414i	$-0.785338\pi$
$383$	−11709.3	−1.56219	−0.781094	+	0.624414i	$0.785338\pi$
$384$	7231.79	0.961056
$385$	0	0
$386$	10589.0	1.39628
$387$	1028.80	0.135134
$388$	−8712.20	−1.13994
$389$	−1278.24	−0.166605	−0.0833023	−	0.996524i	$-0.526547\pi$
$389$	−1278.24	−0.166605	−0.0833023	+	0.996524i	$0.526547\pi$
$390$	0	0
$391$	1181.95	0.152874
$392$	1167.05	0.150370
$393$	−5025.85	−0.645090
$394$	−16062.7	−2.05387
$395$	0	0
$396$	−6278.50	−0.796733
$397$	−2101.91	−0.265723	−0.132862	−	0.991135i	$-0.542417\pi$
$397$	−2101.91	−0.265723	−0.132862	+	0.991135i	$0.542417\pi$
$398$	3784.25	0.476602
$399$	568.949	0.0713862
$400$	0	0
$401$	−11239.5	−1.39969	−0.699846	−	0.714294i	$-0.746748\pi$
$401$	−11239.5	−1.39969	−0.699846	+	0.714294i	$0.746748\pi$
$402$	9943.21	1.23364
$403$	−4423.04	−0.546718
$404$	2466.92	0.303797
$405$	0	0
$406$	5392.58	0.659186
$407$	16503.7	2.00997
$408$	4394.86	0.533279
$409$	6751.66	0.816254	0.408127	−	0.912925i	$-0.366182\pi$
$409$	6751.66	0.816254	0.408127	+	0.912925i	$0.366182\pi$
$410$	0	0
$411$	−8036.34	−0.964486
$412$	−14180.9	−1.69573
$413$	4238.72	0.505021
$414$	−795.853	−0.0944784
$415$	0	0
$416$	3356.63	0.395607
$417$	−3096.32	−0.363615
$418$	−6601.02	−0.772408
$419$	−8449.93	−0.985218	−0.492609	−	0.870251i	$-0.663957\pi$
$419$	−8449.93	−0.985218	−0.492609	+	0.870251i	$0.663957\pi$
$420$	0	0
$421$	2267.58	0.262506	0.131253	−	0.991349i	$-0.458100\pi$
$421$	2267.58	0.262506	0.131253	+	0.991349i	$0.458100\pi$
$422$	−10693.4	−1.23352
$423$	1871.27	0.215093
$424$	−5416.14	−0.620356
$425$	0	0
$426$	776.906	0.0883597
$427$	2211.86	0.250678
$428$	−20971.3	−2.36842
$429$	−3113.63	−0.350414
$430$	0	0
$431$	−3172.81	−0.354591	−0.177296	−	0.984158i	$-0.556735\pi$
$431$	−3172.81	−0.354591	−0.177296	+	0.984158i	$0.556735\pi$
$432$	−113.261	−0.0126141
$433$	13569.4	1.50601	0.753006	−	0.658013i	$-0.228603\pi$
$433$	13569.4	1.50601	0.753006	+	0.658013i	$0.228603\pi$
$434$	−7268.25	−0.803888
$435$	0	0
$436$	22481.7	2.46944
$437$	−520.625	−0.0569905
$438$	−15936.1	−1.73848
$439$	−12021.7	−1.30698	−0.653488	−	0.756936i	$-0.726695\pi$
$439$	−12021.7	−1.30698	−0.653488	+	0.756936i	$0.726695\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	5548.26	0.597068
$443$	−2072.36	−0.222259	−0.111129	−	0.993806i	$-0.535447\pi$
$443$	−2072.36	−0.222259	−0.111129	+	0.993806i	$0.535447\pi$
$444$	12320.8	1.31694
$445$	0	0
$446$	11624.5	1.23416
$447$	−3561.98	−0.376903
$448$	5750.77	0.606469
$449$	154.727	0.0162628	0.00813142	−	0.999967i	$-0.497412\pi$
$449$	154.727	0.0162628	0.00813142	+	0.999967i	$0.497412\pi$
$450$	0	0
$451$	−678.092	−0.0707984
$452$	7378.44	0.767815
$453$	415.453	0.0430898
$454$	−18057.4	−1.86669
$455$	0	0
$456$	−1935.84	−0.198803
$457$	−4347.26	−0.444981	−0.222491	−	0.974935i	$-0.571419\pi$
$457$	−4347.26	−0.444981	−0.222491	+	0.974935i	$0.571419\pi$
$458$	16527.3	1.68618
$459$	1660.70	0.168878
$460$	0	0
$461$	−5510.76	−0.556750	−0.278375	−	0.960473i	$-0.589796\pi$
$461$	−5510.76	−0.556750	−0.278375	+	0.960473i	$0.589796\pi$
$462$	−5116.54	−0.515244
$463$	−9166.14	−0.920058	−0.460029	−	0.887904i	$-0.652161\pi$
$463$	−9166.14	−0.920058	−0.460029	+	0.887904i	$0.652161\pi$
$464$	−702.257	−0.0702618
$465$	0	0
$466$	−22813.3	−2.26783
$467$	−9008.92	−0.892683	−0.446342	−	0.894863i	$-0.647273\pi$
$467$	−9008.92	−0.892683	−0.446342	+	0.894863i	$0.647273\pi$
$468$	−2324.49	−0.229593
$469$	5041.78	0.496392
$470$	0	0
$471$	1324.73	0.129597
$472$	−14422.2	−1.40643
$473$	−6052.39	−0.588349
$474$	−15914.5	−1.54215
$475$	0	0
$476$	5672.87	0.546251
$477$	−2046.62	−0.196453
$478$	−26062.1	−2.49384
$479$	10128.1	0.966108	0.483054	−	0.875590i	$-0.339527\pi$
$479$	10128.1	0.966108	0.483054	+	0.875590i	$0.339527\pi$
$480$	0	0
$481$	6110.15	0.579208
$482$	−10338.4	−0.976973
$483$	−403.543	−0.0380163
$484$	19399.1	1.82185
$485$	0	0
$486$	−1118.22	−0.104369
$487$	954.443	0.0888089	0.0444045	−	0.999014i	$-0.485861\pi$
$487$	954.443	0.0888089	0.0444045	+	0.999014i	$0.485861\pi$
$488$	−7525.85	−0.698113
$489$	6349.86	0.587220
$490$	0	0
$491$	7336.23	0.674296	0.337148	−	0.941452i	$-0.390538\pi$
$491$	7336.23	0.674296	0.337148	+	0.941452i	$0.390538\pi$
$492$	−506.231	−0.0463875
$493$	10296.9	0.940670
$494$	−2443.90	−0.222583
$495$	0	0
$496$	946.519	0.0856854
$497$	393.936	0.0355542
$498$	9561.18	0.860335
$499$	16572.9	1.48679	0.743393	−	0.668855i	$-0.233215\pi$
$499$	16572.9	1.48679	0.743393	+	0.668855i	$0.233215\pi$
$500$	0	0
$501$	2530.70	0.225675
$502$	−15565.3	−1.38390
$503$	−13628.8	−1.20811	−0.604053	−	0.796944i	$-0.706448\pi$
$503$	−13628.8	−1.20811	−0.604053	+	0.796944i	$0.706448\pi$
$504$	−1500.50	−0.132614
$505$	0	0
$506$	4681.96	0.411341
$507$	5438.24	0.476372
$508$	−27655.4	−2.41537
$509$	−8370.77	−0.728935	−0.364467	−	0.931216i	$-0.618749\pi$
$509$	−8370.77	−0.728935	−0.364467	+	0.931216i	$0.618749\pi$
$510$	0	0
$511$	−8080.50	−0.699531
$512$	−1517.60	−0.130994
$513$	−731.506	−0.0629567
$514$	−37352.0	−3.20530
$515$	0	0
$516$	−4518.43	−0.385489
$517$	−11008.6	−0.936473
$518$	10040.6	0.851660
$519$	−12957.0	−1.09586
$520$	0	0
$521$	−705.491	−0.0593246	−0.0296623	−	0.999560i	$-0.509443\pi$
$521$	−705.491	−0.0593246	−0.0296623	+	0.999560i	$0.509443\pi$
$522$	−6933.32	−0.581347
$523$	4556.70	0.380977	0.190488	−	0.981689i	$-0.438993\pi$
$523$	4556.70	0.380977	0.190488	+	0.981689i	$0.438993\pi$
$524$	22073.1	1.84021
$525$	0	0
$526$	−16766.5	−1.38984
$527$	−13878.4	−1.14716
$528$	666.309	0.0549193
$529$	−11797.7	−0.969650
$530$	0	0
$531$	−5449.78	−0.445387
$532$	−2498.78	−0.203639
$533$	−251.050	−0.0204018
$534$	−19571.5	−1.58604
$535$	0	0
$536$	−17154.6	−1.38240
$537$	1265.23	0.101674
$538$	34555.4	2.76912
$539$	−2594.38	−0.207324
$540$	0	0
$541$	−21336.7	−1.69563	−0.847817	−	0.530289i	$-0.822084\pi$
$541$	−21336.7	−1.69563	−0.847817	+	0.530289i	$0.822084\pi$
$542$	−38424.9	−3.04518
$543$	−2373.61	−0.187590
$544$	10532.3	0.830090
$545$	0	0
$546$	−1894.30	−0.148477
$547$	957.875	0.0748734	0.0374367	−	0.999299i	$-0.488081\pi$
$547$	957.875	0.0748734	0.0374367	+	0.999299i	$0.488081\pi$
$548$	35295.0	2.75133
$549$	−2843.83	−0.221077
$550$	0	0
$551$	−4535.58	−0.350676
$552$	1373.05	0.105871
$553$	−8069.58	−0.620531
$554$	−15460.4	−1.18565
$555$	0	0
$556$	13598.8	1.03726
$557$	13357.8	1.01614	0.508069	−	0.861316i	$-0.330359\pi$
$557$	13357.8	1.01614	0.508069	+	0.861316i	$0.330359\pi$
$558$	9344.89	0.708962
$559$	−2240.78	−0.169543
$560$	0	0
$561$	−9769.83	−0.735263
$562$	29466.3	2.21167
$563$	−14988.5	−1.12201	−0.561003	−	0.827814i	$-0.689584\pi$
$563$	−14988.5	−1.12201	−0.561003	+	0.827814i	$0.689584\pi$
$564$	−8218.48	−0.613582
$565$	0	0
$566$	26978.0	2.00348
$567$	−567.000	−0.0419961
$568$	−1340.36	−0.0990148
$569$	−12901.9	−0.950572	−0.475286	−	0.879831i	$-0.657655\pi$
$569$	−12901.9	−0.950572	−0.475286	+	0.879831i	$0.657655\pi$
$570$	0	0
$571$	−19768.4	−1.44883	−0.724414	−	0.689365i	$-0.757890\pi$
$571$	−19768.4	−1.44883	−0.724414	+	0.689365i	$0.757890\pi$
$572$	13674.8	0.999604
$573$	−5370.67	−0.391559
$574$	−412.543	−0.0299986
$575$	0	0
$576$	−7393.84	−0.534855
$577$	5598.68	0.403945	0.201972	−	0.979391i	$-0.435265\pi$
$577$	5598.68	0.403945	0.201972	+	0.979391i	$0.435265\pi$
$578$	−5199.10	−0.374142
$579$	−6903.30	−0.495494
$580$	0	0
$581$	4848.07	0.346182
$582$	9128.36	0.650142
$583$	12040.2	0.855321
$584$	27493.8	1.94812
$585$	0	0
$586$	−17790.4	−1.25412
$587$	−2915.30	−0.204987	−0.102494	−	0.994734i	$-0.532682\pi$
$587$	−2915.30	−0.204987	−0.102494	+	0.994734i	$0.532682\pi$
$588$	−1936.84	−0.135840
$589$	6113.17	0.427655
$590$	0	0
$591$	10471.8	0.728851
$592$	−1307.56	−0.0907774
$593$	6126.89	0.424285	0.212143	−	0.977239i	$-0.431956\pi$
$593$	6126.89	0.424285	0.212143	+	0.977239i	$0.431956\pi$
$594$	6578.40	0.454403
$595$	0	0
$596$	15643.9	1.07517
$597$	−2467.07	−0.169130
$598$	1733.40	0.118535
$599$	−13503.6	−0.921104	−0.460552	−	0.887633i	$-0.652348\pi$
$599$	−13503.6	−0.921104	−0.460552	+	0.887633i	$0.652348\pi$
$600$	0	0
$601$	−28199.1	−1.91392	−0.956960	−	0.290220i	$-0.906272\pi$
$601$	−28199.1	−1.91392	−0.956960	+	0.290220i	$0.906272\pi$
$602$	−3682.20	−0.249294
$603$	−6482.29	−0.437776
$604$	−1824.64	−0.122920
$605$	0	0
$606$	−2584.76	−0.173265
$607$	13557.2	0.906543	0.453272	−	0.891372i	$-0.350257\pi$
$607$	13557.2	0.906543	0.453272	+	0.891372i	$0.350257\pi$
$608$	−4639.27	−0.309452
$609$	−3515.59	−0.233923
$610$	0	0
$611$	−4075.70	−0.269861
$612$	−7293.69	−0.481748
$613$	10357.7	0.682452	0.341226	−	0.939981i	$-0.389158\pi$
$613$	10357.7	0.682452	0.341226	+	0.939981i	$0.389158\pi$
$614$	−32602.3	−2.14287
$615$	0	0
$616$	8827.35	0.577377
$617$	9433.74	0.615540	0.307770	−	0.951461i	$-0.400417\pi$
$617$	9433.74	0.615540	0.307770	+	0.951461i	$0.400417\pi$
$618$	14858.3	0.967131
$619$	−5686.15	−0.369217	−0.184609	−	0.982812i	$-0.559102\pi$
$619$	−5686.15	−0.369217	−0.184609	+	0.982812i	$0.559102\pi$
$620$	0	0
$621$	518.841	0.0335272
$622$	−48854.4	−3.14933
$623$	−9923.89	−0.638190
$624$	246.688	0.0158260
$625$	0	0
$626$	33352.2	2.12943
$627$	4303.41	0.274101
$628$	−5818.10	−0.369693
$629$	19172.2	1.21533
$630$	0	0
$631$	24059.8	1.51792	0.758959	−	0.651138i	$-0.225708\pi$
$631$	24059.8	1.51792	0.758959	+	0.651138i	$0.225708\pi$
$632$	27456.7	1.72811
$633$	6971.37	0.437736
$634$	−29028.7	−1.81842
$635$	0	0
$636$	8988.60	0.560411
$637$	−960.517	−0.0597442
$638$	40788.3	2.53107
$639$	−506.489	−0.0313559
$640$	0	0
$641$	18543.7	1.14264	0.571319	−	0.820728i	$-0.306432\pi$
$641$	18543.7	1.14264	0.571319	+	0.820728i	$0.306432\pi$
$642$	21973.0	1.35079
$643$	−6491.72	−0.398147	−0.199073	−	0.979985i	$-0.563793\pi$
$643$	−6491.72	−0.398147	−0.199073	+	0.979985i	$0.563793\pi$
$644$	1772.33	0.108447
$645$	0	0
$646$	−7668.36	−0.467040
$647$	12238.1	0.743630	0.371815	−	0.928307i	$-0.378736\pi$
$647$	12238.1	0.743630	0.371815	+	0.928307i	$0.378736\pi$
$648$	1929.21	0.116955
$649$	32060.7	1.93913
$650$	0	0
$651$	4738.40	0.285273
$652$	−27888.1	−1.67513
$653$	2507.00	0.150239	0.0751197	−	0.997175i	$-0.476066\pi$
$653$	2507.00	0.150239	0.0751197	+	0.997175i	$0.476066\pi$
$654$	−23555.6	−1.40840
$655$	0	0
$656$	53.7240	0.00319752
$657$	10389.2	0.616928
$658$	−6697.48	−0.396801
$659$	−16501.0	−0.975398	−0.487699	−	0.873012i	$-0.662164\pi$
$659$	−16501.0	−0.975398	−0.487699	+	0.873012i	$0.662164\pi$
$660$	0	0
$661$	19086.4	1.12311	0.561554	−	0.827440i	$-0.310204\pi$
$661$	19086.4	1.12311	0.561554	+	0.827440i	$0.310204\pi$
$662$	30630.2	1.79830
$663$	−3617.09	−0.211879
$664$	−16495.5	−0.964081
$665$	0	0
$666$	−12909.4	−0.751093
$667$	3216.99	0.186750
$668$	−11114.6	−0.643770
$669$	−7578.38	−0.437963
$670$	0	0
$671$	16730.1	0.962529
$672$	−3595.96	−0.206424
$673$	28326.1	1.62242	0.811211	−	0.584754i	$-0.198809\pi$
$673$	28326.1	1.62242	0.811211	+	0.584754i	$0.198809\pi$
$674$	54952.8	3.14051
$675$	0	0
$676$	−23884.3	−1.35892
$677$	−9012.99	−0.511665	−0.255833	−	0.966721i	$-0.582350\pi$
$677$	−9012.99	−0.511665	−0.255833	+	0.966721i	$0.582350\pi$
$678$	−7730.88	−0.437910
$679$	4628.60	0.261604
$680$	0	0
$681$	11772.2	0.662424
$682$	−54975.5	−3.08669
$683$	3111.96	0.174343	0.0871713	−	0.996193i	$-0.472217\pi$
$683$	3111.96	0.174343	0.0871713	+	0.996193i	$0.472217\pi$
$684$	3212.72	0.179593
$685$	0	0
$686$	−1578.39	−0.0878471
$687$	−10774.6	−0.598367
$688$	479.520	0.0265720
$689$	4457.63	0.246476
$690$	0	0
$691$	21905.5	1.20597	0.602984	−	0.797753i	$-0.293978\pi$
$691$	21905.5	1.20597	0.602984	+	0.797753i	$0.293978\pi$
$692$	56906.3	3.12609
$693$	3335.63	0.182843
$694$	29408.0	1.60852
$695$	0	0
$696$	11961.8	0.651451
$697$	−787.735	−0.0428086
$698$	−35711.8	−1.93655
$699$	14872.7	0.804775
$700$	0	0
$701$	−13337.0	−0.718592	−0.359296	−	0.933224i	$-0.616983\pi$
$701$	−13337.0	−0.718592	−0.359296	+	0.933224i	$0.616983\pi$
$702$	2435.52	0.130944
$703$	−8444.95	−0.453069
$704$	43497.5	2.32866
$705$	0	0
$706$	−24925.0	−1.32870
$707$	−1310.62	−0.0697185
$708$	23935.0	1.27053
$709$	31571.2	1.67233	0.836164	−	0.548480i	$-0.184793\pi$
$709$	31571.2	1.67233	0.836164	+	0.548480i	$0.184793\pi$
$710$	0	0
$711$	10375.2	0.547257
$712$	33766.0	1.77729
$713$	−4335.94	−0.227745
$714$	−5943.85	−0.311545
$715$	0	0
$716$	−5556.80	−0.290038
$717$	16990.7	0.884979
$718$	−11795.4	−0.613091
$719$	−1983.80	−0.102897	−0.0514486	−	0.998676i	$-0.516384\pi$
$719$	−1983.80	−0.102897	−0.0514486	+	0.998676i	$0.516384\pi$
$720$	0	0
$721$	7533.99	0.389155
$722$	−28185.4	−1.45284
$723$	6739.92	0.346695
$724$	10424.7	0.535127
$725$	0	0
$726$	−20325.7	−1.03906
$727$	−28855.0	−1.47204	−0.736020	−	0.676960i	$-0.763297\pi$
$727$	−28855.0	−1.47204	−0.736020	+	0.676960i	$0.763297\pi$
$728$	3268.15	0.166381
$729$	729.000	0.0370370
$730$	0	0
$731$	−7031.02	−0.355748
$732$	12489.9	0.630654
$733$	−2709.63	−0.136538	−0.0682691	−	0.997667i	$-0.521748\pi$
$733$	−2709.63	−0.136538	−0.0682691	+	0.997667i	$0.521748\pi$
$734$	46563.8	2.34156
$735$	0	0
$736$	3290.53	0.164797
$737$	38134.9	1.90599
$738$	530.412	0.0264563
$739$	−24900.6	−1.23949	−0.619746	−	0.784802i	$-0.712764\pi$
$739$	−24900.6	−1.23949	−0.619746	+	0.784802i	$0.712764\pi$
$740$	0	0
$741$	1593.25	0.0789872
$742$	7325.08	0.362415
$743$	14522.6	0.717070	0.358535	−	0.933516i	$-0.383276\pi$
$743$	14522.6	0.717070	0.358535	+	0.933516i	$0.383276\pi$
$744$	−16122.4	−0.794455
$745$	0	0
$746$	−8608.16	−0.422476
$747$	−6233.23	−0.305304
$748$	42908.4	2.09744
$749$	11141.6	0.543531
$750$	0	0
$751$	−14360.6	−0.697768	−0.348884	−	0.937166i	$-0.613439\pi$
$751$	−14360.6	−0.697768	−0.348884	+	0.937166i	$0.613439\pi$
$752$	872.190	0.0422945
$753$	10147.5	0.491098
$754$	15101.1	0.729374
$755$	0	0
$756$	2490.22	0.119800
$757$	6200.35	0.297695	0.148848	−	0.988860i	$-0.452444\pi$
$757$	6200.35	0.297695	0.148848	+	0.988860i	$0.452444\pi$
$758$	17508.9	0.838988
$759$	−3052.31	−0.145971
$760$	0	0
$761$	−22218.6	−1.05838	−0.529188	−	0.848505i	$-0.677503\pi$
$761$	−22218.6	−1.05838	−0.529188	+	0.848505i	$0.677503\pi$
$762$	28976.4	1.37756
$763$	−11944.0	−0.566714
$764$	23587.6	1.11698
$765$	0	0
$766$	−53882.9	−2.54160
$767$	11869.8	0.558795
$768$	13561.7	0.637195
$769$	9799.16	0.459515	0.229757	−	0.973248i	$-0.426207\pi$
$769$	9799.16	0.459515	0.229757	+	0.973248i	$0.426207\pi$
$770$	0	0
$771$	24350.9	1.13745
$772$	30318.8	1.41347
$773$	23174.1	1.07828	0.539141	−	0.842215i	$-0.318749\pi$
$773$	23174.1	1.07828	0.539141	+	0.842215i	$0.318749\pi$
$774$	4734.26	0.219857
$775$	0	0
$776$	−15748.8	−0.728541
$777$	−6545.80	−0.302225
$778$	−5882.08	−0.271058
$779$	346.981	0.0159588
$780$	0	0
$781$	2979.65	0.136517
$782$	5439.00	0.248719
$783$	4520.05	0.206300
$784$	205.548	0.00936352
$785$	0	0
$786$	−23127.5	−1.04953
$787$	−13218.3	−0.598707	−0.299353	−	0.954142i	$-0.596771\pi$
$787$	−13218.3	−0.598707	−0.299353	+	0.954142i	$0.596771\pi$
$788$	−45991.2	−2.07915
$789$	10930.6	0.493206
$790$	0	0
$791$	−3920.00	−0.176206
$792$	−11349.4	−0.509198
$793$	6193.97	0.277370
$794$	−9672.41	−0.432319
$795$	0	0
$796$	10835.2	0.482466
$797$	−1421.60	−0.0631815	−0.0315908	−	0.999501i	$-0.510057\pi$
$797$	−1421.60	−0.0631815	−0.0315908	+	0.999501i	$0.510057\pi$
$798$	2618.14	0.116142
$799$	−12788.6	−0.566242
$800$	0	0
$801$	12759.3	0.562831
$802$	−51721.2	−2.27723
$803$	−61119.2	−2.68599
$804$	28469.7	1.24882
$805$	0	0
$806$	−20353.6	−0.889484
$807$	−22527.7	−0.982668
$808$	4459.37	0.194159
$809$	11669.3	0.507131	0.253566	−	0.967318i	$-0.418397\pi$
$809$	11669.3	0.507131	0.253566	+	0.967318i	$0.418397\pi$
$810$	0	0
$811$	2165.42	0.0937587	0.0468793	−	0.998901i	$-0.485072\pi$
$811$	2165.42	0.0937587	0.0468793	+	0.998901i	$0.485072\pi$
$812$	15440.2	0.667297
$813$	25050.4	1.08063
$814$	75945.1	3.27012
$815$	0	0
$816$	774.046	0.0332072
$817$	3097.02	0.132621
$818$	31069.2	1.32801
$819$	1234.95	0.0526894
$820$	0	0
$821$	−37709.7	−1.60302	−0.801509	−	0.597982i	$-0.795969\pi$
$821$	−37709.7	−1.60302	−0.801509	+	0.597982i	$0.795969\pi$
$822$	−36981.0	−1.56917
$823$	−20883.1	−0.884497	−0.442248	−	0.896893i	$-0.645819\pi$
$823$	−20883.1	−0.884497	−0.442248	+	0.896893i	$0.645819\pi$
$824$	−25634.3	−1.08376
$825$	0	0
$826$	19505.4	0.821645
$827$	13018.1	0.547379	0.273690	−	0.961818i	$-0.411756\pi$
$827$	13018.1	0.547379	0.273690	+	0.961818i	$0.411756\pi$
$828$	−2278.71	−0.0956410
$829$	1309.94	0.0548807	0.0274404	−	0.999623i	$-0.491264\pi$
$829$	1309.94	0.0548807	0.0274404	+	0.999623i	$0.491264\pi$
$830$	0	0
$831$	10079.1	0.420746
$832$	16104.1	0.671045
$833$	−3013.87	−0.125359
$834$	−14248.4	−0.591585
$835$	0	0
$836$	−18900.2	−0.781912
$837$	−6092.23	−0.251587
$838$	−38884.2	−1.60290
$839$	35223.2	1.44939	0.724696	−	0.689069i	$-0.241980\pi$
$839$	35223.2	1.44939	0.724696	+	0.689069i	$0.241980\pi$
$840$	0	0
$841$	3636.81	0.149117
$842$	10434.8	0.427085
$843$	−19210.0	−0.784849
$844$	−30617.7	−1.24870
$845$	0	0
$846$	8611.05	0.349946
$847$	−10306.3	−0.418098
$848$	−953.920	−0.0386294
$849$	−17587.8	−0.710967
$850$	0	0
$851$	5989.82	0.241279
$852$	2224.46	0.0894470
$853$	14907.6	0.598390	0.299195	−	0.954192i	$-0.403282\pi$
$853$	14907.6	0.598390	0.299195	+	0.954192i	$0.403282\pi$
$854$	10178.4	0.407841
$855$	0	0
$856$	−37909.1	−1.51368
$857$	30377.1	1.21081	0.605404	−	0.795919i	$-0.293012\pi$
$857$	30377.1	1.21081	0.605404	+	0.795919i	$0.293012\pi$
$858$	−14328.0	−0.570106
$859$	24031.5	0.954535	0.477267	−	0.878758i	$-0.341627\pi$
$859$	24031.5	0.954535	0.477267	+	0.878758i	$0.341627\pi$
$860$	0	0
$861$	268.949	0.0106455
$862$	−14600.4	−0.576903
$863$	33363.7	1.31601	0.658004	−	0.753015i	$-0.271401\pi$
$863$	33363.7	1.31601	0.658004	+	0.753015i	$0.271401\pi$
$864$	4623.37	0.182049
$865$	0	0
$866$	62442.5	2.45021
$867$	3389.45	0.132770
$868$	−20810.7	−0.813780
$869$	−61036.6	−2.38265
$870$	0	0
$871$	14118.7	0.549246
$872$	40639.5	1.57824
$873$	−5951.06	−0.230713
$874$	−2395.77	−0.0927208
$875$	0	0
$876$	−45628.7	−1.75987
$877$	−4392.19	−0.169115	−0.0845574	−	0.996419i	$-0.526948\pi$
$877$	−4392.19	−0.169115	−0.0845574	+	0.996419i	$0.526948\pi$
$878$	−55320.3	−2.12639
$879$	11598.1	0.445045
$880$	0	0
$881$	−8132.72	−0.311008	−0.155504	−	0.987835i	$-0.549700\pi$
$881$	−8132.72	−0.311008	−0.155504	+	0.987835i	$0.549700\pi$
$882$	2029.36	0.0774739
$883$	16582.9	0.632003	0.316002	−	0.948759i	$-0.397659\pi$
$883$	16582.9	0.632003	0.316002	+	0.948759i	$0.397659\pi$
$884$	15886.0	0.604415
$885$	0	0
$886$	−9536.40	−0.361605
$887$	9140.65	0.346012	0.173006	−	0.984921i	$-0.444652\pi$
$887$	9140.65	0.346012	0.173006	+	0.984921i	$0.444652\pi$
$888$	22272.0	0.841667
$889$	14692.7	0.554305
$890$	0	0
$891$	−4288.67	−0.161252
$892$	33283.7	1.24935
$893$	5633.11	0.211092
$894$	−16391.2	−0.613203
$895$	0	0
$896$	16874.2	0.629159
$897$	−1130.06	−0.0420642
$898$	712.009	0.0264589
$899$	−37773.9	−1.40137
$900$	0	0
$901$	13987.0	0.517173
$902$	−3120.38	−0.115186
$903$	2400.54	0.0884662
$904$	13337.8	0.490717
$905$	0	0
$906$	1911.80	0.0701051
$907$	−25359.3	−0.928382	−0.464191	−	0.885735i	$-0.653655\pi$
$907$	−25359.3	−0.928382	−0.464191	+	0.885735i	$0.653655\pi$
$908$	−51702.5	−1.88966
$909$	1685.08	0.0614859
$910$	0	0
$911$	33308.6	1.21138	0.605688	−	0.795702i	$-0.292898\pi$
$911$	33308.6	1.21138	0.605688	+	0.795702i	$0.292898\pi$
$912$	−340.951	−0.0123794
$913$	36669.7	1.32923
$914$	−20004.9	−0.723963
$915$	0	0
$916$	47321.4	1.70692
$917$	−11727.0	−0.422311
$918$	7642.09	0.274757
$919$	−5827.47	−0.209174	−0.104587	−	0.994516i	$-0.533352\pi$
$919$	−5827.47	−0.209174	−0.104587	+	0.994516i	$0.533352\pi$
$920$	0	0
$921$	21254.4	0.760431
$922$	−25358.9	−0.905805
$923$	1103.15	0.0393400
$924$	−14649.8	−0.521584
$925$	0	0
$926$	−42180.0	−1.49689
$927$	−9686.56	−0.343202
$928$	28666.5	1.01403
$929$	19224.3	0.678934	0.339467	−	0.940618i	$-0.389753\pi$
$929$	19224.3	0.678934	0.339467	+	0.940618i	$0.389753\pi$
$930$	0	0
$931$	1327.55	0.0467332
$932$	−65319.8	−2.29573
$933$	31849.7	1.11759
$934$	−41456.5	−1.45235
$935$	0	0
$936$	−4201.91	−0.146735
$937$	37202.8	1.29708	0.648539	−	0.761181i	$-0.275380\pi$
$937$	37202.8	1.29708	0.648539	+	0.761181i	$0.275380\pi$
$938$	23200.8	0.807605
$939$	−21743.4	−0.755663
$940$	0	0
$941$	43155.8	1.49505	0.747523	−	0.664236i	$-0.231243\pi$
$941$	43155.8	1.49505	0.747523	+	0.664236i	$0.231243\pi$
$942$	6096.01	0.210848
$943$	−246.106	−0.00849874
$944$	−2540.12	−0.0875781
$945$	0	0
$946$	−27851.4	−0.957215
$947$	1442.59	0.0495014	0.0247507	−	0.999694i	$-0.492121\pi$
$947$	1442.59	0.0495014	0.0247507	+	0.999694i	$0.492121\pi$
$948$	−45567.0	−1.56113
$949$	−22628.2	−0.774016
$950$	0	0
$951$	18924.7	0.645295
$952$	10254.7	0.349113
$953$	−2964.70	−0.100773	−0.0503863	−	0.998730i	$-0.516045\pi$
$953$	−2964.70	−0.100773	−0.0503863	+	0.998730i	$0.516045\pi$
$954$	−9417.96	−0.319620
$955$	0	0
$956$	−74621.9	−2.52452
$957$	−26591.2	−0.898193
$958$	46606.7	1.57181
$959$	−18751.5	−0.631404
$960$	0	0
$961$	21121.5	0.708990
$962$	28117.2	0.942343
$963$	−14324.9	−0.479349
$964$	−29601.2	−0.988994
$965$	0	0
$966$	−1856.99	−0.0618506
$967$	59088.1	1.96499	0.982494	−	0.186292i	$-0.0596471\pi$
$967$	59088.1	1.96499	0.982494	+	0.186292i	$0.0596471\pi$
$968$	35067.1	1.16436
$969$	4999.24	0.165737
$970$	0	0
$971$	−6678.75	−0.220732	−0.110366	−	0.993891i	$-0.535202\pi$
$971$	−6678.75	−0.220732	−0.110366	+	0.993891i	$0.535202\pi$
$972$	−3201.71	−0.105653
$973$	−7224.76	−0.238042
$974$	4392.07	0.144488
$975$	0	0
$976$	−1325.49	−0.0434713
$977$	−46454.6	−1.52120	−0.760600	−	0.649220i	$-0.775095\pi$
$977$	−46454.6	−1.52120	−0.760600	+	0.649220i	$0.775095\pi$
$978$	29220.3	0.955379
$979$	−75062.2	−2.45046
$980$	0	0
$981$	15356.6	0.499795
$982$	33759.2	1.09705
$983$	3154.80	0.102363	0.0511814	−	0.998689i	$-0.483701\pi$
$983$	3154.80	0.102363	0.0511814	+	0.998689i	$0.483701\pi$
$984$	−915.098	−0.0296466
$985$	0	0
$986$	47383.5	1.53042
$987$	4366.30	0.140811
$988$	−6997.44	−0.225322
$989$	−2196.65	−0.0706262
$990$	0	0
$991$	36207.7	1.16062	0.580311	−	0.814395i	$-0.302931\pi$
$991$	36207.7	1.16062	0.580311	+	0.814395i	$0.302931\pi$
$992$	−38637.3	−1.23663
$993$	−19968.8	−0.638157
$994$	1812.78	0.0578450
$995$	0	0
$996$	27375.9	0.870922
$997$	−18452.9	−0.586166	−0.293083	−	0.956087i	$-0.594681\pi$
$997$	−18452.9	−0.586166	−0.293083	+	0.956087i	$0.594681\pi$
$998$	76263.9	2.41893
$999$	8416.02	0.266538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.a.t.1.4	✓	4
3.2	odd	2		1575.4.a.bj.1.1		4
5.2	odd	4		525.4.d.n.274.7		8
5.3	odd	4		525.4.d.n.274.2		8
5.4	even	2		525.4.a.u.1.1	yes	4
15.14	odd	2		1575.4.a.bk.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.t.1.4	✓	4	1.1	even	1	trivial
525.4.a.u.1.1	yes	4	5.4	even	2
525.4.d.n.274.2		8	5.3	odd	4
525.4.d.n.274.7		8	5.2	odd	4
1575.4.a.bj.1.1		4	3.2	odd	2
1575.4.a.bk.1.4		4	15.14	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+4.60171 q^{2} -3.00000 q^{3} +13.1758 q^{4} -13.8051 q^{6} -7.00000 q^{7} +23.8174 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.60171 q^{2} -3.00000 q^{3} +13.1758 q^{4} -13.8051 q^{6} -7.00000 q^{7} +23.8174 q^{8} +9.00000 q^{9} -52.9465 q^{11} -39.5273 q^{12} -19.6024 q^{13} -32.2120 q^{14} +4.19486 q^{16} -61.5076 q^{17} +41.4154 q^{18} +27.0928 q^{19} +21.0000 q^{21} -243.645 q^{22} -19.2163 q^{23} -71.4523 q^{24} -90.2046 q^{26} -27.0000 q^{27} -92.2304 q^{28} -167.409 q^{29} +225.638 q^{31} -171.236 q^{32} +158.839 q^{33} -283.040 q^{34} +118.582 q^{36} -311.705 q^{37} +124.673 q^{38} +58.8071 q^{39} +12.8071 q^{41} +96.6360 q^{42} +114.311 q^{43} -697.611 q^{44} -88.4281 q^{46} +207.919 q^{47} -12.5846 q^{48} +49.0000 q^{49} +184.523 q^{51} -258.277 q^{52} -227.402 q^{53} -124.246 q^{54} -166.722 q^{56} -81.2784 q^{57} -770.369 q^{58} -605.531 q^{59} -315.981 q^{61} +1038.32 q^{62} -63.0000 q^{63} -821.538 q^{64} +730.934 q^{66} -720.254 q^{67} -810.410 q^{68} +57.6490 q^{69} -56.2766 q^{71} +214.357 q^{72} +1154.36 q^{73} -1434.38 q^{74} +356.969 q^{76} +370.625 q^{77} +270.614 q^{78} +1152.80 q^{79} +81.0000 q^{81} +58.9347 q^{82} -692.581 q^{83} +276.691 q^{84} +526.029 q^{86} +502.227 q^{87} -1261.05 q^{88} +1417.70 q^{89} +137.217 q^{91} -253.190 q^{92} -676.914 q^{93} +956.783 q^{94} +513.708 q^{96} -661.229 q^{97} +225.484 q^{98} -476.518 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 16 * q^4 - 28 * q^7 - 9 * q^8 + 36 * q^9	\( 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9} + 21 q^{11} - 48 q^{12} - 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 84 q^{21} - 221 q^{22} - 102 q^{23} + 27 q^{24} + 129 q^{26} - 108 q^{27} - 112 q^{28} - 240 q^{29} + 351 q^{31} - 72 q^{32} - 63 q^{33} - 285 q^{34} + 144 q^{36} - 399 q^{37} - 324 q^{38} + 15 q^{39} + 381 q^{41} - 460 q^{43} - 975 q^{44} + 550 q^{46} - 60 q^{47} - 216 q^{48} + 196 q^{49} + 297 q^{51} - 223 q^{52} - 873 q^{53} + 63 q^{56} - 216 q^{57} - 1408 q^{58} - 855 q^{59} + 687 q^{61} + 477 q^{62} - 252 q^{63} - 1285 q^{64} + 663 q^{66} + 503 q^{67} - 1725 q^{68} + 306 q^{69} - 681 q^{71} - 81 q^{72} - 1228 q^{73} - 3369 q^{74} + 1910 q^{76} - 147 q^{77} - 387 q^{78} + 345 q^{79} + 324 q^{81} + 495 q^{82} - 1509 q^{83} + 336 q^{84} - 540 q^{86} + 720 q^{87} - 2266 q^{88} - 198 q^{89} + 35 q^{91} - 2994 q^{92} - 1053 q^{93} - 1732 q^{94} + 216 q^{96} + 372 q^{97} + 189 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 16 * q^4 - 28 * q^7 - 9 * q^8 + 36 * q^9 + 21 * q^11 - 48 * q^12 - 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 84 * q^21 - 221 * q^22 - 102 * q^23 + 27 * q^24 + 129 * q^26 - 108 * q^27 - 112 * q^28 - 240 * q^29 + 351 * q^31 - 72 * q^32 - 63 * q^33 - 285 * q^34 + 144 * q^36 - 399 * q^37 - 324 * q^38 + 15 * q^39 + 381 * q^41 - 460 * q^43 - 975 * q^44 + 550 * q^46 - 60 * q^47 - 216 * q^48 + 196 * q^49 + 297 * q^51 - 223 * q^52 - 873 * q^53 + 63 * q^56 - 216 * q^57 - 1408 * q^58 - 855 * q^59 + 687 * q^61 + 477 * q^62 - 252 * q^63 - 1285 * q^64 + 663 * q^66 + 503 * q^67 - 1725 * q^68 + 306 * q^69 - 681 * q^71 - 81 * q^72 - 1228 * q^73 - 3369 * q^74 + 1910 * q^76 - 147 * q^77 - 387 * q^78 + 345 * q^79 + 324 * q^81 + 495 * q^82 - 1509 * q^83 + 336 * q^84 - 540 * q^86 + 720 * q^87 - 2266 * q^88 - 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1053 * q^93 - 1732 * q^94 + 216 * q^96 + 372 * q^97 + 189 * q^99

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