LMFDB - Embedded newform 2025.4.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,4,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2025.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:	$119.478867762$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 $ x^6 - 2x^5 - 38x^4 + 42x^3 + 393x^2 - 72*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.11734$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.11734 q^{2} +8.95250 q^{4} +20.0229 q^{7} -3.92177 q^{8} +O(q^{10})$	$q-4.11734 q^{2} +8.95250 q^{4} +20.0229 q^{7} -3.92177 q^{8} +1.67827 q^{11} -11.1123 q^{13} -82.4412 q^{14} -55.4727 q^{16} +8.98682 q^{17} -50.6419 q^{19} -6.91000 q^{22} +214.491 q^{23} +45.7531 q^{26} +179.255 q^{28} +76.9088 q^{29} -273.537 q^{31} +259.774 q^{32} -37.0018 q^{34} +137.283 q^{37} +208.510 q^{38} +53.3457 q^{41} -295.270 q^{43} +15.0247 q^{44} -883.134 q^{46} -194.983 q^{47} +57.9173 q^{49} -99.4829 q^{52} -450.173 q^{53} -78.5252 q^{56} -316.660 q^{58} -481.145 q^{59} +675.854 q^{61} +1126.25 q^{62} -625.798 q^{64} -894.589 q^{67} +80.4545 q^{68} -721.947 q^{71} -915.175 q^{73} -565.241 q^{74} -453.372 q^{76} +33.6038 q^{77} +59.6642 q^{79} -219.643 q^{82} +742.875 q^{83} +1215.73 q^{86} -6.58177 q^{88} -1540.72 q^{89} -222.501 q^{91} +1920.23 q^{92} +802.812 q^{94} +1121.57 q^{97} -238.465 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8}+O(q^{10}) $ 6 * q + 4 * q^2 + 34 * q^4 - 40 * q^7 + 66 * q^8	$ 6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8} - 88 q^{11} - 20 q^{13} - 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 74 q^{22} + 210 q^{23} - 4 q^{26} - 352 q^{28} - 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 204 q^{37} - 20 q^{38} - 344 q^{41} - 512 q^{43} - 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 468 q^{52} + 850 q^{53} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} - 88 q^{67} + 236 q^{68} - 1364 q^{71} - 836 q^{73} - 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} - 1742 q^{82} + 2148 q^{83} - 2872 q^{86} - 1296 q^{88} - 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} + 612 q^{97} + 1982 q^{98}+O(q^{100}) $ 6 * q + 4 * q^2 + 34 * q^4 - 40 * q^7 + 66 * q^8 - 88 * q^11 - 20 * q^13 - 180 * q^14 + 58 * q^16 + 124 * q^17 - 46 * q^19 + 74 * q^22 + 210 * q^23 - 4 * q^26 - 352 * q^28 - 296 * q^29 - 104 * q^31 + 722 * q^32 - 428 * q^34 + 204 * q^37 - 20 * q^38 - 344 * q^41 - 512 * q^43 - 716 * q^44 - 186 * q^46 + 238 * q^47 + 68 * q^49 + 468 * q^52 + 850 * q^53 - 2316 * q^56 - 890 * q^58 - 1840 * q^59 - 364 * q^61 + 1038 * q^62 - 990 * q^64 - 88 * q^67 + 236 * q^68 - 1364 * q^71 - 836 * q^73 - 1316 * q^74 - 2106 * q^76 + 840 * q^77 - 680 * q^79 - 1742 * q^82 + 2148 * q^83 - 2872 * q^86 - 1296 * q^88 - 3000 * q^89 - 3058 * q^91 + 1002 * q^92 - 3662 * q^94 + 612 * q^97 + 1982 * 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 $

$2$	−4.11734	−1.45570	−0.727850	−	0.685736i	$-0.759480\pi$
$2$	−4.11734	−1.45570	−0.727850	+	0.685736i	$0.759480\pi$
$3$	0	0
$3$	0	0
$4$	8.95250	1.11906
$5$	0	0
$5$	0	0
$6$	0	0
$7$	20.0229	1.08114	0.540568	−	0.841300i	$-0.318209\pi$
$7$	20.0229	1.08114	0.540568	+	0.841300i	$0.318209\pi$
$8$	−3.92177	−0.173319
$9$	0	0
$10$	0	0
$11$	1.67827	0.0460015	0.0230007	−	0.999735i	$-0.492678\pi$
$11$	1.67827	0.0460015	0.0230007	+	0.999735i	$0.492678\pi$
$12$	0	0
$13$	−11.1123	−0.237077	−0.118538	−	0.992949i	$-0.537821\pi$
$13$	−11.1123	−0.237077	−0.118538	+	0.992949i	$0.537821\pi$
$14$	−82.4412	−1.57381
$15$	0	0
$16$	−55.4727	−0.866762
$17$	8.98682	0.128213	0.0641066	−	0.997943i	$-0.479580\pi$
$17$	8.98682	0.128213	0.0641066	+	0.997943i	$0.479580\pi$
$18$	0	0
$19$	−50.6419	−0.611477	−0.305738	−	0.952116i	$-0.598903\pi$
$19$	−50.6419	−0.611477	−0.305738	+	0.952116i	$0.598903\pi$
$20$	0	0
$21$	0	0
$22$	−6.91000	−0.0669644
$23$	214.491	1.94454	0.972272	−	0.233853i	$-0.0751334\pi$
$23$	214.491	1.94454	0.972272	+	0.233853i	$0.0751334\pi$
$24$	0	0
$25$	0	0
$26$	45.7531	0.345113
$27$	0	0
$28$	179.255	1.20986
$29$	76.9088	0.492469	0.246235	−	0.969210i	$-0.420807\pi$
$29$	76.9088	0.492469	0.246235	+	0.969210i	$0.420807\pi$
$30$	0	0
$31$	−273.537	−1.58480	−0.792399	−	0.610003i	$-0.791168\pi$
$31$	−273.537	−1.58480	−0.792399	+	0.610003i	$0.791168\pi$
$32$	259.774	1.43506
$33$	0	0
$34$	−37.0018	−0.186640
$35$	0	0
$36$	0	0
$37$	137.283	0.609978	0.304989	−	0.952356i	$-0.401347\pi$
$37$	137.283	0.609978	0.304989	+	0.952356i	$0.401347\pi$
$38$	208.510	0.890126
$39$	0	0
$40$	0	0
$41$	53.3457	0.203200	0.101600	−	0.994825i	$-0.467604\pi$
$41$	53.3457	0.203200	0.101600	+	0.994825i	$0.467604\pi$
$42$	0	0
$43$	−295.270	−1.04717	−0.523584	−	0.851974i	$-0.675405\pi$
$43$	−295.270	−1.04717	−0.523584	+	0.851974i	$0.675405\pi$
$44$	15.0247	0.0514785
$45$	0	0
$46$	−883.134	−2.83067
$47$	−194.983	−0.605132	−0.302566	−	0.953128i	$-0.597843\pi$
$47$	−194.983	−0.605132	−0.302566	+	0.953128i	$0.597843\pi$
$48$	0	0
$49$	57.9173	0.168855
$50$	0	0
$51$	0	0
$52$	−99.4829	−0.265304
$53$	−450.173	−1.16672	−0.583359	−	0.812214i	$-0.698262\pi$
$53$	−450.173	−1.16672	−0.583359	+	0.812214i	$0.698262\pi$
$54$	0	0
$55$	0	0
$56$	−78.5252	−0.187382
$57$	0	0
$58$	−316.660	−0.716888
$59$	−481.145	−1.06169	−0.530845	−	0.847469i	$-0.678125\pi$
$59$	−481.145	−1.06169	−0.530845	+	0.847469i	$0.678125\pi$
$60$	0	0
$61$	675.854	1.41859	0.709297	−	0.704910i	$-0.249012\pi$
$61$	675.854	1.41859	0.709297	+	0.704910i	$0.249012\pi$
$62$	1126.25	2.30699
$63$	0	0
$64$	−625.798	−1.22226
$65$	0	0
$66$	0	0
$67$	−894.589	−1.63122	−0.815608	−	0.578605i	$-0.803597\pi$
$67$	−894.589	−1.63122	−0.815608	+	0.578605i	$0.803597\pi$
$68$	80.4545	0.143479
$69$	0	0
$70$	0	0
$71$	−721.947	−1.20675	−0.603376	−	0.797457i	$-0.706178\pi$
$71$	−721.947	−1.20675	−0.603376	+	0.797457i	$0.706178\pi$
$72$	0	0
$73$	−915.175	−1.46730	−0.733651	−	0.679526i	$-0.762185\pi$
$73$	−915.175	−1.46730	−0.733651	+	0.679526i	$0.762185\pi$
$74$	−565.241	−0.887945
$75$	0	0
$76$	−453.372	−0.684280
$77$	33.6038	0.0497339
$78$	0	0
$79$	59.6642	0.0849715	0.0424858	−	0.999097i	$-0.486472\pi$
$79$	59.6642	0.0849715	0.0424858	+	0.999097i	$0.486472\pi$
$80$	0	0
$81$	0	0
$82$	−219.643	−0.295798
$83$	742.875	0.982423	0.491212	−	0.871040i	$-0.336554\pi$
$83$	742.875	0.982423	0.491212	+	0.871040i	$0.336554\pi$
$84$	0	0
$85$	0	0
$86$	1215.73	1.52436
$87$	0	0
$88$	−6.58177	−0.00797294
$89$	−1540.72	−1.83501	−0.917505	−	0.397723i	$-0.869800\pi$
$89$	−1540.72	−1.83501	−0.917505	+	0.397723i	$0.869800\pi$
$90$	0	0
$91$	−222.501	−0.256312
$92$	1920.23	2.17607
$93$	0	0
$94$	802.812	0.880891
$95$	0	0
$96$	0	0
$97$	1121.57	1.17400	0.586999	−	0.809588i	$-0.300309\pi$
$97$	1121.57	1.17400	0.586999	+	0.809588i	$0.300309\pi$
$98$	−238.465	−0.245802
$99$	0	0
$100$	0	0
$101$	1758.47	1.73242	0.866211	−	0.499679i	$-0.166549\pi$
$101$	1758.47	1.73242	0.866211	+	0.499679i	$0.166549\pi$
$102$	0	0
$103$	−1056.48	−1.01066	−0.505332	−	0.862925i	$-0.668630\pi$
$103$	−1056.48	−1.01066	−0.505332	+	0.862925i	$0.668630\pi$
$104$	43.5799	0.0410900
$105$	0	0
$106$	1853.52	1.69839
$107$	1470.03	1.32816	0.664081	−	0.747661i	$-0.268823\pi$
$107$	1470.03	1.32816	0.664081	+	0.747661i	$0.268823\pi$
$108$	0	0
$109$	918.889	0.807464	0.403732	−	0.914877i	$-0.367713\pi$
$109$	918.889	0.807464	0.403732	+	0.914877i	$0.367713\pi$
$110$	0	0
$111$	0	0
$112$	−1110.73	−0.937087
$113$	987.217	0.821855	0.410927	−	0.911668i	$-0.365205\pi$
$113$	987.217	0.821855	0.410927	+	0.911668i	$0.365205\pi$
$114$	0	0
$115$	0	0
$116$	688.526	0.551104
$117$	0	0
$118$	1981.04	1.54550
$119$	179.942	0.138616
$120$	0	0
$121$	−1328.18	−0.997884
$122$	−2782.72	−2.06505
$123$	0	0
$124$	−2448.84	−1.77349
$125$	0	0
$126$	0	0
$127$	−1480.10	−1.03415	−0.517077	−	0.855939i	$-0.672980\pi$
$127$	−1480.10	−1.03415	−0.517077	+	0.855939i	$0.672980\pi$
$128$	498.428	0.344182
$129$	0	0
$130$	0	0
$131$	506.595	0.337874	0.168937	−	0.985627i	$-0.445967\pi$
$131$	506.595	0.337874	0.168937	+	0.985627i	$0.445967\pi$
$132$	0	0
$133$	−1014.00	−0.661089
$134$	3683.33	2.37456
$135$	0	0
$136$	−35.2442	−0.0222218
$137$	975.723	0.608479	0.304240	−	0.952596i	$-0.401598\pi$
$137$	975.723	0.608479	0.304240	+	0.952596i	$0.401598\pi$
$138$	0	0
$139$	−1888.83	−1.15258	−0.576288	−	0.817247i	$-0.695499\pi$
$139$	−1888.83	−1.15258	−0.576288	+	0.817247i	$0.695499\pi$
$140$	0	0
$141$	0	0
$142$	2972.50	1.75667
$143$	−18.6494	−0.0109059
$144$	0	0
$145$	0	0
$146$	3768.09	2.13595
$147$	0	0
$148$	1229.03	0.682604
$149$	−2582.29	−1.41980	−0.709898	−	0.704305i	$-0.751259\pi$
$149$	−2582.29	−1.41980	−0.709898	+	0.704305i	$0.751259\pi$
$150$	0	0
$151$	2537.92	1.36777	0.683885	−	0.729590i	$-0.260289\pi$
$151$	2537.92	1.36777	0.683885	+	0.729590i	$0.260289\pi$
$152$	198.606	0.105981
$153$	0	0
$154$	−138.358	−0.0723976
$155$	0	0
$156$	0	0
$157$	−105.035	−0.0533930	−0.0266965	−	0.999644i	$-0.508499\pi$
$157$	−105.035	−0.0533930	−0.0266965	+	0.999644i	$0.508499\pi$
$158$	−245.658	−0.123693
$159$	0	0
$160$	0	0
$161$	4294.74	2.10232
$162$	0	0
$163$	−2228.88	−1.07104	−0.535519	−	0.844523i	$-0.679884\pi$
$163$	−2228.88	−1.07104	−0.535519	+	0.844523i	$0.679884\pi$
$164$	477.578	0.227394
$165$	0	0
$166$	−3058.67	−1.43011
$167$	2813.80	1.30382	0.651911	−	0.758296i	$-0.273968\pi$
$167$	2813.80	1.30382	0.651911	+	0.758296i	$0.273968\pi$
$168$	0	0
$169$	−2073.52	−0.943795
$170$	0	0
$171$	0	0
$172$	−2643.40	−1.17185
$173$	−3284.45	−1.44342	−0.721711	−	0.692195i	$-0.756644\pi$
$173$	−3284.45	−1.44342	−0.721711	+	0.692195i	$0.756644\pi$
$174$	0	0
$175$	0	0
$176$	−93.0981	−0.0398723
$177$	0	0
$178$	6343.67	2.67123
$179$	−1268.62	−0.529726	−0.264863	−	0.964286i	$-0.585327\pi$
$179$	−1268.62	−0.529726	−0.264863	+	0.964286i	$0.585327\pi$
$180$	0	0
$181$	1080.10	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$181$	1080.10	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$182$	916.111	0.373114
$183$	0	0
$184$	−841.185	−0.337027
$185$	0	0
$186$	0	0
$187$	15.0823	0.00589800
$188$	−1745.59	−0.677181
$189$	0	0
$190$	0	0
$191$	−286.714	−0.108617	−0.0543087	−	0.998524i	$-0.517296\pi$
$191$	−286.714	−0.108617	−0.0543087	+	0.998524i	$0.517296\pi$
$192$	0	0
$193$	4650.70	1.73453	0.867267	−	0.497843i	$-0.165875\pi$
$193$	4650.70	1.73453	0.867267	+	0.497843i	$0.165875\pi$
$194$	−4617.87	−1.70899
$195$	0	0
$196$	518.505	0.188959
$197$	1694.82	0.612947	0.306474	−	0.951879i	$-0.400851\pi$
$197$	1694.82	0.612947	0.306474	+	0.951879i	$0.400851\pi$
$198$	0	0
$199$	−3249.77	−1.15764	−0.578819	−	0.815456i	$-0.696486\pi$
$199$	−3249.77	−1.15764	−0.578819	+	0.815456i	$0.696486\pi$
$200$	0	0
$201$	0	0
$202$	−7240.23	−2.52189
$203$	1539.94	0.532426
$204$	0	0
$205$	0	0
$206$	4349.90	1.47122
$207$	0	0
$208$	616.430	0.205489
$209$	−84.9906	−0.0281288
$210$	0	0
$211$	−1078.56	−0.351900	−0.175950	−	0.984399i	$-0.556300\pi$
$211$	−1078.56	−0.351900	−0.175950	+	0.984399i	$0.556300\pi$
$212$	−4030.18	−1.30563
$213$	0	0
$214$	−6052.62	−1.93340
$215$	0	0
$216$	0	0
$217$	−5477.01	−1.71338
$218$	−3783.38	−1.17543
$219$	0	0
$220$	0	0
$221$	−99.8643	−0.0303964
$222$	0	0
$223$	5143.28	1.54448	0.772241	−	0.635330i	$-0.219136\pi$
$223$	5143.28	1.54448	0.772241	+	0.635330i	$0.219136\pi$
$224$	5201.44	1.55150
$225$	0	0
$226$	−4064.71	−1.19637
$227$	2909.66	0.850754	0.425377	−	0.905016i	$-0.360142\pi$
$227$	2909.66	0.850754	0.425377	+	0.905016i	$0.360142\pi$
$228$	0	0
$229$	1725.82	0.498015	0.249007	−	0.968502i	$-0.419896\pi$
$229$	1725.82	0.498015	0.249007	+	0.968502i	$0.419896\pi$
$230$	0	0
$231$	0	0
$232$	−301.619	−0.0853544
$233$	1087.48	0.305765	0.152882	−	0.988244i	$-0.451144\pi$
$233$	1087.48	0.305765	0.152882	+	0.988244i	$0.451144\pi$
$234$	0	0
$235$	0	0
$236$	−4307.45	−1.18810
$237$	0	0
$238$	−740.884	−0.201783
$239$	319.345	0.0864297	0.0432149	−	0.999066i	$-0.486240\pi$
$239$	319.345	0.0864297	0.0432149	+	0.999066i	$0.486240\pi$
$240$	0	0
$241$	3645.83	0.974475	0.487238	−	0.873269i	$-0.338005\pi$
$241$	3645.83	0.974475	0.487238	+	0.873269i	$0.338005\pi$
$242$	5468.58	1.45262
$243$	0	0
$244$	6050.58	1.58750
$245$	0	0
$246$	0	0
$247$	562.748	0.144967
$248$	1072.75	0.274676
$249$	0	0
$250$	0	0
$251$	572.874	0.144062	0.0720309	−	0.997402i	$-0.477052\pi$
$251$	572.874	0.144062	0.0720309	+	0.997402i	$0.477052\pi$
$252$	0	0
$253$	359.973	0.0894519
$254$	6094.07	1.50542
$255$	0	0
$256$	2954.18	0.721236
$257$	−3677.66	−0.892630	−0.446315	−	0.894876i	$-0.647264\pi$
$257$	−3677.66	−0.892630	−0.446315	+	0.894876i	$0.647264\pi$
$258$	0	0
$259$	2748.81	0.659469
$260$	0	0
$261$	0	0
$262$	−2085.83	−0.491842
$263$	2001.71	0.469319	0.234659	−	0.972078i	$-0.424603\pi$
$263$	2001.71	0.469319	0.234659	+	0.972078i	$0.424603\pi$
$264$	0	0
$265$	0	0
$266$	4174.98	0.962348
$267$	0	0
$268$	−8008.81	−1.82543
$269$	20.1629	0.00457009	0.00228504	−	0.999997i	$-0.499273\pi$
$269$	20.1629	0.00457009	0.00228504	+	0.999997i	$0.499273\pi$
$270$	0	0
$271$	4733.12	1.06095	0.530474	−	0.847701i	$-0.322014\pi$
$271$	4733.12	1.06095	0.530474	+	0.847701i	$0.322014\pi$
$272$	−498.524	−0.111130
$273$	0	0
$274$	−4017.38	−0.885763
$275$	0	0
$276$	0	0
$277$	−5029.12	−1.09087	−0.545434	−	0.838154i	$-0.683635\pi$
$277$	−5029.12	−1.09087	−0.545434	+	0.838154i	$0.683635\pi$
$278$	7776.94	1.67780
$279$	0	0
$280$	0	0
$281$	−5808.31	−1.23308	−0.616539	−	0.787325i	$-0.711466\pi$
$281$	−5808.31	−1.23308	−0.616539	+	0.787325i	$0.711466\pi$
$282$	0	0
$283$	−213.023	−0.0447453	−0.0223726	−	0.999750i	$-0.507122\pi$
$283$	−213.023	−0.0447453	−0.0223726	+	0.999750i	$0.507122\pi$
$284$	−6463.23	−1.35043
$285$	0	0
$286$	76.7859	0.0158757
$287$	1068.14	0.219687
$288$	0	0
$289$	−4832.24	−0.983561
$290$	0	0
$291$	0	0
$292$	−8193.10	−1.64200
$293$	−3593.09	−0.716419	−0.358210	−	0.933641i	$-0.616613\pi$
$293$	−3593.09	−0.716419	−0.358210	+	0.933641i	$0.616613\pi$
$294$	0	0
$295$	0	0
$296$	−538.392	−0.105721
$297$	0	0
$298$	10632.2	2.06680
$299$	−2383.49	−0.461006
$300$	0	0
$301$	−5912.17	−1.13213
$302$	−10449.5	−1.99106
$303$	0	0
$304$	2809.25	0.530004
$305$	0	0
$306$	0	0
$307$	−2403.23	−0.446774	−0.223387	−	0.974730i	$-0.571711\pi$
$307$	−2403.23	−0.446774	−0.223387	+	0.974730i	$0.571711\pi$
$308$	300.838	0.0556553
$309$	0	0
$310$	0	0
$311$	1608.16	0.293217	0.146608	−	0.989195i	$-0.453164\pi$
$311$	1608.16	0.293217	0.146608	+	0.989195i	$0.453164\pi$
$312$	0	0
$313$	−2780.54	−0.502127	−0.251063	−	0.967971i	$-0.580780\pi$
$313$	−2780.54	−0.502127	−0.251063	+	0.967971i	$0.580780\pi$
$314$	432.465	0.0777242
$315$	0	0
$316$	534.144	0.0950885
$317$	5535.70	0.980807	0.490403	−	0.871496i	$-0.336850\pi$
$317$	5535.70	0.980807	0.490403	+	0.871496i	$0.336850\pi$
$318$	0	0
$319$	129.074	0.0226543
$320$	0	0
$321$	0	0
$322$	−17682.9	−3.06034
$323$	−455.110	−0.0783994
$324$	0	0
$325$	0	0
$326$	9177.06	1.55911
$327$	0	0
$328$	−209.209	−0.0352185
$329$	−3904.13	−0.654230
$330$	0	0
$331$	−9616.75	−1.59693	−0.798466	−	0.602040i	$-0.794355\pi$
$331$	−9616.75	−1.59693	−0.798466	+	0.602040i	$0.794355\pi$
$332$	6650.59	1.09939
$333$	0	0
$334$	−11585.4	−1.89797
$335$	0	0
$336$	0	0
$337$	2084.46	0.336936	0.168468	−	0.985707i	$-0.446118\pi$
$337$	2084.46	0.336936	0.168468	+	0.985707i	$0.446118\pi$
$338$	8537.38	1.37388
$339$	0	0
$340$	0	0
$341$	−459.068	−0.0729030
$342$	0	0
$343$	−5708.19	−0.898581
$344$	1157.98	0.181494
$345$	0	0
$346$	13523.2	2.10119
$347$	−6113.42	−0.945780	−0.472890	−	0.881121i	$-0.656789\pi$
$347$	−6113.42	−0.945780	−0.472890	+	0.881121i	$0.656789\pi$
$348$	0	0
$349$	2189.06	0.335753	0.167877	−	0.985808i	$-0.446309\pi$
$349$	2189.06	0.335753	0.167877	+	0.985808i	$0.446309\pi$
$350$	0	0
$351$	0	0
$352$	435.971	0.0660151
$353$	−2500.20	−0.376976	−0.188488	−	0.982076i	$-0.560359\pi$
$353$	−2500.20	−0.376976	−0.188488	+	0.982076i	$0.560359\pi$
$354$	0	0
$355$	0	0
$356$	−13793.3	−2.05349
$357$	0	0
$358$	5223.33	0.771122
$359$	−841.607	−0.123728	−0.0618639	−	0.998085i	$-0.519704\pi$
$359$	−841.607	−0.123728	−0.0618639	+	0.998085i	$0.519704\pi$
$360$	0	0
$361$	−4294.40	−0.626096
$362$	−4447.14	−0.645681
$363$	0	0
$364$	−1991.94	−0.286829
$365$	0	0
$366$	0	0
$367$	5614.42	0.798557	0.399278	−	0.916830i	$-0.369261\pi$
$367$	5614.42	0.798557	0.399278	+	0.916830i	$0.369261\pi$
$368$	−11898.4	−1.68546
$369$	0	0
$370$	0	0
$371$	−9013.78	−1.26138
$372$	0	0
$373$	−1319.17	−0.183121	−0.0915607	−	0.995799i	$-0.529186\pi$
$373$	−1319.17	−0.183121	−0.0915607	+	0.995799i	$0.529186\pi$
$374$	−62.0989	−0.00858572
$375$	0	0
$376$	764.678	0.104881
$377$	−854.634	−0.116753
$378$	0	0
$379$	−5002.07	−0.677939	−0.338970	−	0.940797i	$-0.610078\pi$
$379$	−5002.07	−0.677939	−0.338970	+	0.940797i	$0.610078\pi$
$380$	0	0
$381$	0	0
$382$	1180.50	0.158114
$383$	2242.45	0.299174	0.149587	−	0.988749i	$-0.452206\pi$
$383$	2242.45	0.299174	0.149587	+	0.988749i	$0.452206\pi$
$384$	0	0
$385$	0	0
$386$	−19148.5	−2.52496
$387$	0	0
$388$	10040.8	1.31378
$389$	−11393.8	−1.48506	−0.742529	−	0.669814i	$-0.766374\pi$
$389$	−11393.8	−1.48506	−0.742529	+	0.669814i	$0.766374\pi$
$390$	0	0
$391$	1927.59	0.249316
$392$	−227.138	−0.0292659
$393$	0	0
$394$	−6978.13	−0.892267
$395$	0	0
$396$	0	0
$397$	14926.0	1.88694	0.943472	−	0.331454i	$-0.107539\pi$
$397$	14926.0	1.88694	0.943472	+	0.331454i	$0.107539\pi$
$398$	13380.4	1.68517
$399$	0	0
$400$	0	0
$401$	−7165.63	−0.892356	−0.446178	−	0.894944i	$-0.647215\pi$
$401$	−7165.63	−0.892356	−0.446178	+	0.894944i	$0.647215\pi$
$402$	0	0
$403$	3039.63	0.375719
$404$	15742.7	1.93869
$405$	0	0
$406$	−6340.46	−0.775053
$407$	230.397	0.0280599
$408$	0	0
$409$	−11324.1	−1.36905	−0.684524	−	0.728990i	$-0.739990\pi$
$409$	−11324.1	−1.36905	−0.684524	+	0.728990i	$0.739990\pi$
$410$	0	0
$411$	0	0
$412$	−9458.16	−1.13100
$413$	−9633.93	−1.14783
$414$	0	0
$415$	0	0
$416$	−2886.69	−0.340220
$417$	0	0
$418$	349.936	0.0409471
$419$	−12699.8	−1.48073	−0.740365	−	0.672205i	$-0.765347\pi$
$419$	−12699.8	−1.48073	−0.740365	+	0.672205i	$0.765347\pi$
$420$	0	0
$421$	−17162.8	−1.98685	−0.993424	−	0.114494i	$-0.963475\pi$
$421$	−17162.8	−1.98685	−0.993424	+	0.114494i	$0.963475\pi$
$422$	4440.79	0.512261
$423$	0	0
$424$	1765.48	0.202215
$425$	0	0
$426$	0	0
$427$	13532.6	1.53369
$428$	13160.5	1.48630
$429$	0	0
$430$	0	0
$431$	−11130.9	−1.24399	−0.621994	−	0.783022i	$-0.713677\pi$
$431$	−11130.9	−1.24399	−0.621994	+	0.783022i	$0.713677\pi$
$432$	0	0
$433$	−7675.55	−0.851878	−0.425939	−	0.904752i	$-0.640056\pi$
$433$	−7675.55	−0.851878	−0.425939	+	0.904752i	$0.640056\pi$
$434$	22550.7	2.49417
$435$	0	0
$436$	8226.35	0.903603
$437$	−10862.2	−1.18904
$438$	0	0
$439$	3363.80	0.365707	0.182853	−	0.983140i	$-0.441467\pi$
$439$	3363.80	0.365707	0.182853	+	0.983140i	$0.441467\pi$
$440$	0	0
$441$	0	0
$442$	411.175	0.0442480
$443$	−7894.06	−0.846633	−0.423316	−	0.905982i	$-0.639134\pi$
$443$	−7894.06	−0.846633	−0.423316	+	0.905982i	$0.639134\pi$
$444$	0	0
$445$	0	0
$446$	−21176.6	−2.24830
$447$	0	0
$448$	−12530.3	−1.32143
$449$	2244.52	0.235914	0.117957	−	0.993019i	$-0.462366\pi$
$449$	2244.52	0.235914	0.117957	+	0.993019i	$0.462366\pi$
$450$	0	0
$451$	89.5283	0.00934750
$452$	8838.06	0.919707
$453$	0	0
$454$	−11980.1	−1.23844
$455$	0	0
$456$	0	0
$457$	−14749.3	−1.50972	−0.754862	−	0.655884i	$-0.772296\pi$
$457$	−14749.3	−1.50972	−0.754862	+	0.655884i	$0.772296\pi$
$458$	−7105.79	−0.724960
$459$	0	0
$460$	0	0
$461$	12479.5	1.26080	0.630400	−	0.776271i	$-0.282891\pi$
$461$	12479.5	1.26080	0.630400	+	0.776271i	$0.282891\pi$
$462$	0	0
$463$	9679.12	0.971549	0.485774	−	0.874084i	$-0.338538\pi$
$463$	9679.12	0.971549	0.485774	+	0.874084i	$0.338538\pi$
$464$	−4266.34	−0.426854
$465$	0	0
$466$	−4477.53	−0.445102
$467$	−6714.33	−0.665315	−0.332658	−	0.943048i	$-0.607945\pi$
$467$	−6714.33	−0.665315	−0.332658	+	0.943048i	$0.607945\pi$
$468$	0	0
$469$	−17912.3	−1.76357
$470$	0	0
$471$	0	0
$472$	1886.94	0.184011
$473$	−495.542	−0.0481713
$474$	0	0
$475$	0	0
$476$	1610.93	0.155120
$477$	0	0
$478$	−1314.85	−0.125816
$479$	2127.48	0.202937	0.101469	−	0.994839i	$-0.467646\pi$
$479$	2127.48	0.202937	0.101469	+	0.994839i	$0.467646\pi$
$480$	0	0
$481$	−1525.53	−0.144612
$482$	−15011.1	−1.41854
$483$	0	0
$484$	−11890.6	−1.11669
$485$	0	0
$486$	0	0
$487$	6549.04	0.609375	0.304687	−	0.952452i	$-0.401448\pi$
$487$	6549.04	0.609375	0.304687	+	0.952452i	$0.401448\pi$
$488$	−2650.54	−0.245870
$489$	0	0
$490$	0	0
$491$	12933.7	1.18878	0.594389	−	0.804177i	$-0.297394\pi$
$491$	12933.7	1.18878	0.594389	+	0.804177i	$0.297394\pi$
$492$	0	0
$493$	691.166	0.0631411
$494$	−2317.03	−0.211028
$495$	0	0
$496$	15173.9	1.37364
$497$	−14455.5	−1.30466
$498$	0	0
$499$	5120.66	0.459383	0.229692	−	0.973263i	$-0.426228\pi$
$499$	5120.66	0.459383	0.229692	+	0.973263i	$0.426228\pi$
$500$	0	0
$501$	0	0
$502$	−2358.72	−0.209711
$503$	−10809.6	−0.958204	−0.479102	−	0.877759i	$-0.659038\pi$
$503$	−10809.6	−0.958204	−0.479102	+	0.877759i	$0.659038\pi$
$504$	0	0
$505$	0	0
$506$	−1482.13	−0.130215
$507$	0	0
$508$	−13250.6	−1.15728
$509$	14756.9	1.28505	0.642524	−	0.766266i	$-0.277887\pi$
$509$	14756.9	1.28505	0.642524	+	0.766266i	$0.277887\pi$
$510$	0	0
$511$	−18324.5	−1.58635
$512$	−16150.8	−1.39409
$513$	0	0
$514$	15142.2	1.29940
$515$	0	0
$516$	0	0
$517$	−327.234	−0.0278370
$518$	−11317.8	−0.959989
$519$	0	0
$520$	0	0
$521$	−1742.68	−0.146542	−0.0732709	−	0.997312i	$-0.523344\pi$
$521$	−1742.68	−0.146542	−0.0732709	+	0.997312i	$0.523344\pi$
$522$	0	0
$523$	−10304.8	−0.861565	−0.430782	−	0.902456i	$-0.641762\pi$
$523$	−10304.8	−0.861565	−0.430782	+	0.902456i	$0.641762\pi$
$524$	4535.29	0.378102
$525$	0	0
$526$	−8241.73	−0.683187
$527$	−2458.23	−0.203192
$528$	0	0
$529$	33839.5	2.78125
$530$	0	0
$531$	0	0
$532$	−9077.83	−0.739800
$533$	−592.794	−0.0481740
$534$	0	0
$535$	0	0
$536$	3508.37	0.282721
$537$	0	0
$538$	−83.0175	−0.00665267
$539$	97.2007	0.00776759
$540$	0	0
$541$	3966.86	0.315247	0.157623	−	0.987499i	$-0.449617\pi$
$541$	3966.86	0.315247	0.157623	+	0.987499i	$0.449617\pi$
$542$	−19487.9	−1.54442
$543$	0	0
$544$	2334.55	0.183994
$545$	0	0
$546$	0	0
$547$	5808.72	0.454045	0.227023	−	0.973889i	$-0.427101\pi$
$547$	5808.72	0.454045	0.227023	+	0.973889i	$0.427101\pi$
$548$	8735.16	0.680926
$549$	0	0
$550$	0	0
$551$	−3894.81	−0.301133
$552$	0	0
$553$	1194.65	0.0918658
$554$	20706.6	1.58798
$555$	0	0
$556$	−16909.7	−1.28980
$557$	3563.91	0.271109	0.135554	−	0.990770i	$-0.456718\pi$
$557$	3563.91	0.271109	0.135554	+	0.990770i	$0.456718\pi$
$558$	0	0
$559$	3281.13	0.248259
$560$	0	0
$561$	0	0
$562$	23914.8	1.79499
$563$	−511.544	−0.0382931	−0.0191466	−	0.999817i	$-0.506095\pi$
$563$	−511.544	−0.0382931	−0.0191466	+	0.999817i	$0.506095\pi$
$564$	0	0
$565$	0	0
$566$	877.089	0.0651357
$567$	0	0
$568$	2831.31	0.209153
$569$	−8116.54	−0.598002	−0.299001	−	0.954253i	$-0.596653\pi$
$569$	−8116.54	−0.598002	−0.299001	+	0.954253i	$0.596653\pi$
$570$	0	0
$571$	15078.1	1.10508	0.552539	−	0.833487i	$-0.313659\pi$
$571$	15078.1	1.10508	0.552539	+	0.833487i	$0.313659\pi$
$572$	−166.959	−0.0122044
$573$	0	0
$574$	−4397.88	−0.319798
$575$	0	0
$576$	0	0
$577$	−27101.5	−1.95537	−0.977685	−	0.210077i	$-0.932628\pi$
$577$	−27101.5	−1.95537	−0.977685	+	0.210077i	$0.932628\pi$
$578$	19896.0	1.43177
$579$	0	0
$580$	0	0
$581$	14874.5	1.06213
$582$	0	0
$583$	−755.511	−0.0536708
$584$	3589.10	0.254312
$585$	0	0
$586$	14794.0	1.04289
$587$	102.082	0.00717782	0.00358891	−	0.999994i	$-0.498858\pi$
$587$	102.082	0.00717782	0.00358891	+	0.999994i	$0.498858\pi$
$588$	0	0
$589$	13852.4	0.969067
$590$	0	0
$591$	0	0
$592$	−7615.47	−0.528706
$593$	16467.2	1.14035	0.570174	−	0.821524i	$-0.306876\pi$
$593$	16467.2	1.14035	0.570174	+	0.821524i	$0.306876\pi$
$594$	0	0
$595$	0	0
$596$	−23118.0	−1.58884
$597$	0	0
$598$	9813.65	0.671087
$599$	−23216.2	−1.58362	−0.791809	−	0.610769i	$-0.790860\pi$
$599$	−23216.2	−1.58362	−0.791809	+	0.610769i	$0.790860\pi$
$600$	0	0
$601$	−1678.29	−0.113909	−0.0569543	−	0.998377i	$-0.518139\pi$
$601$	−1678.29	−0.113909	−0.0569543	+	0.998377i	$0.518139\pi$
$602$	24342.4	1.64804
$603$	0	0
$604$	22720.8	1.53062
$605$	0	0
$606$	0	0
$607$	−19307.3	−1.29103	−0.645517	−	0.763746i	$-0.723358\pi$
$607$	−19307.3	−1.29103	−0.645517	+	0.763746i	$0.723358\pi$
$608$	−13155.5	−0.877508
$609$	0	0
$610$	0	0
$611$	2166.71	0.143463
$612$	0	0
$613$	−13659.9	−0.900032	−0.450016	−	0.893020i	$-0.648582\pi$
$613$	−13659.9	−0.900032	−0.450016	+	0.893020i	$0.648582\pi$
$614$	9894.93	0.650369
$615$	0	0
$616$	−131.786	−0.00861984
$617$	−14603.7	−0.952872	−0.476436	−	0.879209i	$-0.658072\pi$
$617$	−14603.7	−0.952872	−0.476436	+	0.879209i	$0.658072\pi$
$618$	0	0
$619$	−26388.3	−1.71346	−0.856732	−	0.515762i	$-0.827509\pi$
$619$	−26388.3	−1.71346	−0.856732	+	0.515762i	$0.827509\pi$
$620$	0	0
$621$	0	0
$622$	−6621.35	−0.426836
$623$	−30849.7	−1.98390
$624$	0	0
$625$	0	0
$626$	11448.5	0.730946
$627$	0	0
$628$	−940.326	−0.0597501
$629$	1233.74	0.0782072
$630$	0	0
$631$	14241.1	0.898459	0.449229	−	0.893416i	$-0.351699\pi$
$631$	14241.1	0.898459	0.449229	+	0.893416i	$0.351699\pi$
$632$	−233.989	−0.0147272
$633$	0	0
$634$	−22792.4	−1.42776
$635$	0	0
$636$	0	0
$637$	−643.595	−0.0400316
$638$	−531.440	−0.0329779
$639$	0	0
$640$	0	0
$641$	−18960.5	−1.16832	−0.584160	−	0.811638i	$-0.698576\pi$
$641$	−18960.5	−1.16832	−0.584160	+	0.811638i	$0.698576\pi$
$642$	0	0
$643$	−11054.3	−0.677974	−0.338987	−	0.940791i	$-0.610084\pi$
$643$	−11054.3	−0.677974	−0.338987	+	0.940791i	$0.610084\pi$
$644$	38448.7	2.35262
$645$	0	0
$646$	1873.84	0.114126
$647$	−19310.1	−1.17335	−0.586677	−	0.809821i	$-0.699564\pi$
$647$	−19310.1	−1.17335	−0.586677	+	0.809821i	$0.699564\pi$
$648$	0	0
$649$	−807.489	−0.0488393
$650$	0	0
$651$	0	0
$652$	−19954.0	−1.19856
$653$	−6934.26	−0.415557	−0.207778	−	0.978176i	$-0.566623\pi$
$653$	−6934.26	−0.415557	−0.207778	+	0.978176i	$0.566623\pi$
$654$	0	0
$655$	0	0
$656$	−2959.23	−0.176126
$657$	0	0
$658$	16074.6	0.952363
$659$	−15878.7	−0.938613	−0.469307	−	0.883035i	$-0.655496\pi$
$659$	−15878.7	−0.938613	−0.469307	+	0.883035i	$0.655496\pi$
$660$	0	0
$661$	−9418.32	−0.554206	−0.277103	−	0.960840i	$-0.589374\pi$
$661$	−9418.32	−0.554206	−0.277103	+	0.960840i	$0.589374\pi$
$662$	39595.4	2.32465
$663$	0	0
$664$	−2913.38	−0.170273
$665$	0	0
$666$	0	0
$667$	16496.3	0.957628
$668$	25190.5	1.45906
$669$	0	0
$670$	0	0
$671$	1134.26	0.0652574
$672$	0	0
$673$	18392.3	1.05345	0.526726	−	0.850035i	$-0.323419\pi$
$673$	18392.3	1.05345	0.526726	+	0.850035i	$0.323419\pi$
$674$	−8582.41	−0.490478
$675$	0	0
$676$	−18563.2	−1.05617
$677$	−10570.7	−0.600095	−0.300048	−	0.953924i	$-0.597003\pi$
$677$	−10570.7	−0.600095	−0.300048	+	0.953924i	$0.597003\pi$
$678$	0	0
$679$	22457.0	1.26925
$680$	0	0
$681$	0	0
$682$	1890.14	0.106125
$683$	23975.0	1.34316	0.671579	−	0.740933i	$-0.265616\pi$
$683$	23975.0	1.34316	0.671579	+	0.740933i	$0.265616\pi$
$684$	0	0
$685$	0	0
$686$	23502.6	1.30806
$687$	0	0
$688$	16379.4	0.907645
$689$	5002.46	0.276602
$690$	0	0
$691$	19641.0	1.08130	0.540650	−	0.841248i	$-0.318178\pi$
$691$	19641.0	1.08130	0.540650	+	0.841248i	$0.318178\pi$
$692$	−29404.0	−1.61528
$693$	0	0
$694$	25171.1	1.37677
$695$	0	0
$696$	0	0
$697$	479.408	0.0260529
$698$	−9013.12	−0.488756
$699$	0	0
$700$	0	0
$701$	36098.2	1.94495	0.972475	−	0.233007i	$-0.0748566\pi$
$701$	36098.2	1.94495	0.972475	+	0.233007i	$0.0748566\pi$
$702$	0	0
$703$	−6952.28	−0.372987
$704$	−1050.26	−0.0562258
$705$	0	0
$706$	10294.2	0.548764
$707$	35209.8	1.87298
$708$	0	0
$709$	35234.1	1.86635	0.933177	−	0.359418i	$-0.117025\pi$
$709$	35234.1	1.86635	0.933177	+	0.359418i	$0.117025\pi$
$710$	0	0
$711$	0	0
$712$	6042.34	0.318043
$713$	−58671.3	−3.08171
$714$	0	0
$715$	0	0
$716$	−11357.3	−0.592797
$717$	0	0
$718$	3465.18	0.180111
$719$	−22089.7	−1.14577	−0.572885	−	0.819636i	$-0.694176\pi$
$719$	−22089.7	−1.14577	−0.572885	+	0.819636i	$0.694176\pi$
$720$	0	0
$721$	−21153.9	−1.09267
$722$	17681.5	0.911409
$723$	0	0
$724$	9669.60	0.496364
$725$	0	0
$726$	0	0
$727$	19435.2	0.991488	0.495744	−	0.868469i	$-0.334895\pi$
$727$	19435.2	0.991488	0.495744	+	0.868469i	$0.334895\pi$
$728$	872.596	0.0444238
$729$	0	0
$730$	0	0
$731$	−2653.54	−0.134261
$732$	0	0
$733$	−14974.2	−0.754550	−0.377275	−	0.926101i	$-0.623139\pi$
$733$	−14974.2	−0.754550	−0.377275	+	0.926101i	$0.623139\pi$
$734$	−23116.5	−1.16246
$735$	0	0
$736$	55719.3	2.79055
$737$	−1501.36	−0.0750384
$738$	0	0
$739$	−30663.4	−1.52635	−0.763174	−	0.646193i	$-0.776360\pi$
$739$	−30663.4	−1.52635	−0.763174	+	0.646193i	$0.776360\pi$
$740$	0	0
$741$	0	0
$742$	37112.8	1.83619
$743$	−3414.68	−0.168604	−0.0843018	−	0.996440i	$-0.526866\pi$
$743$	−3414.68	−0.168604	−0.0843018	+	0.996440i	$0.526866\pi$
$744$	0	0
$745$	0	0
$746$	5431.49	0.266570
$747$	0	0
$748$	135.024	0.00660023
$749$	29434.3	1.43592
$750$	0	0
$751$	−746.259	−0.0362601	−0.0181301	−	0.999836i	$-0.505771\pi$
$751$	−746.259	−0.0362601	−0.0181301	+	0.999836i	$0.505771\pi$
$752$	10816.2	0.524505
$753$	0	0
$754$	3518.82	0.169957
$755$	0	0
$756$	0	0
$757$	22929.5	1.10091	0.550454	−	0.834865i	$-0.314454\pi$
$757$	22929.5	1.10091	0.550454	+	0.834865i	$0.314454\pi$
$758$	20595.2	0.986876
$759$	0	0
$760$	0	0
$761$	−21176.6	−1.00874	−0.504370	−	0.863487i	$-0.668275\pi$
$761$	−21176.6	−1.00874	−0.504370	+	0.863487i	$0.668275\pi$
$762$	0	0
$763$	18398.8	0.872978
$764$	−2566.81	−0.121550
$765$	0	0
$766$	−9232.92	−0.435508
$767$	5346.63	0.251702
$768$	0	0
$769$	2774.18	0.130091	0.0650453	−	0.997882i	$-0.479281\pi$
$769$	2774.18	0.130091	0.0650453	+	0.997882i	$0.479281\pi$
$770$	0	0
$771$	0	0
$772$	41635.4	1.94105
$773$	−28891.7	−1.34433	−0.672163	−	0.740403i	$-0.734635\pi$
$773$	−28891.7	−1.34433	−0.672163	+	0.740403i	$0.734635\pi$
$774$	0	0
$775$	0	0
$776$	−4398.52	−0.203476
$777$	0	0
$778$	46912.1	2.16180
$779$	−2701.53	−0.124252
$780$	0	0
$781$	−1211.62	−0.0555124
$782$	−7936.56	−0.362930
$783$	0	0
$784$	−3212.83	−0.146357
$785$	0	0
$786$	0	0
$787$	−8808.75	−0.398981	−0.199490	−	0.979900i	$-0.563929\pi$
$787$	−8808.75	−0.398981	−0.199490	+	0.979900i	$0.563929\pi$
$788$	15172.8	0.685926
$789$	0	0
$790$	0	0
$791$	19767.0	0.888537
$792$	0	0
$793$	−7510.29	−0.336316
$794$	−61455.6	−2.74682
$795$	0	0
$796$	−29093.5	−1.29547
$797$	780.503	0.0346886	0.0173443	−	0.999850i	$-0.494479\pi$
$797$	780.503	0.0346886	0.0173443	+	0.999850i	$0.494479\pi$
$798$	0	0
$799$	−1752.28	−0.0775859
$800$	0	0
$801$	0	0
$802$	29503.4	1.29900
$803$	−1535.91	−0.0674981
$804$	0	0
$805$	0	0
$806$	−12515.2	−0.546934
$807$	0	0
$808$	−6896.32	−0.300262
$809$	15234.0	0.662051	0.331025	−	0.943622i	$-0.392605\pi$
$809$	15234.0	0.662051	0.331025	+	0.943622i	$0.392605\pi$
$810$	0	0
$811$	10825.2	0.468711	0.234356	−	0.972151i	$-0.424702\pi$
$811$	10825.2	0.468711	0.234356	+	0.972151i	$0.424702\pi$
$812$	13786.3	0.595818
$813$	0	0
$814$	−948.625	−0.0408468
$815$	0	0
$816$	0	0
$817$	14953.0	0.640319
$818$	46625.2	1.99292
$819$	0	0
$820$	0	0
$821$	39971.2	1.69915	0.849577	−	0.527464i	$-0.176857\pi$
$821$	39971.2	1.69915	0.849577	+	0.527464i	$0.176857\pi$
$822$	0	0
$823$	−28620.6	−1.21221	−0.606107	−	0.795383i	$-0.707270\pi$
$823$	−28620.6	−1.21221	−0.606107	+	0.795383i	$0.707270\pi$
$824$	4143.28	0.175168
$825$	0	0
$826$	39666.2	1.67090
$827$	−2265.62	−0.0952639	−0.0476320	−	0.998865i	$-0.515167\pi$
$827$	−2265.62	−0.0952639	−0.0476320	+	0.998865i	$0.515167\pi$
$828$	0	0
$829$	−14651.6	−0.613839	−0.306919	−	0.951736i	$-0.599298\pi$
$829$	−14651.6	−0.613839	−0.306919	+	0.951736i	$0.599298\pi$
$830$	0	0
$831$	0	0
$832$	6954.05	0.289770
$833$	520.493	0.0216495
$834$	0	0
$835$	0	0
$836$	−760.879	−0.0314779
$837$	0	0
$838$	52289.4	2.15550
$839$	−22466.4	−0.924466	−0.462233	−	0.886758i	$-0.652952\pi$
$839$	−22466.4	−0.924466	−0.462233	+	0.886758i	$0.652952\pi$
$840$	0	0
$841$	−18474.0	−0.757474
$842$	70665.0	2.89225
$843$	0	0
$844$	−9655.79	−0.393799
$845$	0	0
$846$	0	0
$847$	−26594.1	−1.07885
$848$	24972.4	1.01127
$849$	0	0
$850$	0	0
$851$	29446.0	1.18613
$852$	0	0
$853$	−44471.0	−1.78506	−0.892531	−	0.450986i	$-0.851073\pi$
$853$	−44471.0	−1.78506	−0.892531	+	0.450986i	$0.851073\pi$
$854$	−55718.2	−2.23260
$855$	0	0
$856$	−5765.12	−0.230196
$857$	31480.6	1.25479	0.627396	−	0.778700i	$-0.284121\pi$
$857$	31480.6	1.25479	0.627396	+	0.778700i	$0.284121\pi$
$858$	0	0
$859$	42983.1	1.70729	0.853646	−	0.520853i	$-0.174386\pi$
$859$	42983.1	1.70729	0.853646	+	0.520853i	$0.174386\pi$
$860$	0	0
$861$	0	0
$862$	45829.9	1.81087
$863$	25203.4	0.994131	0.497065	−	0.867713i	$-0.334411\pi$
$863$	25203.4	0.994131	0.497065	+	0.867713i	$0.334411\pi$
$864$	0	0
$865$	0	0
$866$	31602.8	1.24008
$867$	0	0
$868$	−49033.0	−1.91738
$869$	100.132	0.00390882
$870$	0	0
$871$	9940.95	0.386723
$872$	−3603.67	−0.139949
$873$	0	0
$874$	44723.6	1.73089
$875$	0	0
$876$	0	0
$877$	−3472.71	−0.133712	−0.0668558	−	0.997763i	$-0.521297\pi$
$877$	−3472.71	−0.133712	−0.0668558	+	0.997763i	$0.521297\pi$
$878$	−13849.9	−0.532359
$879$	0	0
$880$	0	0
$881$	26066.1	0.996811	0.498405	−	0.866944i	$-0.333919\pi$
$881$	26066.1	0.996811	0.498405	+	0.866944i	$0.333919\pi$
$882$	0	0
$883$	3032.93	0.115590	0.0577952	−	0.998328i	$-0.481593\pi$
$883$	3032.93	0.115590	0.0577952	+	0.998328i	$0.481593\pi$
$884$	−894.035	−0.0340154
$885$	0	0
$886$	32502.5	1.23244
$887$	42339.4	1.60273	0.801363	−	0.598179i	$-0.204109\pi$
$887$	42339.4	1.60273	0.801363	+	0.598179i	$0.204109\pi$
$888$	0	0
$889$	−29635.9	−1.11806
$890$	0	0
$891$	0	0
$892$	46045.2	1.72837
$893$	9874.32	0.370024
$894$	0	0
$895$	0	0
$896$	9979.99	0.372107
$897$	0	0
$898$	−9241.44	−0.343419
$899$	−21037.4	−0.780464
$900$	0	0
$901$	−4045.63	−0.149589
$902$	−368.619	−0.0136072
$903$	0	0
$904$	−3871.64	−0.142443
$905$	0	0
$906$	0	0
$907$	20209.8	0.739861	0.369931	−	0.929059i	$-0.379381\pi$
$907$	20209.8	0.739861	0.369931	+	0.929059i	$0.379381\pi$
$908$	26048.8	0.952047
$909$	0	0
$910$	0	0
$911$	−11254.6	−0.409310	−0.204655	−	0.978834i	$-0.565607\pi$
$911$	−11254.6	−0.409310	−0.204655	+	0.978834i	$0.565607\pi$
$912$	0	0
$913$	1246.74	0.0451929
$914$	60727.9	2.19770
$915$	0	0
$916$	15450.4	0.557309
$917$	10143.5	0.365287
$918$	0	0
$919$	−17825.9	−0.639851	−0.319925	−	0.947443i	$-0.603658\pi$
$919$	−17825.9	−0.639851	−0.319925	+	0.947443i	$0.603658\pi$
$920$	0	0
$921$	0	0
$922$	−51382.4	−1.83535
$923$	8022.49	0.286093
$924$	0	0
$925$	0	0
$926$	−39852.3	−1.41428
$927$	0	0
$928$	19978.9	0.706725
$929$	2084.24	0.0736077	0.0368039	−	0.999323i	$-0.488282\pi$
$929$	2084.24	0.0736077	0.0368039	+	0.999323i	$0.488282\pi$
$930$	0	0
$931$	−2933.04	−0.103251
$932$	9735.67	0.342170
$933$	0	0
$934$	27645.2	0.968499
$935$	0	0
$936$	0	0
$937$	5419.51	0.188952	0.0944758	−	0.995527i	$-0.469883\pi$
$937$	5419.51	0.188952	0.0944758	+	0.995527i	$0.469883\pi$
$938$	73751.0	2.56722
$939$	0	0
$940$	0	0
$941$	46413.8	1.60792	0.803958	−	0.594687i	$-0.202724\pi$
$941$	46413.8	1.60792	0.803958	+	0.594687i	$0.202724\pi$
$942$	0	0
$943$	11442.2	0.395131
$944$	26690.4	0.920232
$945$	0	0
$946$	2040.31	0.0701230
$947$	18783.1	0.644527	0.322264	−	0.946650i	$-0.395556\pi$
$947$	18783.1	0.644527	0.322264	+	0.946650i	$0.395556\pi$
$948$	0	0
$949$	10169.7	0.347863
$950$	0	0
$951$	0	0
$952$	−705.692	−0.0240248
$953$	−11796.7	−0.400979	−0.200489	−	0.979696i	$-0.564253\pi$
$953$	−11796.7	−0.400979	−0.200489	+	0.979696i	$0.564253\pi$
$954$	0	0
$955$	0	0
$956$	2858.94	0.0967203
$957$	0	0
$958$	−8759.55	−0.295416
$959$	19536.8	0.657849
$960$	0	0
$961$	45031.6	1.51158
$962$	6281.13	0.210511
$963$	0	0
$964$	32639.3	1.09050
$965$	0	0
$966$	0	0
$967$	10161.7	0.337932	0.168966	−	0.985622i	$-0.445957\pi$
$967$	10161.7	0.337932	0.168966	+	0.985622i	$0.445957\pi$
$968$	5208.83	0.172953
$969$	0	0
$970$	0	0
$971$	−36614.9	−1.21012	−0.605060	−	0.796180i	$-0.706851\pi$
$971$	−36614.9	−1.21012	−0.605060	+	0.796180i	$0.706851\pi$
$972$	0	0
$973$	−37819.8	−1.24609
$974$	−26964.6	−0.887067
$975$	0	0
$976$	−37491.5	−1.22958
$977$	−50990.9	−1.66975	−0.834874	−	0.550441i	$-0.814459\pi$
$977$	−50990.9	−1.66975	−0.834874	+	0.550441i	$0.814459\pi$
$978$	0	0
$979$	−2585.74	−0.0844132
$980$	0	0
$981$	0	0
$982$	−53252.5	−1.73051
$983$	−21783.8	−0.706812	−0.353406	−	0.935470i	$-0.614977\pi$
$983$	−21783.8	−0.706812	−0.353406	+	0.935470i	$0.614977\pi$
$984$	0	0
$985$	0	0
$986$	−2845.77	−0.0919145
$987$	0	0
$988$	5038.00	0.162227
$989$	−63332.8	−2.03626
$990$	0	0
$991$	−50955.6	−1.63336	−0.816679	−	0.577092i	$-0.804187\pi$
$991$	−50955.6	−1.63336	−0.816679	+	0.577092i	$0.804187\pi$
$992$	−71057.9	−2.27429
$993$	0	0
$994$	59518.2	1.89920
$995$	0	0
$996$	0	0
$997$	19162.8	0.608719	0.304360	−	0.952557i	$-0.401558\pi$
$997$	19162.8	0.608719	0.304360	+	0.952557i	$0.401558\pi$
$998$	−21083.5	−0.668724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.z.1.1		6
3.2	odd	2		2025.4.a.y.1.6		6
5.4	even	2		405.4.a.k.1.6	✓	6
15.14	odd	2		405.4.a.l.1.1	yes	6
45.4	even	6		405.4.e.x.136.1		12
45.14	odd	6		405.4.e.w.136.6		12
45.29	odd	6		405.4.e.w.271.6		12
45.34	even	6		405.4.e.x.271.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.6	✓	6	5.4	even	2
405.4.a.l.1.1	yes	6	15.14	odd	2
405.4.e.w.136.6		12	45.14	odd	6
405.4.e.w.271.6		12	45.29	odd	6
405.4.e.x.136.1		12	45.4	even	6
405.4.e.x.271.1		12	45.34	even	6
2025.4.a.y.1.6		6	3.2	odd	2
2025.4.a.z.1.1		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q-4.11734 q^{2} +8.95250 q^{4} +20.0229 q^{7} -3.92177 q^{8} +O(q^{10})\)	\(q-4.11734 q^{2} +8.95250 q^{4} +20.0229 q^{7} -3.92177 q^{8} +1.67827 q^{11} -11.1123 q^{13} -82.4412 q^{14} -55.4727 q^{16} +8.98682 q^{17} -50.6419 q^{19} -6.91000 q^{22} +214.491 q^{23} +45.7531 q^{26} +179.255 q^{28} +76.9088 q^{29} -273.537 q^{31} +259.774 q^{32} -37.0018 q^{34} +137.283 q^{37} +208.510 q^{38} +53.3457 q^{41} -295.270 q^{43} +15.0247 q^{44} -883.134 q^{46} -194.983 q^{47} +57.9173 q^{49} -99.4829 q^{52} -450.173 q^{53} -78.5252 q^{56} -316.660 q^{58} -481.145 q^{59} +675.854 q^{61} +1126.25 q^{62} -625.798 q^{64} -894.589 q^{67} +80.4545 q^{68} -721.947 q^{71} -915.175 q^{73} -565.241 q^{74} -453.372 q^{76} +33.6038 q^{77} +59.6642 q^{79} -219.643 q^{82} +742.875 q^{83} +1215.73 q^{86} -6.58177 q^{88} -1540.72 q^{89} -222.501 q^{91} +1920.23 q^{92} +802.812 q^{94} +1121.57 q^{97} -238.465 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8}+O(q^{10}) \) 6 * q + 4 * q^2 + 34 * q^4 - 40 * q^7 + 66 * q^8	\( 6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8} - 88 q^{11} - 20 q^{13} - 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 74 q^{22} + 210 q^{23} - 4 q^{26} - 352 q^{28} - 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 204 q^{37} - 20 q^{38} - 344 q^{41} - 512 q^{43} - 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 468 q^{52} + 850 q^{53} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} - 88 q^{67} + 236 q^{68} - 1364 q^{71} - 836 q^{73} - 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} - 1742 q^{82} + 2148 q^{83} - 2872 q^{86} - 1296 q^{88} - 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} + 612 q^{97} + 1982 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 + 34 * q^4 - 40 * q^7 + 66 * q^8 - 88 * q^11 - 20 * q^13 - 180 * q^14 + 58 * q^16 + 124 * q^17 - 46 * q^19 + 74 * q^22 + 210 * q^23 - 4 * q^26 - 352 * q^28 - 296 * q^29 - 104 * q^31 + 722 * q^32 - 428 * q^34 + 204 * q^37 - 20 * q^38 - 344 * q^41 - 512 * q^43 - 716 * q^44 - 186 * q^46 + 238 * q^47 + 68 * q^49 + 468 * q^52 + 850 * q^53 - 2316 * q^56 - 890 * q^58 - 1840 * q^59 - 364 * q^61 + 1038 * q^62 - 990 * q^64 - 88 * q^67 + 236 * q^68 - 1364 * q^71 - 836 * q^73 - 1316 * q^74 - 2106 * q^76 + 840 * q^77 - 680 * q^79 - 1742 * q^82 + 2148 * q^83 - 2872 * q^86 - 1296 * q^88 - 3000 * q^89 - 3058 * q^91 + 1002 * q^92 - 3662 * q^94 + 612 * q^97 + 1982 * q^98

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