LMFDB - Embedded newform 2025.4.a.be.1.11

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + \cdots + 73008 $ x^12 - 4x^11 - 64x^10 + 241x^9 + 1452x^8 - 5130x^7 - 13834x^6 + 45247x^5 + 52669x^4 - 144610x^3 - 96816x^2 + 143136*x + 73008
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-4.33545$ 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)

$f(q)$	$=$	$q+4.33545 q^{2} +10.7961 q^{4} -13.0015 q^{7} +12.1226 q^{8} +O(q^{10})$	$q+4.33545 q^{2} +10.7961 q^{4} -13.0015 q^{7} +12.1226 q^{8} +34.5521 q^{11} +7.55767 q^{13} -56.3674 q^{14} -33.8124 q^{16} -82.0901 q^{17} +146.406 q^{19} +149.799 q^{22} -187.890 q^{23} +32.7659 q^{26} -140.366 q^{28} -43.7178 q^{29} -58.6782 q^{31} -243.572 q^{32} -355.898 q^{34} -329.358 q^{37} +634.736 q^{38} +172.969 q^{41} +156.687 q^{43} +373.030 q^{44} -814.589 q^{46} -169.246 q^{47} -173.961 q^{49} +81.5937 q^{52} -609.837 q^{53} -157.612 q^{56} -189.536 q^{58} +594.430 q^{59} +325.077 q^{61} -254.396 q^{62} -785.498 q^{64} +688.382 q^{67} -886.257 q^{68} +515.170 q^{71} -1088.37 q^{73} -1427.92 q^{74} +1580.62 q^{76} -449.229 q^{77} -525.792 q^{79} +749.899 q^{82} -358.013 q^{83} +679.311 q^{86} +418.861 q^{88} -1517.67 q^{89} -98.2610 q^{91} -2028.49 q^{92} -733.758 q^{94} +395.361 q^{97} -754.200 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) $ 12 * q - 4 * q^2 + 48 * q^4 - 6 * q^7 - 45 * q^8	$ 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8} - 29 q^{11} - 24 q^{13} + 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} + 18 q^{22} - 318 q^{23} + 154 q^{26} - 96 q^{28} - 106 q^{29} + 60 q^{31} - 914 q^{32} - 108 q^{34} + 84 q^{37} - 640 q^{38} + 353 q^{41} + 426 q^{43} - 571 q^{44} + 270 q^{46} - 1210 q^{47} + 666 q^{49} + 75 q^{52} - 448 q^{53} + 570 q^{56} - 594 q^{58} - 482 q^{59} + 402 q^{61} - 2544 q^{62} + 975 q^{64} + 201 q^{67} - 3437 q^{68} + 944 q^{71} + 453 q^{73} - 10 q^{74} - 462 q^{76} - 2652 q^{77} + 258 q^{79} - 873 q^{82} - 3012 q^{83} - 1952 q^{86} + 216 q^{88} + 738 q^{89} - 618 q^{91} - 5232 q^{92} + 63 q^{94} + 318 q^{97} - 7511 q^{98}+O(q^{100}) $ 12 * q - 4 * q^2 + 48 * q^4 - 6 * q^7 - 45 * q^8 - 29 * q^11 - 24 * q^13 + 69 * q^14 + 192 * q^16 - 79 * q^17 - 75 * q^19 + 18 * q^22 - 318 * q^23 + 154 * q^26 - 96 * q^28 - 106 * q^29 + 60 * q^31 - 914 * q^32 - 108 * q^34 + 84 * q^37 - 640 * q^38 + 353 * q^41 + 426 * q^43 - 571 * q^44 + 270 * q^46 - 1210 * q^47 + 666 * q^49 + 75 * q^52 - 448 * q^53 + 570 * q^56 - 594 * q^58 - 482 * q^59 + 402 * q^61 - 2544 * q^62 + 975 * q^64 + 201 * q^67 - 3437 * q^68 + 944 * q^71 + 453 * q^73 - 10 * q^74 - 462 * q^76 - 2652 * q^77 + 258 * q^79 - 873 * q^82 - 3012 * q^83 - 1952 * q^86 + 216 * q^88 + 738 * q^89 - 618 * q^91 - 5232 * q^92 + 63 * q^94 + 318 * q^97 - 7511 * 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 $

$2$	4.33545	1.53281	0.766407	−	0.642355i	$-0.222043\pi$
$2$	4.33545	1.53281	0.766407	+	0.642355i	$0.222043\pi$
$3$	0	0
$3$	0	0
$4$	10.7961	1.34952
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−13.0015	−0.702015	−0.351007	−	0.936373i	$-0.614161\pi$
$7$	−13.0015	−0.702015	−0.351007	+	0.936373i	$0.614161\pi$
$8$	12.1226	0.535747
$9$	0	0
$10$	0	0
$11$	34.5521	0.947078	0.473539	−	0.880773i	$-0.342976\pi$
$11$	34.5521	0.947078	0.473539	+	0.880773i	$0.342976\pi$
$12$	0	0
$13$	7.55767	0.161240	0.0806200	−	0.996745i	$-0.474310\pi$
$13$	7.55767	0.161240	0.0806200	+	0.996745i	$0.474310\pi$
$14$	−56.3674	−1.07606
$15$	0	0
$16$	−33.8124	−0.528318
$17$	−82.0901	−1.17116	−0.585581	−	0.810614i	$-0.699134\pi$
$17$	−82.0901	−1.17116	−0.585581	+	0.810614i	$0.699134\pi$
$18$	0	0
$19$	146.406	1.76778	0.883890	−	0.467696i	$-0.154916\pi$
$19$	146.406	1.76778	0.883890	+	0.467696i	$0.154916\pi$
$20$	0	0
$21$	0	0
$22$	149.799	1.45169
$23$	−187.890	−1.70338	−0.851691	−	0.524044i	$-0.824423\pi$
$23$	−187.890	−1.70338	−0.851691	+	0.524044i	$0.824423\pi$
$24$	0	0
$25$	0	0
$26$	32.7659	0.247151
$27$	0	0
$28$	−140.366	−0.947382
$29$	−43.7178	−0.279938	−0.139969	−	0.990156i	$-0.544700\pi$
$29$	−43.7178	−0.279938	−0.139969	+	0.990156i	$0.544700\pi$
$30$	0	0
$31$	−58.6782	−0.339965	−0.169982	−	0.985447i	$-0.554371\pi$
$31$	−58.6782	−0.339965	−0.169982	+	0.985447i	$0.554371\pi$
$32$	−243.572	−1.34556
$33$	0	0
$34$	−355.898	−1.79517
$35$	0	0
$36$	0	0
$37$	−329.358	−1.46341	−0.731705	−	0.681621i	$-0.761275\pi$
$37$	−329.358	−1.46341	−0.731705	+	0.681621i	$0.761275\pi$
$38$	634.736	2.70968
$39$	0	0
$40$	0	0
$41$	172.969	0.658859	0.329429	−	0.944180i	$-0.393144\pi$
$41$	172.969	0.658859	0.329429	+	0.944180i	$0.393144\pi$
$42$	0	0
$43$	156.687	0.555689	0.277844	−	0.960626i	$-0.410380\pi$
$43$	156.687	0.555689	0.277844	+	0.960626i	$0.410380\pi$
$44$	373.030	1.27810
$45$	0	0
$46$	−814.589	−2.61097
$47$	−169.246	−0.525257	−0.262628	−	0.964897i	$-0.584589\pi$
$47$	−169.246	−0.525257	−0.262628	+	0.964897i	$0.584589\pi$
$48$	0	0
$49$	−173.961	−0.507175
$50$	0	0
$51$	0	0
$52$	81.5937	0.217596
$53$	−609.837	−1.58052	−0.790259	−	0.612772i	$-0.790054\pi$
$53$	−609.837	−1.58052	−0.790259	+	0.612772i	$0.790054\pi$
$54$	0	0
$55$	0	0
$56$	−157.612	−0.376102
$57$	0	0
$58$	−189.536	−0.429092
$59$	594.430	1.31167	0.655833	−	0.754906i	$-0.272318\pi$
$59$	594.430	1.31167	0.655833	+	0.754906i	$0.272318\pi$
$60$	0	0
$61$	325.077	0.682325	0.341163	−	0.940004i	$-0.389179\pi$
$61$	325.077	0.682325	0.341163	+	0.940004i	$0.389179\pi$
$62$	−254.396	−0.521103
$63$	0	0
$64$	−785.498	−1.53418
$65$	0	0
$66$	0	0
$67$	688.382	1.25521	0.627606	−	0.778531i	$-0.284035\pi$
$67$	688.382	1.25521	0.627606	+	0.778531i	$0.284035\pi$
$68$	−886.257	−1.58051
$69$	0	0
$70$	0	0
$71$	515.170	0.861119	0.430560	−	0.902562i	$-0.358316\pi$
$71$	515.170	0.861119	0.430560	+	0.902562i	$0.358316\pi$
$72$	0	0
$73$	−1088.37	−1.74499	−0.872497	−	0.488619i	$-0.837501\pi$
$73$	−1088.37	−1.74499	−0.872497	+	0.488619i	$0.837501\pi$
$74$	−1427.92	−2.24314
$75$	0	0
$76$	1580.62	2.38565
$77$	−449.229	−0.664863
$78$	0	0
$79$	−525.792	−0.748813	−0.374406	−	0.927265i	$-0.622154\pi$
$79$	−525.792	−0.748813	−0.374406	+	0.927265i	$0.622154\pi$
$80$	0	0
$81$	0	0
$82$	749.899	1.00991
$83$	−358.013	−0.473458	−0.236729	−	0.971576i	$-0.576075\pi$
$83$	−358.013	−0.473458	−0.236729	+	0.971576i	$0.576075\pi$
$84$	0	0
$85$	0	0
$86$	679.311	0.851767
$87$	0	0
$88$	418.861	0.507394
$89$	−1517.67	−1.80756	−0.903779	−	0.427999i	$-0.859219\pi$
$89$	−1517.67	−1.80756	−0.903779	+	0.427999i	$0.859219\pi$
$90$	0	0
$91$	−98.2610	−0.113193
$92$	−2028.49	−2.29875
$93$	0	0
$94$	−733.758	−0.805121
$95$	0	0
$96$	0	0
$97$	395.361	0.413844	0.206922	−	0.978357i	$-0.433655\pi$
$97$	395.361	0.413844	0.206922	+	0.978357i	$0.433655\pi$
$98$	−754.200	−0.777405
$99$	0	0
$100$	0	0
$101$	−333.258	−0.328320	−0.164160	−	0.986434i	$-0.552491\pi$
$101$	−333.258	−0.328320	−0.164160	+	0.986434i	$0.552491\pi$
$102$	0	0
$103$	−1630.89	−1.56016	−0.780078	−	0.625682i	$-0.784821\pi$
$103$	−1630.89	−1.56016	−0.780078	+	0.625682i	$0.784821\pi$
$104$	91.6184	0.0863839
$105$	0	0
$106$	−2643.92	−2.42264
$107$	−688.472	−0.622029	−0.311015	−	0.950405i	$-0.600669\pi$
$107$	−688.472	−0.622029	−0.311015	+	0.950405i	$0.600669\pi$
$108$	0	0
$109$	−1188.95	−1.04478	−0.522390	−	0.852707i	$-0.674959\pi$
$109$	−1188.95	−1.04478	−0.522390	+	0.852707i	$0.674959\pi$
$110$	0	0
$111$	0	0
$112$	439.611	0.370887
$113$	268.415	0.223455	0.111727	−	0.993739i	$-0.464362\pi$
$113$	268.415	0.223455	0.111727	+	0.993739i	$0.464362\pi$
$114$	0	0
$115$	0	0
$116$	−471.984	−0.377781
$117$	0	0
$118$	2577.12	2.01054
$119$	1067.29	0.822174
$120$	0	0
$121$	−137.151	−0.103043
$122$	1409.36	1.04588
$123$	0	0
$124$	−633.498	−0.458789
$125$	0	0
$126$	0	0
$127$	145.302	0.101523	0.0507616	−	0.998711i	$-0.483835\pi$
$127$	145.302	0.101523	0.0507616	+	0.998711i	$0.483835\pi$
$128$	−1456.91	−1.00605
$129$	0	0
$130$	0	0
$131$	269.319	0.179622	0.0898110	−	0.995959i	$-0.471374\pi$
$131$	269.319	0.179622	0.0898110	+	0.995959i	$0.471374\pi$
$132$	0	0
$133$	−1903.49	−1.24101
$134$	2984.45	1.92401
$135$	0	0
$136$	−995.143	−0.627447
$137$	−827.376	−0.515967	−0.257984	−	0.966149i	$-0.583058\pi$
$137$	−827.376	−0.515967	−0.257984	+	0.966149i	$0.583058\pi$
$138$	0	0
$139$	−169.870	−0.103656	−0.0518280	−	0.998656i	$-0.516505\pi$
$139$	−169.870	−0.103656	−0.0518280	+	0.998656i	$0.516505\pi$
$140$	0	0
$141$	0	0
$142$	2233.50	1.31994
$143$	261.134	0.152707
$144$	0	0
$145$	0	0
$146$	−4718.59	−2.67475
$147$	0	0
$148$	−3555.80	−1.97490
$149$	−1746.35	−0.960177	−0.480089	−	0.877220i	$-0.659396\pi$
$149$	−1746.35	−0.960177	−0.480089	+	0.877220i	$0.659396\pi$
$150$	0	0
$151$	342.838	0.184766	0.0923832	−	0.995724i	$-0.470552\pi$
$151$	342.838	0.184766	0.0923832	+	0.995724i	$0.470552\pi$
$152$	1774.82	0.947082
$153$	0	0
$154$	−1947.61	−1.01911
$155$	0	0
$156$	0	0
$157$	3495.70	1.77699	0.888494	−	0.458889i	$-0.151752\pi$
$157$	3495.70	1.77699	0.888494	+	0.458889i	$0.151752\pi$
$158$	−2279.55	−1.14779
$159$	0	0
$160$	0	0
$161$	2442.85	1.19580
$162$	0	0
$163$	−663.542	−0.318851	−0.159425	−	0.987210i	$-0.550964\pi$
$163$	−663.542	−0.318851	−0.159425	+	0.987210i	$0.550964\pi$
$164$	1867.40	0.889142
$165$	0	0
$166$	−1552.15	−0.725723
$167$	2459.64	1.13972	0.569858	−	0.821743i	$-0.306998\pi$
$167$	2459.64	1.13972	0.569858	+	0.821743i	$0.306998\pi$
$168$	0	0
$169$	−2139.88	−0.974002
$170$	0	0
$171$	0	0
$172$	1691.62	0.749912
$173$	−1527.75	−0.671401	−0.335701	−	0.941969i	$-0.608973\pi$
$173$	−1527.75	−0.671401	−0.335701	+	0.941969i	$0.608973\pi$
$174$	0	0
$175$	0	0
$176$	−1168.29	−0.500358
$177$	0	0
$178$	−6579.79	−2.77065
$179$	−1006.17	−0.420137	−0.210069	−	0.977687i	$-0.567369\pi$
$179$	−1006.17	−0.420137	−0.210069	+	0.977687i	$0.567369\pi$
$180$	0	0
$181$	591.861	0.243053	0.121527	−	0.992588i	$-0.461221\pi$
$181$	591.861	0.243053	0.121527	+	0.992588i	$0.461221\pi$
$182$	−426.006	−0.173504
$183$	0	0
$184$	−2277.71	−0.912582
$185$	0	0
$186$	0	0
$187$	−2836.39	−1.10918
$188$	−1827.20	−0.708843
$189$	0	0
$190$	0	0
$191$	−3644.57	−1.38069	−0.690345	−	0.723480i	$-0.742541\pi$
$191$	−3644.57	−1.38069	−0.690345	+	0.723480i	$0.742541\pi$
$192$	0	0
$193$	−13.4016	−0.00499826	−0.00249913	−	0.999997i	$-0.500795\pi$
$193$	−13.4016	−0.00499826	−0.00249913	+	0.999997i	$0.500795\pi$
$194$	1714.07	0.634345
$195$	0	0
$196$	−1878.11	−0.684443
$197$	−63.1983	−0.0228563	−0.0114281	−	0.999935i	$-0.503638\pi$
$197$	−63.1983	−0.0228563	−0.0114281	+	0.999935i	$0.503638\pi$
$198$	0	0
$199$	−4902.79	−1.74648	−0.873240	−	0.487291i	$-0.837985\pi$
$199$	−4902.79	−1.74648	−0.873240	+	0.487291i	$0.837985\pi$
$200$	0	0
$201$	0	0
$202$	−1444.82	−0.503254
$203$	568.397	0.196520
$204$	0	0
$205$	0	0
$206$	−7070.64	−2.39143
$207$	0	0
$208$	−255.543	−0.0851861
$209$	5058.63	1.67422
$210$	0	0
$211$	−3112.22	−1.01542	−0.507711	−	0.861528i	$-0.669508\pi$
$211$	−3112.22	−1.01542	−0.507711	+	0.861528i	$0.669508\pi$
$212$	−6583.89	−2.13294
$213$	0	0
$214$	−2984.84	−0.953455
$215$	0	0
$216$	0	0
$217$	762.904	0.238660
$218$	−5154.65	−1.60145
$219$	0	0
$220$	0	0
$221$	−620.410	−0.188838
$222$	0	0
$223$	4869.43	1.46225	0.731123	−	0.682246i	$-0.238997\pi$
$223$	4869.43	1.46225	0.731123	+	0.682246i	$0.238997\pi$
$224$	3166.81	0.944603
$225$	0	0
$226$	1163.70	0.342515
$227$	6587.19	1.92602	0.963011	−	0.269463i	$-0.0868462\pi$
$227$	6587.19	1.92602	0.963011	+	0.269463i	$0.0868462\pi$
$228$	0	0
$229$	16.4301	0.00474117	0.00237059	−	0.999997i	$-0.499245\pi$
$229$	16.4301	0.00474117	0.00237059	+	0.999997i	$0.499245\pi$
$230$	0	0
$231$	0	0
$232$	−529.972	−0.149976
$233$	−707.542	−0.198938	−0.0994692	−	0.995041i	$-0.531714\pi$
$233$	−707.542	−0.198938	−0.0994692	+	0.995041i	$0.531714\pi$
$234$	0	0
$235$	0	0
$236$	6417.56	1.77012
$237$	0	0
$238$	4627.20	1.26024
$239$	6112.04	1.65420	0.827102	−	0.562052i	$-0.189988\pi$
$239$	6112.04	1.65420	0.827102	+	0.562052i	$0.189988\pi$
$240$	0	0
$241$	3088.23	0.825437	0.412719	−	0.910859i	$-0.364579\pi$
$241$	3088.23	0.825437	0.412719	+	0.910859i	$0.364579\pi$
$242$	−594.611	−0.157946
$243$	0	0
$244$	3509.58	0.920810
$245$	0	0
$246$	0	0
$247$	1106.49	0.285037
$248$	−711.330	−0.182135
$249$	0	0
$250$	0	0
$251$	−2743.28	−0.689859	−0.344929	−	0.938629i	$-0.612097\pi$
$251$	−2743.28	−0.689859	−0.344929	+	0.938629i	$0.612097\pi$
$252$	0	0
$253$	−6492.00	−1.61324
$254$	629.949	0.155616
$255$	0	0
$256$	−32.3777	−0.00790471
$257$	−720.046	−0.174768	−0.0873838	−	0.996175i	$-0.527851\pi$
$257$	−720.046	−0.174768	−0.0873838	+	0.996175i	$0.527851\pi$
$258$	0	0
$259$	4282.15	1.02734
$260$	0	0
$261$	0	0
$262$	1167.62	0.275327
$263$	5228.16	1.22579	0.612894	−	0.790165i	$-0.290005\pi$
$263$	5228.16	1.22579	0.612894	+	0.790165i	$0.290005\pi$
$264$	0	0
$265$	0	0
$266$	−8252.51	−1.90223
$267$	0	0
$268$	7431.87	1.69393
$269$	−320.400	−0.0726213	−0.0363107	−	0.999341i	$-0.511561\pi$
$269$	−320.400	−0.0726213	−0.0363107	+	0.999341i	$0.511561\pi$
$270$	0	0
$271$	−1044.06	−0.234029	−0.117014	−	0.993130i	$-0.537332\pi$
$271$	−1044.06	−0.234029	−0.117014	+	0.993130i	$0.537332\pi$
$272$	2775.66	0.618747
$273$	0	0
$274$	−3587.05	−0.790881
$275$	0	0
$276$	0	0
$277$	3331.32	0.722599	0.361299	−	0.932450i	$-0.382333\pi$
$277$	3331.32	0.722599	0.361299	+	0.932450i	$0.382333\pi$
$278$	−736.463	−0.158885
$279$	0	0
$280$	0	0
$281$	5601.29	1.18913	0.594563	−	0.804049i	$-0.297325\pi$
$281$	5601.29	1.18913	0.594563	+	0.804049i	$0.297325\pi$
$282$	0	0
$283$	6267.84	1.31655	0.658277	−	0.752776i	$-0.271286\pi$
$283$	6267.84	1.31655	0.658277	+	0.752776i	$0.271286\pi$
$284$	5561.85	1.16210
$285$	0	0
$286$	1132.13	0.234071
$287$	−2248.85	−0.462529
$288$	0	0
$289$	1825.78	0.371622
$290$	0	0
$291$	0	0
$292$	−11750.2	−2.35490
$293$	−2102.86	−0.419284	−0.209642	−	0.977778i	$-0.567230\pi$
$293$	−2102.86	−0.419284	−0.209642	+	0.977778i	$0.567230\pi$
$294$	0	0
$295$	0	0
$296$	−3992.67	−0.784018
$297$	0	0
$298$	−7571.21	−1.47177
$299$	−1420.01	−0.274654
$300$	0	0
$301$	−2037.17	−0.390102
$302$	1486.36	0.283213
$303$	0	0
$304$	−4950.33	−0.933950
$305$	0	0
$306$	0	0
$307$	2020.07	0.375543	0.187771	−	0.982213i	$-0.439874\pi$
$307$	2020.07	0.375543	0.187771	+	0.982213i	$0.439874\pi$
$308$	−4849.95	−0.897244
$309$	0	0
$310$	0	0
$311$	−5393.32	−0.983367	−0.491684	−	0.870774i	$-0.663618\pi$
$311$	−5393.32	−0.983367	−0.491684	+	0.870774i	$0.663618\pi$
$312$	0	0
$313$	−3111.77	−0.561941	−0.280971	−	0.959716i	$-0.590656\pi$
$313$	−3111.77	−0.561941	−0.280971	+	0.959716i	$0.590656\pi$
$314$	15155.4	2.72379
$315$	0	0
$316$	−5676.53	−1.01054
$317$	−4933.58	−0.874124	−0.437062	−	0.899431i	$-0.643981\pi$
$317$	−4933.58	−0.874124	−0.437062	+	0.899431i	$0.643981\pi$
$318$	0	0
$319$	−1510.54	−0.265123
$320$	0	0
$321$	0	0
$322$	10590.9	1.83294
$323$	−12018.5	−2.07036
$324$	0	0
$325$	0	0
$326$	−2876.76	−0.488739
$327$	0	0
$328$	2096.83	0.352982
$329$	2200.45	0.368738
$330$	0	0
$331$	8772.78	1.45678	0.728392	−	0.685161i	$-0.240268\pi$
$331$	8772.78	1.45678	0.728392	+	0.685161i	$0.240268\pi$
$332$	−3865.16	−0.638940
$333$	0	0
$334$	10663.7	1.74697
$335$	0	0
$336$	0	0
$337$	10342.1	1.67173	0.835863	−	0.548939i	$-0.184968\pi$
$337$	10342.1	1.67173	0.835863	+	0.548939i	$0.184968\pi$
$338$	−9277.36	−1.49296
$339$	0	0
$340$	0	0
$341$	−2027.46	−0.321973
$342$	0	0
$343$	6721.27	1.05806
$344$	1899.45	0.297708
$345$	0	0
$346$	−6623.47	−1.02913
$347$	1338.32	0.207046	0.103523	−	0.994627i	$-0.466989\pi$
$347$	1338.32	0.207046	0.103523	+	0.994627i	$0.466989\pi$
$348$	0	0
$349$	10112.3	1.55101	0.775503	−	0.631344i	$-0.217496\pi$
$349$	10112.3	1.55101	0.775503	+	0.631344i	$0.217496\pi$
$350$	0	0
$351$	0	0
$352$	−8415.95	−1.27435
$353$	1644.28	0.247921	0.123961	−	0.992287i	$-0.460440\pi$
$353$	1644.28	0.247921	0.123961	+	0.992287i	$0.460440\pi$
$354$	0	0
$355$	0	0
$356$	−16385.0	−2.43933
$357$	0	0
$358$	−4362.20	−0.643992
$359$	−11283.0	−1.65876	−0.829379	−	0.558686i	$-0.811306\pi$
$359$	−11283.0	−1.65876	−0.829379	+	0.558686i	$0.811306\pi$
$360$	0	0
$361$	14575.7	2.12504
$362$	2565.98	0.372556
$363$	0	0
$364$	−1060.84	−0.152756
$365$	0	0
$366$	0	0
$367$	278.022	0.0395440	0.0197720	−	0.999805i	$-0.493706\pi$
$367$	278.022	0.0395440	0.0197720	+	0.999805i	$0.493706\pi$
$368$	6353.01	0.899928
$369$	0	0
$370$	0	0
$371$	7928.79	1.10955
$372$	0	0
$373$	−9833.41	−1.36503	−0.682513	−	0.730874i	$-0.739113\pi$
$373$	−9833.41	−1.36503	−0.682513	+	0.730874i	$0.739113\pi$
$374$	−12297.0	−1.70017
$375$	0	0
$376$	−2051.70	−0.281405
$377$	−330.405	−0.0451372
$378$	0	0
$379$	13511.9	1.83129	0.915644	−	0.401990i	$-0.131681\pi$
$379$	13511.9	1.83129	0.915644	+	0.401990i	$0.131681\pi$
$380$	0	0
$381$	0	0
$382$	−15800.9	−2.11634
$383$	−1389.64	−0.185398	−0.0926991	−	0.995694i	$-0.529549\pi$
$383$	−1389.64	−0.185398	−0.0926991	+	0.995694i	$0.529549\pi$
$384$	0	0
$385$	0	0
$386$	−58.1018	−0.00766141
$387$	0	0
$388$	4268.38	0.558490
$389$	906.737	0.118184	0.0590918	−	0.998253i	$-0.481180\pi$
$389$	906.737	0.118184	0.0590918	+	0.998253i	$0.481180\pi$
$390$	0	0
$391$	15423.9	1.99494
$392$	−2108.86	−0.271718
$393$	0	0
$394$	−273.993	−0.0350345
$395$	0	0
$396$	0	0
$397$	−1904.52	−0.240769	−0.120384	−	0.992727i	$-0.538413\pi$
$397$	−1904.52	−0.240769	−0.120384	+	0.992727i	$0.538413\pi$
$398$	−21255.8	−2.67703
$399$	0	0
$400$	0	0
$401$	10413.8	1.29686	0.648431	−	0.761274i	$-0.275426\pi$
$401$	10413.8	1.29686	0.648431	+	0.761274i	$0.275426\pi$
$402$	0	0
$403$	−443.470	−0.0548159
$404$	−3597.90	−0.443075
$405$	0	0
$406$	2464.26	0.301229
$407$	−11380.0	−1.38596
$408$	0	0
$409$	6249.48	0.755542	0.377771	−	0.925899i	$-0.376691\pi$
$409$	6249.48	0.755542	0.377771	+	0.925899i	$0.376691\pi$
$410$	0	0
$411$	0	0
$412$	−17607.3	−2.10546
$413$	−7728.48	−0.920808
$414$	0	0
$415$	0	0
$416$	−1840.84	−0.216958
$417$	0	0
$418$	21931.5	2.56628
$419$	−2990.27	−0.348650	−0.174325	−	0.984688i	$-0.555774\pi$
$419$	−2990.27	−0.348650	−0.174325	+	0.984688i	$0.555774\pi$
$420$	0	0
$421$	−4692.64	−0.543243	−0.271622	−	0.962404i	$-0.587560\pi$
$421$	−4692.64	−0.543243	−0.271622	+	0.962404i	$0.587560\pi$
$422$	−13492.9	−1.55645
$423$	0	0
$424$	−7392.79	−0.846758
$425$	0	0
$426$	0	0
$427$	−4226.49	−0.479002
$428$	−7432.85	−0.839440
$429$	0	0
$430$	0	0
$431$	−817.709	−0.0913866	−0.0456933	−	0.998956i	$-0.514550\pi$
$431$	−817.709	−0.0913866	−0.0456933	+	0.998956i	$0.514550\pi$
$432$	0	0
$433$	−5547.29	−0.615671	−0.307836	−	0.951440i	$-0.599605\pi$
$433$	−5547.29	−0.615671	−0.307836	+	0.951440i	$0.599605\pi$
$434$	3307.53	0.365822
$435$	0	0
$436$	−12836.1	−1.40995
$437$	−27508.2	−3.01120
$438$	0	0
$439$	13381.6	1.45483	0.727416	−	0.686197i	$-0.240721\pi$
$439$	13381.6	1.45483	0.727416	+	0.686197i	$0.240721\pi$
$440$	0	0
$441$	0	0
$442$	−2689.76	−0.289454
$443$	−10836.9	−1.16225	−0.581123	−	0.813816i	$-0.697387\pi$
$443$	−10836.9	−1.16225	−0.581123	+	0.813816i	$0.697387\pi$
$444$	0	0
$445$	0	0
$446$	21111.2	2.24135
$447$	0	0
$448$	10212.6	1.07701
$449$	−2970.62	−0.312232	−0.156116	−	0.987739i	$-0.549897\pi$
$449$	−2970.62	−0.312232	−0.156116	+	0.987739i	$0.549897\pi$
$450$	0	0
$451$	5976.44	0.623991
$452$	2897.85	0.301556
$453$	0	0
$454$	28558.4	2.95223
$455$	0	0
$456$	0	0
$457$	−16176.3	−1.65579	−0.827896	−	0.560882i	$-0.810462\pi$
$457$	−16176.3	−1.65579	−0.827896	+	0.560882i	$0.810462\pi$
$458$	71.2317	0.00726733
$459$	0	0
$460$	0	0
$461$	−5731.81	−0.579083	−0.289541	−	0.957165i	$-0.593503\pi$
$461$	−5731.81	−0.579083	−0.289541	+	0.957165i	$0.593503\pi$
$462$	0	0
$463$	−14040.1	−1.40928	−0.704641	−	0.709564i	$-0.748892\pi$
$463$	−14040.1	−1.40928	−0.704641	+	0.709564i	$0.748892\pi$
$464$	1478.20	0.147896
$465$	0	0
$466$	−3067.52	−0.304935
$467$	3.48921	0.000345742 0	0.000172871	−	1.00000i	$-0.499945\pi$
$467$	3.48921	0.000345742 0	0.000172871		1.00000i	$0.499945\pi$
$468$	0	0
$469$	−8949.99	−0.881177
$470$	0	0
$471$	0	0
$472$	7206.02	0.702721
$473$	5413.88	0.526280
$474$	0	0
$475$	0	0
$476$	11522.7	1.10954
$477$	0	0
$478$	26498.4	2.53559
$479$	3651.08	0.348272	0.174136	−	0.984722i	$-0.444287\pi$
$479$	3651.08	0.348272	0.174136	+	0.984722i	$0.444287\pi$
$480$	0	0
$481$	−2489.18	−0.235960
$482$	13388.9	1.26524
$483$	0	0
$484$	−1480.70	−0.139059
$485$	0	0
$486$	0	0
$487$	13646.8	1.26981	0.634903	−	0.772592i	$-0.281040\pi$
$487$	13646.8	1.26981	0.634903	+	0.772592i	$0.281040\pi$
$488$	3940.77	0.365554
$489$	0	0
$490$	0	0
$491$	−41.6983	−0.00383263	−0.00191631	−	0.999998i	$-0.500610\pi$
$491$	−41.6983	−0.00383263	−0.00191631	+	0.999998i	$0.500610\pi$
$492$	0	0
$493$	3588.80	0.327853
$494$	4797.12	0.436908
$495$	0	0
$496$	1984.05	0.179610
$497$	−6697.98	−0.604518
$498$	0	0
$499$	−2392.93	−0.214674	−0.107337	−	0.994223i	$-0.534232\pi$
$499$	−2392.93	−0.214674	−0.107337	+	0.994223i	$0.534232\pi$
$500$	0	0
$501$	0	0
$502$	−11893.4	−1.05743
$503$	−1254.89	−0.111238	−0.0556190	−	0.998452i	$-0.517713\pi$
$503$	−1254.89	−0.111238	−0.0556190	+	0.998452i	$0.517713\pi$
$504$	0	0
$505$	0	0
$506$	−28145.8	−2.47279
$507$	0	0
$508$	1568.70	0.137007
$509$	−547.930	−0.0477142	−0.0238571	−	0.999715i	$-0.507595\pi$
$509$	−547.930	−0.0477142	−0.0238571	+	0.999715i	$0.507595\pi$
$510$	0	0
$511$	14150.5	1.22501
$512$	11514.9	0.993929
$513$	0	0
$514$	−3121.73	−0.267886
$515$	0	0
$516$	0	0
$517$	−5847.81	−0.497459
$518$	18565.1	1.57471
$519$	0	0
$520$	0	0
$521$	−12151.4	−1.02181	−0.510903	−	0.859638i	$-0.670689\pi$
$521$	−12151.4	−1.02181	−0.510903	+	0.859638i	$0.670689\pi$
$522$	0	0
$523$	18913.8	1.58134	0.790672	−	0.612241i	$-0.209732\pi$
$523$	18913.8	1.58134	0.790672	+	0.612241i	$0.209732\pi$
$524$	2907.60	0.242403
$525$	0	0
$526$	22666.4	1.87890
$527$	4816.89	0.398154
$528$	0	0
$529$	23135.7	1.90151
$530$	0	0
$531$	0	0
$532$	−20550.4	−1.67476
$533$	1307.24	0.106234
$534$	0	0
$535$	0	0
$536$	8344.96	0.672476
$537$	0	0
$538$	−1389.08	−0.111315
$539$	−6010.73	−0.480335
$540$	0	0
$541$	−9936.16	−0.789628	−0.394814	−	0.918761i	$-0.629191\pi$
$541$	−9936.16	−0.789628	−0.394814	+	0.918761i	$0.629191\pi$
$542$	−4526.45	−0.358723
$543$	0	0
$544$	19994.9	1.57587
$545$	0	0
$546$	0	0
$547$	12719.7	0.994250	0.497125	−	0.867679i	$-0.334389\pi$
$547$	12719.7	0.994250	0.497125	+	0.867679i	$0.334389\pi$
$548$	−8932.47	−0.696307
$549$	0	0
$550$	0	0
$551$	−6400.54	−0.494868
$552$	0	0
$553$	6836.08	0.525678
$554$	14442.8	1.10761
$555$	0	0
$556$	−1833.94	−0.139886
$557$	16006.5	1.21763	0.608813	−	0.793314i	$-0.291646\pi$
$557$	16006.5	1.21763	0.608813	+	0.793314i	$0.291646\pi$
$558$	0	0
$559$	1184.19	0.0895993
$560$	0	0
$561$	0	0
$562$	24284.1	1.82271
$563$	−4626.58	−0.346336	−0.173168	−	0.984892i	$-0.555400\pi$
$563$	−4626.58	−0.346336	−0.173168	+	0.984892i	$0.555400\pi$
$564$	0	0
$565$	0	0
$566$	27173.9	2.01803
$567$	0	0
$568$	6245.19	0.461342
$569$	−16859.0	−1.24212	−0.621061	−	0.783762i	$-0.713298\pi$
$569$	−16859.0	−1.24212	−0.621061	+	0.783762i	$0.713298\pi$
$570$	0	0
$571$	16149.1	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$571$	16149.1	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$572$	2819.24	0.206081
$573$	0	0
$574$	−9749.80	−0.708970
$575$	0	0
$576$	0	0
$577$	7430.10	0.536082	0.268041	−	0.963408i	$-0.413624\pi$
$577$	7430.10	0.536082	0.268041	+	0.963408i	$0.413624\pi$
$578$	7915.59	0.569628
$579$	0	0
$580$	0	0
$581$	4654.70	0.332375
$582$	0	0
$583$	−21071.1	−1.49687
$584$	−13193.9	−0.934875
$585$	0	0
$586$	−9116.84	−0.642685
$587$	−17152.1	−1.20604	−0.603019	−	0.797727i	$-0.706036\pi$
$587$	−17152.1	−1.20604	−0.603019	+	0.797727i	$0.706036\pi$
$588$	0	0
$589$	−8590.83	−0.600983
$590$	0	0
$591$	0	0
$592$	11136.4	0.773146
$593$	15477.7	1.07183	0.535913	−	0.844273i	$-0.319967\pi$
$593$	15477.7	1.07183	0.535913	+	0.844273i	$0.319967\pi$
$594$	0	0
$595$	0	0
$596$	−18853.8	−1.29578
$597$	0	0
$598$	−6156.39	−0.420993
$599$	−18959.2	−1.29324	−0.646622	−	0.762811i	$-0.723819\pi$
$599$	−18959.2	−1.29324	−0.646622	+	0.762811i	$0.723819\pi$
$600$	0	0
$601$	−2093.61	−0.142096	−0.0710482	−	0.997473i	$-0.522634\pi$
$601$	−2093.61	−0.142096	−0.0710482	+	0.997473i	$0.522634\pi$
$602$	−8832.06	−0.597953
$603$	0	0
$604$	3701.33	0.249346
$605$	0	0
$606$	0	0
$607$	−6135.25	−0.410251	−0.205125	−	0.978736i	$-0.565760\pi$
$607$	−6135.25	−0.410251	−0.205125	+	0.978736i	$0.565760\pi$
$608$	−35660.4	−2.37865
$609$	0	0
$610$	0	0
$611$	−1279.11	−0.0846924
$612$	0	0
$613$	−12762.8	−0.840922	−0.420461	−	0.907311i	$-0.638132\pi$
$613$	−12762.8	−0.840922	−0.420461	+	0.907311i	$0.638132\pi$
$614$	8757.93	0.575637
$615$	0	0
$616$	−5445.81	−0.356198
$617$	−734.080	−0.0478978	−0.0239489	−	0.999713i	$-0.507624\pi$
$617$	−734.080	−0.0478978	−0.0239489	+	0.999713i	$0.507624\pi$
$618$	0	0
$619$	−10649.7	−0.691515	−0.345757	−	0.938324i	$-0.612378\pi$
$619$	−10649.7	−0.691515	−0.345757	+	0.938324i	$0.612378\pi$
$620$	0	0
$621$	0	0
$622$	−23382.5	−1.50732
$623$	19732.0	1.26893
$624$	0	0
$625$	0	0
$626$	−13490.9	−0.861352
$627$	0	0
$628$	37740.1	2.39808
$629$	27037.1	1.71389
$630$	0	0
$631$	−3465.43	−0.218632	−0.109316	−	0.994007i	$-0.534866\pi$
$631$	−3465.43	−0.218632	−0.109316	+	0.994007i	$0.534866\pi$
$632$	−6373.95	−0.401174
$633$	0	0
$634$	−21389.3	−1.33987
$635$	0	0
$636$	0	0
$637$	−1314.74	−0.0817770
$638$	−6548.89	−0.406384
$639$	0	0
$640$	0	0
$641$	29210.1	1.79989	0.899945	−	0.436004i	$-0.143607\pi$
$641$	29210.1	1.79989	0.899945	+	0.436004i	$0.143607\pi$
$642$	0	0
$643$	−17025.7	−1.04421	−0.522107	−	0.852880i	$-0.674854\pi$
$643$	−17025.7	−1.04421	−0.522107	+	0.852880i	$0.674854\pi$
$644$	26373.4	1.61375
$645$	0	0
$646$	−52105.5	−3.17347
$647$	−8673.26	−0.527019	−0.263509	−	0.964657i	$-0.584880\pi$
$647$	−8673.26	−0.527019	−0.263509	+	0.964657i	$0.584880\pi$
$648$	0	0
$649$	20538.8	1.24225
$650$	0	0
$651$	0	0
$652$	−7163.70	−0.430295
$653$	1080.67	0.0647625	0.0323812	−	0.999476i	$-0.489691\pi$
$653$	1080.67	0.0647625	0.0323812	+	0.999476i	$0.489691\pi$
$654$	0	0
$655$	0	0
$656$	−5848.49	−0.348087
$657$	0	0
$658$	9539.95	0.565206
$659$	−6969.61	−0.411984	−0.205992	−	0.978554i	$-0.566042\pi$
$659$	−6969.61	−0.411984	−0.205992	+	0.978554i	$0.566042\pi$
$660$	0	0
$661$	10897.3	0.641236	0.320618	−	0.947209i	$-0.396109\pi$
$661$	10897.3	0.641236	0.320618	+	0.947209i	$0.396109\pi$
$662$	38034.0	2.23298
$663$	0	0
$664$	−4340.04	−0.253654
$665$	0	0
$666$	0	0
$667$	8214.14	0.476841
$668$	26554.6	1.53807
$669$	0	0
$670$	0	0
$671$	11232.1	0.646215
$672$	0	0
$673$	29607.9	1.69584	0.847919	−	0.530126i	$-0.177855\pi$
$673$	29607.9	1.69584	0.847919	+	0.530126i	$0.177855\pi$
$674$	44837.8	2.56244
$675$	0	0
$676$	−23102.5	−1.31443
$677$	−6387.41	−0.362612	−0.181306	−	0.983427i	$-0.558032\pi$
$677$	−6387.41	−0.362612	−0.181306	+	0.983427i	$0.558032\pi$
$678$	0	0
$679$	−5140.28	−0.290524
$680$	0	0
$681$	0	0
$682$	−8789.94	−0.493525
$683$	19929.2	1.11650	0.558251	−	0.829672i	$-0.311473\pi$
$683$	19929.2	1.11650	0.558251	+	0.829672i	$0.311473\pi$
$684$	0	0
$685$	0	0
$686$	29139.7	1.62181
$687$	0	0
$688$	−5297.97	−0.293580
$689$	−4608.94	−0.254843
$690$	0	0
$691$	12947.2	0.712784	0.356392	−	0.934336i	$-0.384007\pi$
$691$	12947.2	0.712784	0.356392	+	0.934336i	$0.384007\pi$
$692$	−16493.8	−0.906069
$693$	0	0
$694$	5802.23	0.317362
$695$	0	0
$696$	0	0
$697$	−14199.0	−0.771631
$698$	43841.6	2.37740
$699$	0	0
$700$	0	0
$701$	10692.9	0.576125	0.288063	−	0.957612i	$-0.406989\pi$
$701$	10692.9	0.576125	0.288063	+	0.957612i	$0.406989\pi$
$702$	0	0
$703$	−48220.0	−2.58699
$704$	−27140.6	−1.45298
$705$	0	0
$706$	7128.70	0.380017
$707$	4332.85	0.230486
$708$	0	0
$709$	−23573.4	−1.24869	−0.624343	−	0.781150i	$-0.714633\pi$
$709$	−23573.4	−1.24869	−0.624343	+	0.781150i	$0.714633\pi$
$710$	0	0
$711$	0	0
$712$	−18398.1	−0.968394
$713$	11025.0	0.579090
$714$	0	0
$715$	0	0
$716$	−10862.7	−0.566983
$717$	0	0
$718$	−48916.9	−2.54257
$719$	−27013.2	−1.40114	−0.700572	−	0.713581i	$-0.747072\pi$
$719$	−27013.2	−1.40114	−0.700572	+	0.713581i	$0.747072\pi$
$720$	0	0
$721$	21204.0	1.09525
$722$	63192.1	3.25730
$723$	0	0
$724$	6389.82	0.328005
$725$	0	0
$726$	0	0
$727$	17854.0	0.910825	0.455412	−	0.890281i	$-0.349492\pi$
$727$	17854.0	0.910825	0.455412	+	0.890281i	$0.349492\pi$
$728$	−1191.18	−0.0606428
$729$	0	0
$730$	0	0
$731$	−12862.5	−0.650802
$732$	0	0
$733$	−9590.81	−0.483280	−0.241640	−	0.970366i	$-0.577685\pi$
$733$	−9590.81	−0.483280	−0.241640	+	0.970366i	$0.577685\pi$
$734$	1205.35	0.0606136
$735$	0	0
$736$	45764.9	2.29200
$737$	23785.1	1.18878
$738$	0	0
$739$	−6237.43	−0.310484	−0.155242	−	0.987876i	$-0.549616\pi$
$739$	−6237.43	−0.310484	−0.155242	+	0.987876i	$0.549616\pi$
$740$	0	0
$741$	0	0
$742$	34374.9	1.70073
$743$	−17134.5	−0.846034	−0.423017	−	0.906122i	$-0.639029\pi$
$743$	−17134.5	−0.846034	−0.423017	+	0.906122i	$0.639029\pi$
$744$	0	0
$745$	0	0
$746$	−42632.3	−2.09233
$747$	0	0
$748$	−30622.1	−1.49686
$749$	8951.17	0.436674
$750$	0	0
$751$	39141.7	1.90186	0.950932	−	0.309400i	$-0.100128\pi$
$751$	39141.7	1.90186	0.950932	+	0.309400i	$0.100128\pi$
$752$	5722.61	0.277503
$753$	0	0
$754$	−1432.45	−0.0691869
$755$	0	0
$756$	0	0
$757$	−3774.26	−0.181212	−0.0906062	−	0.995887i	$-0.528880\pi$
$757$	−3774.26	−0.181212	−0.0906062	+	0.995887i	$0.528880\pi$
$758$	58580.1	2.80702
$759$	0	0
$760$	0	0
$761$	−25997.9	−1.23840	−0.619200	−	0.785233i	$-0.712543\pi$
$761$	−25997.9	−1.23840	−0.619200	+	0.785233i	$0.712543\pi$
$762$	0	0
$763$	15458.2	0.733451
$764$	−39347.3	−1.86327
$765$	0	0
$766$	−6024.73	−0.284181
$767$	4492.51	0.211493
$768$	0	0
$769$	15635.4	0.733193	0.366597	−	0.930380i	$-0.380523\pi$
$769$	15635.4	0.733193	0.366597	+	0.930380i	$0.380523\pi$
$770$	0	0
$771$	0	0
$772$	−144.685	−0.00674525
$773$	−5936.28	−0.276213	−0.138107	−	0.990417i	$-0.544102\pi$
$773$	−5936.28	−0.276213	−0.138107	+	0.990417i	$0.544102\pi$
$774$	0	0
$775$	0	0
$776$	4792.79	0.221715
$777$	0	0
$778$	3931.12	0.181153
$779$	25323.7	1.16472
$780$	0	0
$781$	17800.2	0.815547
$782$	66869.7	3.05787
$783$	0	0
$784$	5882.04	0.267950
$785$	0	0
$786$	0	0
$787$	10387.7	0.470499	0.235249	−	0.971935i	$-0.424409\pi$
$787$	10387.7	0.470499	0.235249	+	0.971935i	$0.424409\pi$
$788$	−682.298	−0.0308450
$789$	0	0
$790$	0	0
$791$	−3489.80	−0.156869
$792$	0	0
$793$	2456.82	0.110018
$794$	−8256.96	−0.369053
$795$	0	0
$796$	−52931.2	−2.35691
$797$	−1394.86	−0.0619931	−0.0309966	−	0.999519i	$-0.509868\pi$
$797$	−1394.86	−0.0619931	−0.0309966	+	0.999519i	$0.509868\pi$
$798$	0	0
$799$	13893.4	0.615161
$800$	0	0
$801$	0	0
$802$	45148.6	1.98785
$803$	−37605.6	−1.65265
$804$	0	0
$805$	0	0
$806$	−1922.64	−0.0840227
$807$	0	0
$808$	−4039.94	−0.175897
$809$	7633.11	0.331726	0.165863	−	0.986149i	$-0.446959\pi$
$809$	7633.11	0.331726	0.165863	+	0.986149i	$0.446959\pi$
$810$	0	0
$811$	−38117.1	−1.65040	−0.825199	−	0.564842i	$-0.808937\pi$
$811$	−38117.1	−1.65040	−0.825199	+	0.564842i	$0.808937\pi$
$812$	6136.50	0.265208
$813$	0	0
$814$	−49337.6	−2.12442
$815$	0	0
$816$	0	0
$817$	22940.0	0.982335
$818$	27094.3	1.15811
$819$	0	0
$820$	0	0
$821$	−3931.07	−0.167107	−0.0835537	−	0.996503i	$-0.526627\pi$
$821$	−3931.07	−0.167107	−0.0835537	+	0.996503i	$0.526627\pi$
$822$	0	0
$823$	21498.2	0.910548	0.455274	−	0.890351i	$-0.349541\pi$
$823$	21498.2	0.910548	0.455274	+	0.890351i	$0.349541\pi$
$824$	−19770.5	−0.835849
$825$	0	0
$826$	−33506.5	−1.41143
$827$	−5272.26	−0.221686	−0.110843	−	0.993838i	$-0.535355\pi$
$827$	−5272.26	−0.221686	−0.110843	+	0.993838i	$0.535355\pi$
$828$	0	0
$829$	−13595.1	−0.569573	−0.284786	−	0.958591i	$-0.591923\pi$
$829$	−13595.1	−0.569573	−0.284786	+	0.958591i	$0.591923\pi$
$830$	0	0
$831$	0	0
$832$	−5936.54	−0.247371
$833$	14280.5	0.593985
$834$	0	0
$835$	0	0
$836$	54613.8	2.25940
$837$	0	0
$838$	−12964.2	−0.534416
$839$	−15476.6	−0.636843	−0.318422	−	0.947949i	$-0.603153\pi$
$839$	−15476.6	−0.636843	−0.318422	+	0.947949i	$0.603153\pi$
$840$	0	0
$841$	−22477.8	−0.921635
$842$	−20344.7	−0.832691
$843$	0	0
$844$	−33600.0	−1.37033
$845$	0	0
$846$	0	0
$847$	1783.16	0.0723380
$848$	20620.0	0.835017
$849$	0	0
$850$	0	0
$851$	61883.2	2.49275
$852$	0	0
$853$	−30310.0	−1.21664	−0.608320	−	0.793692i	$-0.708156\pi$
$853$	−30310.0	−1.21664	−0.608320	+	0.793692i	$0.708156\pi$
$854$	−18323.7	−0.734221
$855$	0	0
$856$	−8346.06	−0.333250
$857$	31153.2	1.24174	0.620872	−	0.783912i	$-0.286779\pi$
$857$	31153.2	1.24174	0.620872	+	0.783912i	$0.286779\pi$
$858$	0	0
$859$	−17911.9	−0.711461	−0.355731	−	0.934589i	$-0.615768\pi$
$859$	−17911.9	−0.711461	−0.355731	+	0.934589i	$0.615768\pi$
$860$	0	0
$861$	0	0
$862$	−3545.14	−0.140079
$863$	−34410.6	−1.35730	−0.678649	−	0.734462i	$-0.737434\pi$
$863$	−34410.6	−1.35730	−0.678649	+	0.734462i	$0.737434\pi$
$864$	0	0
$865$	0	0
$866$	−24050.0	−0.943709
$867$	0	0
$868$	8236.42	0.322077
$869$	−18167.2	−0.709184
$870$	0	0
$871$	5202.56	0.202391
$872$	−14413.2	−0.559738
$873$	0	0
$874$	−119261.	−4.61562
$875$	0	0
$876$	0	0
$877$	28229.3	1.08693	0.543464	−	0.839432i	$-0.317112\pi$
$877$	28229.3	1.08693	0.543464	+	0.839432i	$0.317112\pi$
$878$	58015.5	2.22999
$879$	0	0
$880$	0	0
$881$	20204.7	0.772661	0.386331	−	0.922360i	$-0.373742\pi$
$881$	20204.7	0.772661	0.386331	+	0.922360i	$0.373742\pi$
$882$	0	0
$883$	19916.7	0.759061	0.379530	−	0.925179i	$-0.376086\pi$
$883$	19916.7	0.759061	0.379530	+	0.925179i	$0.376086\pi$
$884$	−6698.04	−0.254841
$885$	0	0
$886$	−46982.7	−1.78151
$887$	−7027.21	−0.266010	−0.133005	−	0.991115i	$-0.542463\pi$
$887$	−7027.21	−0.266010	−0.133005	+	0.991115i	$0.542463\pi$
$888$	0	0
$889$	−1889.14	−0.0712707
$890$	0	0
$891$	0	0
$892$	52571.0	1.97333
$893$	−24778.6	−0.928538
$894$	0	0
$895$	0	0
$896$	18942.0	0.706259
$897$	0	0
$898$	−12879.0	−0.478593
$899$	2565.28	0.0951689
$900$	0	0
$901$	50061.5	1.85104
$902$	25910.6	0.956462
$903$	0	0
$904$	3253.88	0.119715
$905$	0	0
$906$	0	0
$907$	−6372.22	−0.233281	−0.116641	−	0.993174i	$-0.537213\pi$
$907$	−6372.22	−0.233281	−0.116641	+	0.993174i	$0.537213\pi$
$908$	71116.2	2.59920
$909$	0	0
$910$	0	0
$911$	34896.0	1.26911	0.634553	−	0.772879i	$-0.281184\pi$
$911$	34896.0	1.26911	0.634553	+	0.772879i	$0.281184\pi$
$912$	0	0
$913$	−12370.1	−0.448402
$914$	−70131.7	−2.53802
$915$	0	0
$916$	177.381	0.00639830
$917$	−3501.54	−0.126097
$918$	0	0
$919$	−702.234	−0.0252063	−0.0126031	−	0.999921i	$-0.504012\pi$
$919$	−702.234	−0.0252063	−0.0126031	+	0.999921i	$0.504012\pi$
$920$	0	0
$921$	0	0
$922$	−24850.0	−0.887626
$923$	3893.49	0.138847
$924$	0	0
$925$	0	0
$926$	−60870.0	−2.16017
$927$	0	0
$928$	10648.5	0.376673
$929$	30573.4	1.07974	0.539871	−	0.841748i	$-0.318473\pi$
$929$	30573.4	1.07974	0.539871	+	0.841748i	$0.318473\pi$
$930$	0	0
$931$	−25468.9	−0.896574
$932$	−7638.73	−0.268471
$933$	0	0
$934$	15.1273	0.000529958 0
$935$	0	0
$936$	0	0
$937$	13601.7	0.474224	0.237112	−	0.971482i	$-0.423799\pi$
$937$	13601.7	0.474224	0.237112	+	0.971482i	$0.423799\pi$
$938$	−38802.3	−1.35068
$939$	0	0
$940$	0	0
$941$	−47606.2	−1.64922	−0.824611	−	0.565700i	$-0.808606\pi$
$941$	−47606.2	−1.64922	−0.824611	+	0.565700i	$0.808606\pi$
$942$	0	0
$943$	−32499.2	−1.12229
$944$	−20099.1	−0.692976
$945$	0	0
$946$	23471.6	0.806690
$947$	−48534.3	−1.66542	−0.832710	−	0.553710i	$-0.813212\pi$
$947$	−48534.3	−1.66542	−0.832710	+	0.553710i	$0.813212\pi$
$948$	0	0
$949$	−8225.57	−0.281363
$950$	0	0
$951$	0	0
$952$	12938.3	0.440477
$953$	30379.5	1.03262	0.516312	−	0.856401i	$-0.327305\pi$
$953$	30379.5	1.03262	0.516312	+	0.856401i	$0.327305\pi$
$954$	0	0
$955$	0	0
$956$	65986.5	2.23238
$957$	0	0
$958$	15829.1	0.533835
$959$	10757.1	0.362216
$960$	0	0
$961$	−26347.9	−0.884424
$962$	−10791.7	−0.361683
$963$	0	0
$964$	33341.0	1.11394
$965$	0	0
$966$	0	0
$967$	32323.9	1.07494	0.537470	−	0.843283i	$-0.319380\pi$
$967$	32323.9	1.07494	0.537470	+	0.843283i	$0.319380\pi$
$968$	−1662.62	−0.0552052
$969$	0	0
$970$	0	0
$971$	27461.7	0.907609	0.453805	−	0.891101i	$-0.350066\pi$
$971$	27461.7	0.907609	0.453805	+	0.891101i	$0.350066\pi$
$972$	0	0
$973$	2208.56	0.0727681
$974$	59165.0	1.94638
$975$	0	0
$976$	−10991.6	−0.360485
$977$	−43543.8	−1.42589	−0.712943	−	0.701222i	$-0.752638\pi$
$977$	−43543.8	−1.42589	−0.712943	+	0.701222i	$0.752638\pi$
$978$	0	0
$979$	−52438.7	−1.71190
$980$	0	0
$981$	0	0
$982$	−180.781	−0.00587470
$983$	−34369.0	−1.11516	−0.557579	−	0.830124i	$-0.688270\pi$
$983$	−34369.0	−1.11516	−0.557579	+	0.830124i	$0.688270\pi$
$984$	0	0
$985$	0	0
$986$	15559.1	0.502537
$987$	0	0
$988$	11945.8	0.384663
$989$	−29440.0	−0.946550
$990$	0	0
$991$	4360.89	0.139786	0.0698931	−	0.997554i	$-0.477734\pi$
$991$	4360.89	0.139786	0.0698931	+	0.997554i	$0.477734\pi$
$992$	14292.4	0.457443
$993$	0	0
$994$	−29038.8	−0.926614
$995$	0	0
$996$	0	0
$997$	29354.7	0.932470	0.466235	−	0.884661i	$-0.345610\pi$
$997$	29354.7	0.932470	0.466235	+	0.884661i	$0.345610\pi$
$998$	−10374.4	−0.329056
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.be.1.11		12
3.2	odd	2		2025.4.a.bi.1.2		12
5.4	even	2		2025.4.a.bj.1.2		12
9.2	odd	6		225.4.e.e.76.11	✓	24
9.5	odd	6		225.4.e.e.151.11	yes	24
15.14	odd	2		2025.4.a.bf.1.11		12
45.2	even	12		225.4.k.e.49.4		48
45.14	odd	6		225.4.e.f.151.2	yes	24
45.23	even	12		225.4.k.e.124.4		48
45.29	odd	6		225.4.e.f.76.2	yes	24
45.32	even	12		225.4.k.e.124.21		48
45.38	even	12		225.4.k.e.49.21		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.e.e.76.11	✓	24	9.2	odd	6
225.4.e.e.151.11	yes	24	9.5	odd	6
225.4.e.f.76.2	yes	24	45.29	odd	6
225.4.e.f.151.2	yes	24	45.14	odd	6
225.4.k.e.49.4		48	45.2	even	12
225.4.k.e.49.21		48	45.38	even	12
225.4.k.e.124.4		48	45.23	even	12
225.4.k.e.124.21		48	45.32	even	12
2025.4.a.be.1.11		12	1.1	even	1	trivial
2025.4.a.bf.1.11		12	15.14	odd	2
2025.4.a.bi.1.2		12	3.2	odd	2
2025.4.a.bj.1.2		12	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+4.33545 q^{2} +10.7961 q^{4} -13.0015 q^{7} +12.1226 q^{8} +O(q^{10})\)	\(q+4.33545 q^{2} +10.7961 q^{4} -13.0015 q^{7} +12.1226 q^{8} +34.5521 q^{11} +7.55767 q^{13} -56.3674 q^{14} -33.8124 q^{16} -82.0901 q^{17} +146.406 q^{19} +149.799 q^{22} -187.890 q^{23} +32.7659 q^{26} -140.366 q^{28} -43.7178 q^{29} -58.6782 q^{31} -243.572 q^{32} -355.898 q^{34} -329.358 q^{37} +634.736 q^{38} +172.969 q^{41} +156.687 q^{43} +373.030 q^{44} -814.589 q^{46} -169.246 q^{47} -173.961 q^{49} +81.5937 q^{52} -609.837 q^{53} -157.612 q^{56} -189.536 q^{58} +594.430 q^{59} +325.077 q^{61} -254.396 q^{62} -785.498 q^{64} +688.382 q^{67} -886.257 q^{68} +515.170 q^{71} -1088.37 q^{73} -1427.92 q^{74} +1580.62 q^{76} -449.229 q^{77} -525.792 q^{79} +749.899 q^{82} -358.013 q^{83} +679.311 q^{86} +418.861 q^{88} -1517.67 q^{89} -98.2610 q^{91} -2028.49 q^{92} -733.758 q^{94} +395.361 q^{97} -754.200 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) \) 12 * q - 4 * q^2 + 48 * q^4 - 6 * q^7 - 45 * q^8	\( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8} - 29 q^{11} - 24 q^{13} + 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} + 18 q^{22} - 318 q^{23} + 154 q^{26} - 96 q^{28} - 106 q^{29} + 60 q^{31} - 914 q^{32} - 108 q^{34} + 84 q^{37} - 640 q^{38} + 353 q^{41} + 426 q^{43} - 571 q^{44} + 270 q^{46} - 1210 q^{47} + 666 q^{49} + 75 q^{52} - 448 q^{53} + 570 q^{56} - 594 q^{58} - 482 q^{59} + 402 q^{61} - 2544 q^{62} + 975 q^{64} + 201 q^{67} - 3437 q^{68} + 944 q^{71} + 453 q^{73} - 10 q^{74} - 462 q^{76} - 2652 q^{77} + 258 q^{79} - 873 q^{82} - 3012 q^{83} - 1952 q^{86} + 216 q^{88} + 738 q^{89} - 618 q^{91} - 5232 q^{92} + 63 q^{94} + 318 q^{97} - 7511 q^{98}+O(q^{100}) \) 12 * q - 4 * q^2 + 48 * q^4 - 6 * q^7 - 45 * q^8 - 29 * q^11 - 24 * q^13 + 69 * q^14 + 192 * q^16 - 79 * q^17 - 75 * q^19 + 18 * q^22 - 318 * q^23 + 154 * q^26 - 96 * q^28 - 106 * q^29 + 60 * q^31 - 914 * q^32 - 108 * q^34 + 84 * q^37 - 640 * q^38 + 353 * q^41 + 426 * q^43 - 571 * q^44 + 270 * q^46 - 1210 * q^47 + 666 * q^49 + 75 * q^52 - 448 * q^53 + 570 * q^56 - 594 * q^58 - 482 * q^59 + 402 * q^61 - 2544 * q^62 + 975 * q^64 + 201 * q^67 - 3437 * q^68 + 944 * q^71 + 453 * q^73 - 10 * q^74 - 462 * q^76 - 2652 * q^77 + 258 * q^79 - 873 * q^82 - 3012 * q^83 - 1952 * q^86 + 216 * q^88 + 738 * q^89 - 618 * q^91 - 5232 * q^92 + 63 * q^94 + 318 * q^97 - 7511 * q^98

Introduction

L-functions

Modular forms

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Database

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