LMFDB - Embedded newform 2025.4.a.be.1.2

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			1.2
Root			$4.78880$ 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)

$f(q)$	$=$	$q-4.78880 q^{2} +14.9326 q^{4} +30.6698 q^{7} -33.1990 q^{8} +O(q^{10})$	$q-4.78880 q^{2} +14.9326 q^{4} +30.6698 q^{7} -33.1990 q^{8} -42.7904 q^{11} -53.4418 q^{13} -146.872 q^{14} +39.5224 q^{16} +0.609730 q^{17} +94.5419 q^{19} +204.915 q^{22} -152.762 q^{23} +255.922 q^{26} +457.982 q^{28} +143.348 q^{29} +37.3789 q^{31} +76.3269 q^{32} -2.91988 q^{34} +274.029 q^{37} -452.743 q^{38} +273.492 q^{41} -185.362 q^{43} -638.974 q^{44} +731.548 q^{46} -438.013 q^{47} +597.639 q^{49} -798.027 q^{52} +211.483 q^{53} -1018.21 q^{56} -686.463 q^{58} -497.046 q^{59} -209.369 q^{61} -179.000 q^{62} -681.694 q^{64} -967.611 q^{67} +9.10487 q^{68} +401.080 q^{71} -97.6223 q^{73} -1312.27 q^{74} +1411.76 q^{76} -1312.38 q^{77} +1132.32 q^{79} -1309.70 q^{82} -573.219 q^{83} +887.663 q^{86} +1420.60 q^{88} +90.6612 q^{89} -1639.05 q^{91} -2281.14 q^{92} +2097.56 q^{94} +9.24640 q^{97} -2861.98 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.78880	−1.69310	−0.846549	−	0.532311i	$-0.821324\pi$
$2$	−4.78880	−1.69310	−0.846549	+	0.532311i	$0.821324\pi$
$3$	0	0
$3$	0	0
$4$	14.9326	1.86658
$5$	0	0
$5$	0	0
$6$	0	0
$7$	30.6698	1.65602	0.828008	−	0.560716i	$-0.189474\pi$
$7$	30.6698	1.65602	0.828008	+	0.560716i	$0.189474\pi$
$8$	−33.1990	−1.46720
$9$	0	0
$10$	0	0
$11$	−42.7904	−1.17289	−0.586445	−	0.809989i	$-0.699473\pi$
$11$	−42.7904	−1.17289	−0.586445	+	0.809989i	$0.699473\pi$
$12$	0	0
$13$	−53.4418	−1.14016	−0.570081	−	0.821589i	$-0.693088\pi$
$13$	−53.4418	−1.14016	−0.570081	+	0.821589i	$0.693088\pi$
$14$	−146.872	−2.80380
$15$	0	0
$16$	39.5224	0.617538
$17$	0.609730	0.00869890	0.00434945	−	0.999991i	$-0.498616\pi$
$17$	0.609730	0.00869890	0.00434945	+	0.999991i	$0.498616\pi$
$18$	0	0
$19$	94.5419	1.14155	0.570774	−	0.821107i	$-0.306643\pi$
$19$	94.5419	1.14155	0.570774	+	0.821107i	$0.306643\pi$
$20$	0	0
$21$	0	0
$22$	204.915	1.98582
$23$	−152.762	−1.38492	−0.692459	−	0.721457i	$-0.743473\pi$
$23$	−152.762	−1.38492	−0.692459	+	0.721457i	$0.743473\pi$
$24$	0	0
$25$	0	0
$26$	255.922	1.93040
$27$	0	0
$28$	457.982	3.09108
$29$	143.348	0.917896	0.458948	−	0.888463i	$-0.348227\pi$
$29$	143.348	0.917896	0.458948	+	0.888463i	$0.348227\pi$
$30$	0	0
$31$	37.3789	0.216563	0.108281	−	0.994120i	$-0.465465\pi$
$31$	37.3789	0.216563	0.108281	+	0.994120i	$0.465465\pi$
$32$	76.3269	0.421650
$33$	0	0
$34$	−2.91988	−0.0147281
$35$	0	0
$36$	0	0
$37$	274.029	1.21757	0.608785	−	0.793335i	$-0.291657\pi$
$37$	274.029	1.21757	0.608785	+	0.793335i	$0.291657\pi$
$38$	−452.743	−1.93275
$39$	0	0
$40$	0	0
$41$	273.492	1.04176	0.520881	−	0.853629i	$-0.325603\pi$
$41$	273.492	1.04176	0.520881	+	0.853629i	$0.325603\pi$
$42$	0	0
$43$	−185.362	−0.657383	−0.328691	−	0.944437i	$-0.606608\pi$
$43$	−185.362	−0.657383	−0.328691	+	0.944437i	$0.606608\pi$
$44$	−638.974	−2.18929
$45$	0	0
$46$	731.548	2.34480
$47$	−438.013	−1.35938	−0.679689	−	0.733500i	$-0.737885\pi$
$47$	−438.013	−1.35938	−0.679689	+	0.733500i	$0.737885\pi$
$48$	0	0
$49$	597.639	1.74239
$50$	0	0
$51$	0	0
$52$	−798.027	−2.12820
$53$	211.483	0.548103	0.274051	−	0.961715i	$-0.411636\pi$
$53$	211.483	0.548103	0.274051	+	0.961715i	$0.411636\pi$
$54$	0	0
$55$	0	0
$56$	−1018.21	−2.42971
$57$	0	0
$58$	−686.463	−1.55409
$59$	−497.046	−1.09678	−0.548389	−	0.836223i	$-0.684759\pi$
$59$	−497.046	−1.09678	−0.548389	+	0.836223i	$0.684759\pi$
$60$	0	0
$61$	−209.369	−0.439459	−0.219730	−	0.975561i	$-0.570517\pi$
$61$	−209.369	−0.439459	−0.219730	+	0.975561i	$0.570517\pi$
$62$	−179.000	−0.366662
$63$	0	0
$64$	−681.694	−1.33143
$65$	0	0
$66$	0	0
$67$	−967.611	−1.76437	−0.882183	−	0.470907i	$-0.843927\pi$
$67$	−967.611	−1.76437	−0.882183	+	0.470907i	$0.843927\pi$
$68$	9.10487	0.0162372
$69$	0	0
$70$	0	0
$71$	401.080	0.670415	0.335208	−	0.942144i	$-0.391194\pi$
$71$	401.080	0.670415	0.335208	+	0.942144i	$0.391194\pi$
$72$	0	0
$73$	−97.6223	−0.156518	−0.0782591	−	0.996933i	$-0.524936\pi$
$73$	−97.6223	−0.156518	−0.0782591	+	0.996933i	$0.524936\pi$
$74$	−1312.27	−2.06146
$75$	0	0
$76$	1411.76	2.13079
$77$	−1312.38	−1.94233
$78$	0	0
$79$	1132.32	1.61260	0.806302	−	0.591503i	$-0.201465\pi$
$79$	1132.32	1.61260	0.806302	+	0.591503i	$0.201465\pi$
$80$	0	0
$81$	0	0
$82$	−1309.70	−1.76381
$83$	−573.219	−0.758060	−0.379030	−	0.925384i	$-0.623742\pi$
$83$	−573.219	−0.758060	−0.379030	+	0.925384i	$0.623742\pi$
$84$	0	0
$85$	0	0
$86$	887.663	1.11301
$87$	0	0
$88$	1420.60	1.72087
$89$	90.6612	0.107978	0.0539891	−	0.998542i	$-0.482806\pi$
$89$	90.6612	0.107978	0.0539891	+	0.998542i	$0.482806\pi$
$90$	0	0
$91$	−1639.05	−1.88813
$92$	−2281.14	−2.58506
$93$	0	0
$94$	2097.56	2.30156
$95$	0	0
$96$	0	0
$97$	9.24640	0.00967866	0.00483933	−	0.999988i	$-0.498460\pi$
$97$	9.24640	0.00967866	0.00483933	+	0.999988i	$0.498460\pi$
$98$	−2861.98	−2.95003
$99$	0	0
$100$	0	0
$101$	217.074	0.213858	0.106929	−	0.994267i	$-0.465898\pi$
$101$	217.074	0.213858	0.106929	+	0.994267i	$0.465898\pi$
$102$	0	0
$103$	434.490	0.415646	0.207823	−	0.978166i	$-0.433362\pi$
$103$	434.490	0.415646	0.207823	+	0.978166i	$0.433362\pi$
$104$	1774.22	1.67285
$105$	0	0
$106$	−1012.75	−0.927992
$107$	193.254	0.174604	0.0873018	−	0.996182i	$-0.472176\pi$
$107$	193.254	0.174604	0.0873018	+	0.996182i	$0.472176\pi$
$108$	0	0
$109$	1703.14	1.49662	0.748308	−	0.663351i	$-0.230866\pi$
$109$	1703.14	1.49662	0.748308	+	0.663351i	$0.230866\pi$
$110$	0	0
$111$	0	0
$112$	1212.15	1.02265
$113$	−609.568	−0.507463	−0.253731	−	0.967275i	$-0.581658\pi$
$113$	−609.568	−0.507463	−0.253731	+	0.967275i	$0.581658\pi$
$114$	0	0
$115$	0	0
$116$	2140.56	1.71332
$117$	0	0
$118$	2380.26	1.85695
$119$	18.7003	0.0144055
$120$	0	0
$121$	500.020	0.375673
$122$	1002.63	0.744047
$123$	0	0
$124$	558.165	0.404232
$125$	0	0
$126$	0	0
$127$	−1372.33	−0.958857	−0.479429	−	0.877581i	$-0.659156\pi$
$127$	−1372.33	−0.958857	−0.479429	+	0.877581i	$0.659156\pi$
$128$	2653.88	1.83260
$129$	0	0
$130$	0	0
$131$	−1074.17	−0.716420	−0.358210	−	0.933641i	$-0.616613\pi$
$131$	−1074.17	−0.716420	−0.358210	+	0.933641i	$0.616613\pi$
$132$	0	0
$133$	2899.59	1.89042
$134$	4633.70	2.98724
$135$	0	0
$136$	−20.2424	−0.0127630
$137$	−1983.57	−1.23699	−0.618495	−	0.785788i	$-0.712257\pi$
$137$	−1983.57	−1.23699	−0.618495	+	0.785788i	$0.712257\pi$
$138$	0	0
$139$	−2050.98	−1.25152	−0.625761	−	0.780015i	$-0.715212\pi$
$139$	−2050.98	−1.25152	−0.625761	+	0.780015i	$0.715212\pi$
$140$	0	0
$141$	0	0
$142$	−1920.69	−1.13508
$143$	2286.80	1.33728
$144$	0	0
$145$	0	0
$146$	467.494	0.265001
$147$	0	0
$148$	4091.97	2.27269
$149$	580.288	0.319054	0.159527	−	0.987194i	$-0.449003\pi$
$149$	580.288	0.319054	0.159527	+	0.987194i	$0.449003\pi$
$150$	0	0
$151$	−1609.35	−0.867330	−0.433665	−	0.901074i	$-0.642780\pi$
$151$	−1609.35	−0.867330	−0.433665	+	0.901074i	$0.642780\pi$
$152$	−3138.70	−1.67488
$153$	0	0
$154$	6284.71	3.28855
$155$	0	0
$156$	0	0
$157$	−206.960	−0.105205	−0.0526026	−	0.998616i	$-0.516752\pi$
$157$	−206.960	−0.105205	−0.0526026	+	0.998616i	$0.516752\pi$
$158$	−5422.45	−2.73030
$159$	0	0
$160$	0	0
$161$	−4685.19	−2.29345
$162$	0	0
$163$	−2104.04	−1.01105	−0.505525	−	0.862812i	$-0.668701\pi$
$163$	−2104.04	−1.01105	−0.505525	+	0.862812i	$0.668701\pi$
$164$	4083.96	1.94453
$165$	0	0
$166$	2745.03	1.28347
$167$	−1034.85	−0.479517	−0.239758	−	0.970833i	$-0.577068\pi$
$167$	−1034.85	−0.479517	−0.239758	+	0.970833i	$0.577068\pi$
$168$	0	0
$169$	659.030	0.299968
$170$	0	0
$171$	0	0
$172$	−2767.94	−1.22706
$173$	3521.23	1.54748	0.773741	−	0.633502i	$-0.218383\pi$
$173$	3521.23	1.54748	0.773741	+	0.633502i	$0.218383\pi$
$174$	0	0
$175$	0	0
$176$	−1691.18	−0.724305
$177$	0	0
$178$	−434.158	−0.182818
$179$	−961.672	−0.401557	−0.200779	−	0.979637i	$-0.564347\pi$
$179$	−961.672	−0.401557	−0.200779	+	0.979637i	$0.564347\pi$
$180$	0	0
$181$	1493.35	0.613258	0.306629	−	0.951829i	$-0.400799\pi$
$181$	1493.35	0.613258	0.306629	+	0.951829i	$0.400799\pi$
$182$	7849.10	3.19678
$183$	0	0
$184$	5071.55	2.03195
$185$	0	0
$186$	0	0
$187$	−26.0906	−0.0102029
$188$	−6540.69	−2.53739
$189$	0	0
$190$	0	0
$191$	−1657.44	−0.627895	−0.313948	−	0.949440i	$-0.601652\pi$
$191$	−1657.44	−0.627895	−0.313948	+	0.949440i	$0.601652\pi$
$192$	0	0
$193$	−1016.77	−0.379215	−0.189608	−	0.981860i	$-0.560722\pi$
$193$	−1016.77	−0.379215	−0.189608	+	0.981860i	$0.560722\pi$
$194$	−44.2792	−0.0163869
$195$	0	0
$196$	8924.33	3.25231
$197$	4594.73	1.66173	0.830865	−	0.556475i	$-0.187846\pi$
$197$	4594.73	1.66173	0.830865	+	0.556475i	$0.187846\pi$
$198$	0	0
$199$	−832.746	−0.296642	−0.148321	−	0.988939i	$-0.547387\pi$
$199$	−832.746	−0.296642	−0.148321	+	0.988939i	$0.547387\pi$
$200$	0	0
$201$	0	0
$202$	−1039.53	−0.362083
$203$	4396.45	1.52005
$204$	0	0
$205$	0	0
$206$	−2080.69	−0.703729
$207$	0	0
$208$	−2112.15	−0.704093
$209$	−4045.49	−1.33891
$210$	0	0
$211$	2424.80	0.791137	0.395568	−	0.918437i	$-0.370548\pi$
$211$	2424.80	0.791137	0.395568	+	0.918437i	$0.370548\pi$
$212$	3158.00	1.02308
$213$	0	0
$214$	−925.456	−0.295621
$215$	0	0
$216$	0	0
$217$	1146.40	0.358631
$218$	−8156.00	−2.53392
$219$	0	0
$220$	0	0
$221$	−32.5851	−0.00991815
$222$	0	0
$223$	3010.78	0.904111	0.452056	−	0.891990i	$-0.350691\pi$
$223$	3010.78	0.904111	0.452056	+	0.891990i	$0.350691\pi$
$224$	2340.93	0.698260
$225$	0	0
$226$	2919.10	0.859184
$227$	79.6404	0.0232860	0.0116430	−	0.999932i	$-0.496294\pi$
$227$	79.6404	0.0232860	0.0116430	+	0.999932i	$0.496294\pi$
$228$	0	0
$229$	−2315.95	−0.668308	−0.334154	−	0.942519i	$-0.608450\pi$
$229$	−2315.95	−0.668308	−0.334154	+	0.942519i	$0.608450\pi$
$230$	0	0
$231$	0	0
$232$	−4759.00	−1.34674
$233$	−2055.15	−0.577842	−0.288921	−	0.957353i	$-0.593297\pi$
$233$	−2055.15	−0.577842	−0.288921	+	0.957353i	$0.593297\pi$
$234$	0	0
$235$	0	0
$236$	−7422.21	−2.04722
$237$	0	0
$238$	−89.5522	−0.0243899
$239$	4559.80	1.23410	0.617048	−	0.786926i	$-0.288329\pi$
$239$	4559.80	1.23410	0.617048	+	0.786926i	$0.288329\pi$
$240$	0	0
$241$	3893.23	1.04060	0.520301	−	0.853983i	$-0.325820\pi$
$241$	3893.23	1.04060	0.520301	+	0.853983i	$0.325820\pi$
$242$	−2394.50	−0.636050
$243$	0	0
$244$	−3126.44	−0.820285
$245$	0	0
$246$	0	0
$247$	−5052.49	−1.30155
$248$	−1240.94	−0.317742
$249$	0	0
$250$	0	0
$251$	−1560.52	−0.392426	−0.196213	−	0.980561i	$-0.562864\pi$
$251$	−1560.52	−0.392426	−0.196213	+	0.980561i	$0.562864\pi$
$252$	0	0
$253$	6536.76	1.62436
$254$	6571.83	1.62344
$255$	0	0
$256$	−7255.37	−1.77133
$257$	−6189.73	−1.50235	−0.751177	−	0.660101i	$-0.770513\pi$
$257$	−6189.73	−1.50235	−0.751177	+	0.660101i	$0.770513\pi$
$258$	0	0
$259$	8404.43	2.01632
$260$	0	0
$261$	0	0
$262$	5144.01	1.21297
$263$	−5952.06	−1.39551	−0.697757	−	0.716335i	$-0.745818\pi$
$263$	−5952.06	−1.39551	−0.697757	+	0.716335i	$0.745818\pi$
$264$	0	0
$265$	0	0
$266$	−13885.5	−3.20067
$267$	0	0
$268$	−14449.0	−3.29333
$269$	−5228.45	−1.18507	−0.592536	−	0.805544i	$-0.701873\pi$
$269$	−5228.45	−1.18507	−0.592536	+	0.805544i	$0.701873\pi$
$270$	0	0
$271$	−5868.44	−1.31543	−0.657717	−	0.753265i	$-0.728478\pi$
$271$	−5868.44	−1.31543	−0.657717	+	0.753265i	$0.728478\pi$
$272$	24.0980	0.00537190
$273$	0	0
$274$	9498.92	2.09435
$275$	0	0
$276$	0	0
$277$	1677.68	0.363907	0.181953	−	0.983307i	$-0.441758\pi$
$277$	1677.68	0.363907	0.181953	+	0.983307i	$0.441758\pi$
$278$	9821.72	2.11895
$279$	0	0
$280$	0	0
$281$	−1178.56	−0.250202	−0.125101	−	0.992144i	$-0.539925\pi$
$281$	−1178.56	−0.250202	−0.125101	+	0.992144i	$0.539925\pi$
$282$	0	0
$283$	3740.55	0.785699	0.392849	−	0.919603i	$-0.371489\pi$
$283$	3740.55	0.785699	0.392849	+	0.919603i	$0.371489\pi$
$284$	5989.19	1.25138
$285$	0	0
$286$	−10951.0	−2.26415
$287$	8387.96	1.72518
$288$	0	0
$289$	−4912.63	−0.999924
$290$	0	0
$291$	0	0
$292$	−1457.76	−0.292154
$293$	−2716.14	−0.541565	−0.270783	−	0.962641i	$-0.587282\pi$
$293$	−2716.14	−0.541565	−0.270783	+	0.962641i	$0.587282\pi$
$294$	0	0
$295$	0	0
$296$	−9097.49	−1.78642
$297$	0	0
$298$	−2778.88	−0.540189
$299$	8163.89	1.57903
$300$	0	0
$301$	−5685.03	−1.08864
$302$	7706.85	1.46847
$303$	0	0
$304$	3736.53	0.704949
$305$	0	0
$306$	0	0
$307$	4308.59	0.800992	0.400496	−	0.916299i	$-0.368838\pi$
$307$	4308.59	0.800992	0.400496	+	0.916299i	$0.368838\pi$
$308$	−19597.2	−3.62550
$309$	0	0
$310$	0	0
$311$	6306.22	1.14982	0.574908	−	0.818218i	$-0.305038\pi$
$311$	6306.22	1.14982	0.574908	+	0.818218i	$0.305038\pi$
$312$	0	0
$313$	4763.61	0.860239	0.430120	−	0.902772i	$-0.358471\pi$
$313$	4763.61	0.860239	0.430120	+	0.902772i	$0.358471\pi$
$314$	991.091	0.178123
$315$	0	0
$316$	16908.5	3.01005
$317$	1462.95	0.259203	0.129601	−	0.991566i	$-0.458630\pi$
$317$	1462.95	0.259203	0.129601	+	0.991566i	$0.458630\pi$
$318$	0	0
$319$	−6133.90	−1.07659
$320$	0	0
$321$	0	0
$322$	22436.5	3.88303
$323$	57.6450	0.00993020
$324$	0	0
$325$	0	0
$326$	10075.8	1.71181
$327$	0	0
$328$	−9079.66	−1.52848
$329$	−13433.8	−2.25115
$330$	0	0
$331$	−5777.17	−0.959342	−0.479671	−	0.877448i	$-0.659244\pi$
$331$	−5777.17	−0.959342	−0.479671	+	0.877448i	$0.659244\pi$
$332$	−8559.67	−1.41498
$333$	0	0
$334$	4955.71	0.811869
$335$	0	0
$336$	0	0
$337$	−2937.53	−0.474829	−0.237415	−	0.971408i	$-0.576300\pi$
$337$	−2937.53	−0.474829	−0.237415	+	0.971408i	$0.576300\pi$
$338$	−3155.97	−0.507875
$339$	0	0
$340$	0	0
$341$	−1599.46	−0.254004
$342$	0	0
$343$	7809.75	1.22941
$344$	6153.84	0.964514
$345$	0	0
$346$	−16862.5	−2.62004
$347$	6991.12	1.08157	0.540783	−	0.841162i	$-0.318128\pi$
$347$	6991.12	1.08157	0.540783	+	0.841162i	$0.318128\pi$
$348$	0	0
$349$	−5352.31	−0.820924	−0.410462	−	0.911878i	$-0.634633\pi$
$349$	−5352.31	−0.820924	−0.410462	+	0.911878i	$0.634633\pi$
$350$	0	0
$351$	0	0
$352$	−3266.06	−0.494550
$353$	−1034.16	−0.155928	−0.0779639	−	0.996956i	$-0.524842\pi$
$353$	−1034.16	−0.155928	−0.0779639	+	0.996956i	$0.524842\pi$
$354$	0	0
$355$	0	0
$356$	1353.81	0.201550
$357$	0	0
$358$	4605.26	0.679875
$359$	7870.36	1.15705	0.578526	−	0.815664i	$-0.303628\pi$
$359$	7870.36	1.15705	0.578526	+	0.815664i	$0.303628\pi$
$360$	0	0
$361$	2079.17	0.303130
$362$	−7151.35	−1.03831
$363$	0	0
$364$	−24475.4	−3.52434
$365$	0	0
$366$	0	0
$367$	−13251.7	−1.88483	−0.942416	−	0.334444i	$-0.891452\pi$
$367$	−13251.7	−1.88483	−0.942416	+	0.334444i	$0.891452\pi$
$368$	−6037.53	−0.855239
$369$	0	0
$370$	0	0
$371$	6486.16	0.907667
$372$	0	0
$373$	−2089.84	−0.290101	−0.145051	−	0.989424i	$-0.546334\pi$
$373$	−2089.84	−0.290101	−0.145051	+	0.989424i	$0.546334\pi$
$374$	124.943	0.0172744
$375$	0	0
$376$	14541.6	1.99448
$377$	−7660.76	−1.04655
$378$	0	0
$379$	−4197.61	−0.568909	−0.284455	−	0.958690i	$-0.591812\pi$
$379$	−4197.61	−0.568909	−0.284455	+	0.958690i	$0.591812\pi$
$380$	0	0
$381$	0	0
$382$	7937.14	1.06309
$383$	−7713.33	−1.02907	−0.514534	−	0.857470i	$-0.672035\pi$
$383$	−7713.33	−1.02907	−0.514534	+	0.857470i	$0.672035\pi$
$384$	0	0
$385$	0	0
$386$	4869.10	0.642048
$387$	0	0
$388$	138.073	0.0180660
$389$	−1143.59	−0.149055	−0.0745276	−	0.997219i	$-0.523745\pi$
$389$	−1143.59	−0.149055	−0.0745276	+	0.997219i	$0.523745\pi$
$390$	0	0
$391$	−93.1436	−0.0120473
$392$	−19841.0	−2.55644
$393$	0	0
$394$	−22003.2	−2.81347
$395$	0	0
$396$	0	0
$397$	−2122.39	−0.268311	−0.134156	−	0.990960i	$-0.542832\pi$
$397$	−2122.39	−0.268311	−0.134156	+	0.990960i	$0.542832\pi$
$398$	3987.86	0.502244
$399$	0	0
$400$	0	0
$401$	−329.564	−0.0410415	−0.0205208	−	0.999789i	$-0.506532\pi$
$401$	−329.564	−0.0410415	−0.0205208	+	0.999789i	$0.506532\pi$
$402$	0	0
$403$	−1997.60	−0.246917
$404$	3241.49	0.399183
$405$	0	0
$406$	−21053.7	−2.57359
$407$	−11725.8	−1.42808
$408$	0	0
$409$	−1187.60	−0.143577	−0.0717884	−	0.997420i	$-0.522871\pi$
$409$	−1187.60	−0.143577	−0.0717884	+	0.997420i	$0.522871\pi$
$410$	0	0
$411$	0	0
$412$	6488.07	0.775836
$413$	−15244.3	−1.81628
$414$	0	0
$415$	0	0
$416$	−4079.05	−0.480750
$417$	0	0
$418$	19373.0	2.26691
$419$	11973.7	1.39606	0.698032	−	0.716066i	$-0.254059\pi$
$419$	11973.7	1.39606	0.698032	+	0.716066i	$0.254059\pi$
$420$	0	0
$421$	−8636.46	−0.999799	−0.499900	−	0.866083i	$-0.666630\pi$
$421$	−8636.46	−0.999799	−0.499900	+	0.866083i	$0.666630\pi$
$422$	−11611.9	−1.33947
$423$	0	0
$424$	−7021.03	−0.804178
$425$	0	0
$426$	0	0
$427$	−6421.33	−0.727751
$428$	2885.79	0.325911
$429$	0	0
$430$	0	0
$431$	4589.22	0.512889	0.256445	−	0.966559i	$-0.417449\pi$
$431$	4589.22	0.512889	0.256445	+	0.966559i	$0.417449\pi$
$432$	0	0
$433$	−16279.0	−1.80674	−0.903369	−	0.428864i	$-0.858914\pi$
$433$	−16279.0	−1.80674	−0.903369	+	0.428864i	$0.858914\pi$
$434$	−5489.91	−0.607198
$435$	0	0
$436$	25432.4	2.79355
$437$	−14442.4	−1.58095
$438$	0	0
$439$	−10523.0	−1.14404	−0.572020	−	0.820240i	$-0.693840\pi$
$439$	−10523.0	−1.14404	−0.572020	+	0.820240i	$0.693840\pi$
$440$	0	0
$441$	0	0
$442$	156.044	0.0167924
$443$	15757.9	1.69003	0.845014	−	0.534744i	$-0.179592\pi$
$443$	15757.9	1.69003	0.845014	+	0.534744i	$0.179592\pi$
$444$	0	0
$445$	0	0
$446$	−14418.0	−1.53075
$447$	0	0
$448$	−20907.4	−2.20487
$449$	−10964.2	−1.15241	−0.576206	−	0.817305i	$-0.695467\pi$
$449$	−10964.2	−1.15241	−0.576206	+	0.817305i	$0.695467\pi$
$450$	0	0
$451$	−11702.8	−1.22187
$452$	−9102.45	−0.947220
$453$	0	0
$454$	−381.382	−0.0394255
$455$	0	0
$456$	0	0
$457$	−13478.3	−1.37962	−0.689811	−	0.723989i	$-0.742307\pi$
$457$	−13478.3	−1.37962	−0.689811	+	0.723989i	$0.742307\pi$
$458$	11090.6	1.13151
$459$	0	0
$460$	0	0
$461$	10358.3	1.04650	0.523248	−	0.852181i	$-0.324720\pi$
$461$	10358.3	1.04650	0.523248	+	0.852181i	$0.324720\pi$
$462$	0	0
$463$	−14398.6	−1.44527	−0.722634	−	0.691231i	$-0.757069\pi$
$463$	−14398.6	−1.44527	−0.722634	+	0.691231i	$0.757069\pi$
$464$	5665.45	0.566836
$465$	0	0
$466$	9841.70	0.978343
$467$	−18838.8	−1.86671	−0.933356	−	0.358953i	$-0.883134\pi$
$467$	−18838.8	−1.86671	−0.933356	+	0.358953i	$0.883134\pi$
$468$	0	0
$469$	−29676.5	−2.92182
$470$	0	0
$471$	0	0
$472$	16501.4	1.60920
$473$	7931.72	0.771038
$474$	0	0
$475$	0	0
$476$	279.245	0.0268890
$477$	0	0
$478$	−21836.0	−2.08944
$479$	−6363.92	−0.607046	−0.303523	−	0.952824i	$-0.598163\pi$
$479$	−6363.92	−0.607046	−0.303523	+	0.952824i	$0.598163\pi$
$480$	0	0
$481$	−14644.6	−1.38823
$482$	−18643.9	−1.76184
$483$	0	0
$484$	7466.62	0.701223
$485$	0	0
$486$	0	0
$487$	−1556.53	−0.144832	−0.0724158	−	0.997375i	$-0.523071\pi$
$487$	−1556.53	−0.144832	−0.0724158	+	0.997375i	$0.523071\pi$
$488$	6950.86	0.644776
$489$	0	0
$490$	0	0
$491$	13634.7	1.25321	0.626605	−	0.779337i	$-0.284444\pi$
$491$	13634.7	1.25321	0.626605	+	0.779337i	$0.284444\pi$
$492$	0	0
$493$	87.4033	0.00798468
$494$	24195.4	2.20365
$495$	0	0
$496$	1477.30	0.133736
$497$	12301.1	1.11022
$498$	0	0
$499$	2264.91	0.203189	0.101594	−	0.994826i	$-0.467606\pi$
$499$	2264.91	0.203189	0.101594	+	0.994826i	$0.467606\pi$
$500$	0	0
$501$	0	0
$502$	7473.01	0.664416
$503$	−20004.6	−1.77329	−0.886643	−	0.462455i	$-0.846969\pi$
$503$	−20004.6	−1.77329	−0.886643	+	0.462455i	$0.846969\pi$
$504$	0	0
$505$	0	0
$506$	−31303.2	−2.75019
$507$	0	0
$508$	−20492.5	−1.78978
$509$	723.933	0.0630408	0.0315204	−	0.999503i	$-0.489965\pi$
$509$	723.933	0.0630408	0.0315204	+	0.999503i	$0.489965\pi$
$510$	0	0
$511$	−2994.06	−0.259197
$512$	13513.5	1.16644
$513$	0	0
$514$	29641.4	2.54363
$515$	0	0
$516$	0	0
$517$	18742.8	1.59440
$518$	−40247.1	−3.41382
$519$	0	0
$520$	0	0
$521$	2150.30	0.180818	0.0904090	−	0.995905i	$-0.471183\pi$
$521$	2150.30	0.180818	0.0904090	+	0.995905i	$0.471183\pi$
$522$	0	0
$523$	−12696.4	−1.06152	−0.530760	−	0.847522i	$-0.678093\pi$
$523$	−12696.4	−1.06152	−0.530760	+	0.847522i	$0.678093\pi$
$524$	−16040.2	−1.33725
$525$	0	0
$526$	28503.3	2.36274
$527$	22.7910	0.00188386
$528$	0	0
$529$	11169.3	0.917996
$530$	0	0
$531$	0	0
$532$	43298.4	3.52862
$533$	−14615.9	−1.18778
$534$	0	0
$535$	0	0
$536$	32123.7	2.58868
$537$	0	0
$538$	25038.0	2.00644
$539$	−25573.2	−2.04363
$540$	0	0
$541$	−12233.1	−0.972167	−0.486083	−	0.873912i	$-0.661575\pi$
$541$	−12233.1	−0.972167	−0.486083	+	0.873912i	$0.661575\pi$
$542$	28102.8	2.22716
$543$	0	0
$544$	46.5388	0.00366789
$545$	0	0
$546$	0	0
$547$	18086.3	1.41374	0.706868	−	0.707345i	$-0.250107\pi$
$547$	18086.3	1.41374	0.706868	+	0.707345i	$0.250107\pi$
$548$	−29619.9	−2.30894
$549$	0	0
$550$	0	0
$551$	13552.3	1.04782
$552$	0	0
$553$	34728.0	2.67050
$554$	−8034.09	−0.616129
$555$	0	0
$556$	−30626.5	−2.33607
$557$	9106.96	0.692773	0.346386	−	0.938092i	$-0.387409\pi$
$557$	9106.96	0.692773	0.346386	+	0.938092i	$0.387409\pi$
$558$	0	0
$559$	9906.09	0.749523
$560$	0	0
$561$	0	0
$562$	5643.87	0.423616
$563$	14711.8	1.10129	0.550647	−	0.834738i	$-0.314381\pi$
$563$	14711.8	1.10129	0.550647	+	0.834738i	$0.314381\pi$
$564$	0	0
$565$	0	0
$566$	−17912.8	−1.33026
$567$	0	0
$568$	−13315.5	−0.983635
$569$	−1973.65	−0.145412	−0.0727062	−	0.997353i	$-0.523164\pi$
$569$	−1973.65	−0.145412	−0.0727062	+	0.997353i	$0.523164\pi$
$570$	0	0
$571$	−26194.0	−1.91976	−0.959881	−	0.280407i	$-0.909531\pi$
$571$	−26194.0	−1.91976	−0.959881	+	0.280407i	$0.909531\pi$
$572$	34147.9	2.49615
$573$	0	0
$574$	−40168.3	−2.92089
$575$	0	0
$576$	0	0
$577$	−8674.57	−0.625870	−0.312935	−	0.949775i	$-0.601312\pi$
$577$	−8674.57	−0.625870	−0.312935	+	0.949775i	$0.601312\pi$
$578$	23525.6	1.69297
$579$	0	0
$580$	0	0
$581$	−17580.5	−1.25536
$582$	0	0
$583$	−9049.46	−0.642865
$584$	3240.96	0.229644
$585$	0	0
$586$	13007.1	0.916923
$587$	−24383.8	−1.71453	−0.857263	−	0.514879i	$-0.827837\pi$
$587$	−24383.8	−1.71453	−0.857263	+	0.514879i	$0.827837\pi$
$588$	0	0
$589$	3533.87	0.247217
$590$	0	0
$591$	0	0
$592$	10830.3	0.751896
$593$	12980.5	0.898895	0.449447	−	0.893307i	$-0.351621\pi$
$593$	12980.5	0.898895	0.449447	+	0.893307i	$0.351621\pi$
$594$	0	0
$595$	0	0
$596$	8665.22	0.595539
$597$	0	0
$598$	−39095.3	−2.67345
$599$	25004.7	1.70562	0.852808	−	0.522224i	$-0.174897\pi$
$599$	25004.7	1.70562	0.852808	+	0.522224i	$0.174897\pi$
$600$	0	0
$601$	8309.00	0.563946	0.281973	−	0.959422i	$-0.409011\pi$
$601$	8309.00	0.563946	0.281973	+	0.959422i	$0.409011\pi$
$602$	27224.5	1.84317
$603$	0	0
$604$	−24031.8	−1.61894
$605$	0	0
$606$	0	0
$607$	23783.6	1.59035	0.795177	−	0.606377i	$-0.207378\pi$
$607$	23783.6	1.59035	0.795177	+	0.606377i	$0.207378\pi$
$608$	7216.09	0.481334
$609$	0	0
$610$	0	0
$611$	23408.2	1.54991
$612$	0	0
$613$	13900.8	0.915903	0.457951	−	0.888977i	$-0.348583\pi$
$613$	13900.8	0.915903	0.457951	+	0.888977i	$0.348583\pi$
$614$	−20633.0	−1.35616
$615$	0	0
$616$	43569.6	2.84979
$617$	20201.5	1.31812	0.659061	−	0.752089i	$-0.270954\pi$
$617$	20201.5	1.31812	0.659061	+	0.752089i	$0.270954\pi$
$618$	0	0
$619$	11440.9	0.742890	0.371445	−	0.928455i	$-0.378862\pi$
$619$	11440.9	0.742890	0.371445	+	0.928455i	$0.378862\pi$
$620$	0	0
$621$	0	0
$622$	−30199.2	−1.94675
$623$	2780.56	0.178814
$624$	0	0
$625$	0	0
$626$	−22812.0	−1.45647
$627$	0	0
$628$	−3090.46	−0.196374
$629$	167.084	0.0105915
$630$	0	0
$631$	−7670.95	−0.483955	−0.241978	−	0.970282i	$-0.577796\pi$
$631$	−7670.95	−0.483955	−0.241978	+	0.970282i	$0.577796\pi$
$632$	−37591.9	−2.36602
$633$	0	0
$634$	−7005.76	−0.438855
$635$	0	0
$636$	0	0
$637$	−31938.9	−1.98660
$638$	29374.0	1.82277
$639$	0	0
$640$	0	0
$641$	−23983.9	−1.47786	−0.738930	−	0.673783i	$-0.764668\pi$
$641$	−23983.9	−1.47786	−0.738930	+	0.673783i	$0.764668\pi$
$642$	0	0
$643$	−28590.4	−1.75349	−0.876747	−	0.480951i	$-0.840291\pi$
$643$	−28590.4	−1.75349	−0.876747	+	0.480951i	$0.840291\pi$
$644$	−69962.2	−4.28090
$645$	0	0
$646$	−276.051	−0.0168128
$647$	−25703.8	−1.56186	−0.780928	−	0.624621i	$-0.785254\pi$
$647$	−25703.8	−1.56186	−0.780928	+	0.624621i	$0.785254\pi$
$648$	0	0
$649$	21268.8	1.28640
$650$	0	0
$651$	0	0
$652$	−31418.9	−1.88721
$653$	−3643.22	−0.218331	−0.109165	−	0.994024i	$-0.534818\pi$
$653$	−3643.22	−0.218331	−0.109165	+	0.994024i	$0.534818\pi$
$654$	0	0
$655$	0	0
$656$	10809.1	0.643328
$657$	0	0
$658$	64331.8	3.81142
$659$	19410.3	1.14737	0.573687	−	0.819075i	$-0.305513\pi$
$659$	19410.3	1.14737	0.573687	+	0.819075i	$0.305513\pi$
$660$	0	0
$661$	−21227.3	−1.24909	−0.624544	−	0.780989i	$-0.714715\pi$
$661$	−21227.3	−1.24909	−0.624544	+	0.780989i	$0.714715\pi$
$662$	27665.7	1.62426
$663$	0	0
$664$	19030.3	1.11223
$665$	0	0
$666$	0	0
$667$	−21898.1	−1.27121
$668$	−15453.1	−0.895056
$669$	0	0
$670$	0	0
$671$	8959.00	0.515437
$672$	0	0
$673$	14233.6	0.815253	0.407626	−	0.913149i	$-0.366357\pi$
$673$	14233.6	0.815253	0.407626	+	0.913149i	$0.366357\pi$
$674$	14067.3	0.803932
$675$	0	0
$676$	9841.05	0.559914
$677$	16924.2	0.960782	0.480391	−	0.877054i	$-0.340495\pi$
$677$	16924.2	0.960782	0.480391	+	0.877054i	$0.340495\pi$
$678$	0	0
$679$	283.586	0.0160280
$680$	0	0
$681$	0	0
$682$	7659.49	0.430054
$683$	−3582.98	−0.200730	−0.100365	−	0.994951i	$-0.532001\pi$
$683$	−3582.98	−0.200730	−0.100365	+	0.994951i	$0.532001\pi$
$684$	0	0
$685$	0	0
$686$	−37399.3	−2.08151
$687$	0	0
$688$	−7325.96	−0.405959
$689$	−11302.1	−0.624926
$690$	0	0
$691$	−17858.2	−0.983154	−0.491577	−	0.870834i	$-0.663579\pi$
$691$	−17858.2	−0.983154	−0.491577	+	0.870834i	$0.663579\pi$
$692$	52581.3	2.88850
$693$	0	0
$694$	−33479.1	−1.83120
$695$	0	0
$696$	0	0
$697$	166.756	0.00906219
$698$	25631.2	1.38990
$699$	0	0
$700$	0	0
$701$	19796.1	1.06660	0.533302	−	0.845925i	$-0.320951\pi$
$701$	19796.1	1.06660	0.533302	+	0.845925i	$0.320951\pi$
$702$	0	0
$703$	25907.2	1.38991
$704$	29170.0	1.56163
$705$	0	0
$706$	4952.36	0.264001
$707$	6657.63	0.354153
$708$	0	0
$709$	−16290.7	−0.862921	−0.431460	−	0.902132i	$-0.642002\pi$
$709$	−16290.7	−0.862921	−0.431460	+	0.902132i	$0.642002\pi$
$710$	0	0
$711$	0	0
$712$	−3009.86	−0.158426
$713$	−5710.08	−0.299922
$714$	0	0
$715$	0	0
$716$	−14360.3	−0.749538
$717$	0	0
$718$	−37689.6	−1.95900
$719$	8647.80	0.448551	0.224276	−	0.974526i	$-0.427998\pi$
$719$	8647.80	0.448551	0.224276	+	0.974526i	$0.427998\pi$
$720$	0	0
$721$	13325.7	0.688316
$722$	−9956.74	−0.513229
$723$	0	0
$724$	22299.6	1.14470
$725$	0	0
$726$	0	0
$727$	−10580.0	−0.539740	−0.269870	−	0.962897i	$-0.586981\pi$
$727$	−10580.0	−0.539740	−0.269870	+	0.962897i	$0.586981\pi$
$728$	54414.9	2.77026
$729$	0	0
$730$	0	0
$731$	−113.021	−0.00571850
$732$	0	0
$733$	28197.5	1.42087	0.710435	−	0.703763i	$-0.248498\pi$
$733$	28197.5	1.42087	0.710435	+	0.703763i	$0.248498\pi$
$734$	63459.8	3.19120
$735$	0	0
$736$	−11659.9	−0.583951
$737$	41404.5	2.06941
$738$	0	0
$739$	−27120.4	−1.34998	−0.674992	−	0.737825i	$-0.735853\pi$
$739$	−27120.4	−1.34998	−0.674992	+	0.737825i	$0.735853\pi$
$740$	0	0
$741$	0	0
$742$	−31060.9	−1.53677
$743$	−12665.4	−0.625366	−0.312683	−	0.949857i	$-0.601228\pi$
$743$	−12665.4	−0.625366	−0.312683	+	0.949857i	$0.601228\pi$
$744$	0	0
$745$	0	0
$746$	10007.8	0.491169
$747$	0	0
$748$	−389.601	−0.0190444
$749$	5927.07	0.289146
$750$	0	0
$751$	8649.25	0.420261	0.210130	−	0.977673i	$-0.432611\pi$
$751$	8649.25	0.420261	0.210130	+	0.977673i	$0.432611\pi$
$752$	−17311.4	−0.839468
$753$	0	0
$754$	36685.9	1.77191
$755$	0	0
$756$	0	0
$757$	11715.6	0.562496	0.281248	−	0.959635i	$-0.409252\pi$
$757$	11715.6	0.562496	0.281248	+	0.959635i	$0.409252\pi$
$758$	20101.5	0.963219
$759$	0	0
$760$	0	0
$761$	36513.7	1.73932	0.869658	−	0.493655i	$-0.164340\pi$
$761$	36513.7	1.73932	0.869658	+	0.493655i	$0.164340\pi$
$762$	0	0
$763$	52235.0	2.47842
$764$	−24749.9	−1.17202
$765$	0	0
$766$	36937.6	1.74231
$767$	26563.1	1.25050
$768$	0	0
$769$	−22919.5	−1.07477	−0.537386	−	0.843337i	$-0.680588\pi$
$769$	−22919.5	−1.07477	−0.537386	+	0.843337i	$0.680588\pi$
$770$	0	0
$771$	0	0
$772$	−15183.0	−0.707835
$773$	5112.86	0.237900	0.118950	−	0.992900i	$-0.462047\pi$
$773$	5112.86	0.237900	0.118950	+	0.992900i	$0.462047\pi$
$774$	0	0
$775$	0	0
$776$	−306.971	−0.0142006
$777$	0	0
$778$	5476.44	0.252365
$779$	25856.5	1.18922
$780$	0	0
$781$	−17162.4	−0.786324
$782$	446.046	0.0203972
$783$	0	0
$784$	23620.2	1.07599
$785$	0	0
$786$	0	0
$787$	−34588.7	−1.56665	−0.783324	−	0.621613i	$-0.786478\pi$
$787$	−34588.7	−1.56665	−0.783324	+	0.621613i	$0.786478\pi$
$788$	68611.4	3.10175
$789$	0	0
$790$	0	0
$791$	−18695.3	−0.840367
$792$	0	0
$793$	11189.1	0.501054
$794$	10163.7	0.454277
$795$	0	0
$796$	−12435.1	−0.553706
$797$	−17278.0	−0.767901	−0.383951	−	0.923354i	$-0.625437\pi$
$797$	−17278.0	−0.767901	−0.383951	+	0.923354i	$0.625437\pi$
$798$	0	0
$799$	−267.070	−0.0118251
$800$	0	0
$801$	0	0
$802$	1578.22	0.0694873
$803$	4177.30	0.183579
$804$	0	0
$805$	0	0
$806$	9566.10	0.418054
$807$	0	0
$808$	−7206.65	−0.313773
$809$	−35591.8	−1.54677	−0.773387	−	0.633934i	$-0.781439\pi$
$809$	−35591.8	−1.54677	−0.773387	+	0.633934i	$0.781439\pi$
$810$	0	0
$811$	−23814.4	−1.03112	−0.515558	−	0.856854i	$-0.672415\pi$
$811$	−23814.4	−1.03112	−0.515558	+	0.856854i	$0.672415\pi$
$812$	65650.5	2.83729
$813$	0	0
$814$	56152.6	2.41787
$815$	0	0
$816$	0	0
$817$	−17524.5	−0.750434
$818$	5687.17	0.243089
$819$	0	0
$820$	0	0
$821$	−7965.64	−0.338615	−0.169307	−	0.985563i	$-0.554153\pi$
$821$	−7965.64	−0.338615	−0.169307	+	0.985563i	$0.554153\pi$
$822$	0	0
$823$	35484.1	1.50291	0.751457	−	0.659782i	$-0.229351\pi$
$823$	35484.1	1.50291	0.751457	+	0.659782i	$0.229351\pi$
$824$	−14424.6	−0.609837
$825$	0	0
$826$	73002.1	3.07514
$827$	−37867.6	−1.59224	−0.796122	−	0.605136i	$-0.793119\pi$
$827$	−37867.6	−1.59224	−0.796122	+	0.605136i	$0.793119\pi$
$828$	0	0
$829$	10136.2	0.424662	0.212331	−	0.977198i	$-0.431894\pi$
$829$	10136.2	0.424662	0.212331	+	0.977198i	$0.431894\pi$
$830$	0	0
$831$	0	0
$832$	36431.0	1.51805
$833$	364.399	0.0151569
$834$	0	0
$835$	0	0
$836$	−60409.8	−2.49918
$837$	0	0
$838$	−57339.5	−2.36367
$839$	−31403.4	−1.29221	−0.646106	−	0.763248i	$-0.723604\pi$
$839$	−31403.4	−1.29221	−0.646106	+	0.763248i	$0.723604\pi$
$840$	0	0
$841$	−3840.48	−0.157468
$842$	41358.3	1.69276
$843$	0	0
$844$	36208.6	1.47672
$845$	0	0
$846$	0	0
$847$	15335.5	0.622120
$848$	8358.33	0.338474
$849$	0	0
$850$	0	0
$851$	−41861.2	−1.68623
$852$	0	0
$853$	5563.39	0.223314	0.111657	−	0.993747i	$-0.464384\pi$
$853$	5563.39	0.223314	0.111657	+	0.993747i	$0.464384\pi$
$854$	30750.5	1.23215
$855$	0	0
$856$	−6415.85	−0.256179
$857$	−22710.1	−0.905205	−0.452603	−	0.891712i	$-0.649504\pi$
$857$	−22710.1	−0.905205	−0.452603	+	0.891712i	$0.649504\pi$
$858$	0	0
$859$	34783.7	1.38161	0.690807	−	0.723039i	$-0.257255\pi$
$859$	34783.7	1.38161	0.690807	+	0.723039i	$0.257255\pi$
$860$	0	0
$861$	0	0
$862$	−21976.9	−0.868371
$863$	19361.6	0.763705	0.381853	−	0.924223i	$-0.375286\pi$
$863$	19361.6	0.763705	0.381853	+	0.924223i	$0.375286\pi$
$864$	0	0
$865$	0	0
$866$	77956.8	3.05898
$867$	0	0
$868$	17118.8	0.669414
$869$	−48452.4	−1.89141
$870$	0	0
$871$	51710.9	2.01166
$872$	−56542.6	−2.19584
$873$	0	0
$874$	69161.9	2.67670
$875$	0	0
$876$	0	0
$877$	2827.10	0.108853	0.0544266	−	0.998518i	$-0.482667\pi$
$877$	2827.10	0.108853	0.0544266	+	0.998518i	$0.482667\pi$
$878$	50392.4	1.93697
$879$	0	0
$880$	0	0
$881$	−3106.28	−0.118789	−0.0593946	−	0.998235i	$-0.518917\pi$
$881$	−3106.28	−0.118789	−0.0593946	+	0.998235i	$0.518917\pi$
$882$	0	0
$883$	15219.6	0.580045	0.290023	−	0.957020i	$-0.406337\pi$
$883$	15219.6	0.580045	0.290023	+	0.957020i	$0.406337\pi$
$884$	−486.581	−0.0185130
$885$	0	0
$886$	−75461.7	−2.86138
$887$	−26172.9	−0.990755	−0.495377	−	0.868678i	$-0.664970\pi$
$887$	−26172.9	−0.990755	−0.495377	+	0.868678i	$0.664970\pi$
$888$	0	0
$889$	−42089.2	−1.58788
$890$	0	0
$891$	0	0
$892$	44958.9	1.68760
$893$	−41410.6	−1.55180
$894$	0	0
$895$	0	0
$896$	81394.2	3.03481
$897$	0	0
$898$	52505.4	1.95114
$899$	5358.17	0.198782
$900$	0	0
$901$	128.948	0.00476789
$902$	56042.6	2.06875
$903$	0	0
$904$	20237.0	0.744551
$905$	0	0
$906$	0	0
$907$	20025.9	0.733130	0.366565	−	0.930392i	$-0.380534\pi$
$907$	20025.9	0.733130	0.366565	+	0.930392i	$0.380534\pi$
$908$	1189.24	0.0434651
$909$	0	0
$910$	0	0
$911$	4864.87	0.176927	0.0884633	−	0.996079i	$-0.471804\pi$
$911$	4864.87	0.176927	0.0884633	+	0.996079i	$0.471804\pi$
$912$	0	0
$913$	24528.3	0.889121
$914$	64544.9	2.33584
$915$	0	0
$916$	−34583.3	−1.24745
$917$	−32944.8	−1.18640
$918$	0	0
$919$	16242.7	0.583021	0.291510	−	0.956568i	$-0.405842\pi$
$919$	16242.7	0.583021	0.291510	+	0.956568i	$0.405842\pi$
$920$	0	0
$921$	0	0
$922$	−49603.9	−1.77182
$923$	−21434.5	−0.764382
$924$	0	0
$925$	0	0
$926$	68952.0	2.44698
$927$	0	0
$928$	10941.3	0.387031
$929$	155.698	0.00549868	0.00274934	−	0.999996i	$-0.499125\pi$
$929$	155.698	0.00549868	0.00274934	+	0.999996i	$0.499125\pi$
$930$	0	0
$931$	56502.0	1.98902
$932$	−30688.8	−1.07859
$933$	0	0
$934$	90215.2	3.16052
$935$	0	0
$936$	0	0
$937$	−27195.9	−0.948186	−0.474093	−	0.880475i	$-0.657224\pi$
$937$	−27195.9	−0.948186	−0.474093	+	0.880475i	$0.657224\pi$
$938$	142115.	4.94692
$939$	0	0
$940$	0	0
$941$	−37140.0	−1.28664	−0.643321	−	0.765597i	$-0.722444\pi$
$941$	−37140.0	−1.28664	−0.643321	+	0.765597i	$0.722444\pi$
$942$	0	0
$943$	−41779.2	−1.44276
$944$	−19644.5	−0.677303
$945$	0	0
$946$	−37983.5	−1.30544
$947$	−45751.9	−1.56994	−0.784972	−	0.619532i	$-0.787322\pi$
$947$	−45751.9	−1.56994	−0.784972	+	0.619532i	$0.787322\pi$
$948$	0	0
$949$	5217.12	0.178456
$950$	0	0
$951$	0	0
$952$	−620.832	−0.0211358
$953$	−9011.61	−0.306311	−0.153156	−	0.988202i	$-0.548944\pi$
$953$	−9011.61	−0.306311	−0.153156	+	0.988202i	$0.548944\pi$
$954$	0	0
$955$	0	0
$956$	68089.8	2.30354
$957$	0	0
$958$	30475.5	1.02779
$959$	−60835.7	−2.04848
$960$	0	0
$961$	−28393.8	−0.953101
$962$	70130.2	2.35040
$963$	0	0
$964$	58136.2	1.94237
$965$	0	0
$966$	0	0
$967$	−28396.0	−0.944316	−0.472158	−	0.881514i	$-0.656525\pi$
$967$	−28396.0	−0.944316	−0.472158	+	0.881514i	$0.656525\pi$
$968$	−16600.2	−0.551188
$969$	0	0
$970$	0	0
$971$	10379.7	0.343048	0.171524	−	0.985180i	$-0.445131\pi$
$971$	10379.7	0.343048	0.171524	+	0.985180i	$0.445131\pi$
$972$	0	0
$973$	−62903.1	−2.07254
$974$	7453.90	0.245214
$975$	0	0
$976$	−8274.79	−0.271383
$977$	−31631.7	−1.03581	−0.517905	−	0.855438i	$-0.673288\pi$
$977$	−31631.7	−1.03581	−0.517905	+	0.855438i	$0.673288\pi$
$978$	0	0
$979$	−3879.43	−0.126647
$980$	0	0
$981$	0	0
$982$	−65293.9	−2.12181
$983$	25294.3	0.820715	0.410358	−	0.911925i	$-0.365404\pi$
$983$	25294.3	0.820715	0.410358	+	0.911925i	$0.365404\pi$
$984$	0	0
$985$	0	0
$986$	−418.557	−0.0135188
$987$	0	0
$988$	−75447.0	−2.42944
$989$	28316.3	0.910421
$990$	0	0
$991$	−20172.4	−0.646617	−0.323308	−	0.946294i	$-0.604795\pi$
$991$	−20172.4	−0.646617	−0.323308	+	0.946294i	$0.604795\pi$
$992$	2853.01	0.0913138
$993$	0	0
$994$	−58907.4	−1.87971
$995$	0	0
$996$	0	0
$997$	56355.2	1.79016	0.895079	−	0.445909i	$-0.147119\pi$
$997$	56355.2	1.79016	0.895079	+	0.445909i	$0.147119\pi$
$998$	−10846.2	−0.344018
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.be.1.2		12
3.2	odd	2		2025.4.a.bi.1.11		12
5.4	even	2		2025.4.a.bj.1.11		12
9.2	odd	6		225.4.e.e.76.2	✓	24
9.5	odd	6		225.4.e.e.151.2	yes	24
15.14	odd	2		2025.4.a.bf.1.2		12
45.2	even	12		225.4.k.e.49.22		48
45.14	odd	6		225.4.e.f.151.11	yes	24
45.23	even	12		225.4.k.e.124.22		48
45.29	odd	6		225.4.e.f.76.11	yes	24
45.32	even	12		225.4.k.e.124.3		48
45.38	even	12		225.4.k.e.49.3		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.e.e.76.2	✓	24	9.2	odd	6
225.4.e.e.151.2	yes	24	9.5	odd	6
225.4.e.f.76.11	yes	24	45.29	odd	6
225.4.e.f.151.11	yes	24	45.14	odd	6
225.4.k.e.49.3		48	45.38	even	12
225.4.k.e.49.22		48	45.2	even	12
225.4.k.e.124.3		48	45.32	even	12
225.4.k.e.124.22		48	45.23	even	12
2025.4.a.be.1.2		12	1.1	even	1	trivial
2025.4.a.bf.1.2		12	15.14	odd	2
2025.4.a.bi.1.11		12	3.2	odd	2
2025.4.a.bj.1.11		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)

\(f(q)\)	\(=\)	\(q-4.78880 q^{2} +14.9326 q^{4} +30.6698 q^{7} -33.1990 q^{8} +O(q^{10})\)	\(q-4.78880 q^{2} +14.9326 q^{4} +30.6698 q^{7} -33.1990 q^{8} -42.7904 q^{11} -53.4418 q^{13} -146.872 q^{14} +39.5224 q^{16} +0.609730 q^{17} +94.5419 q^{19} +204.915 q^{22} -152.762 q^{23} +255.922 q^{26} +457.982 q^{28} +143.348 q^{29} +37.3789 q^{31} +76.3269 q^{32} -2.91988 q^{34} +274.029 q^{37} -452.743 q^{38} +273.492 q^{41} -185.362 q^{43} -638.974 q^{44} +731.548 q^{46} -438.013 q^{47} +597.639 q^{49} -798.027 q^{52} +211.483 q^{53} -1018.21 q^{56} -686.463 q^{58} -497.046 q^{59} -209.369 q^{61} -179.000 q^{62} -681.694 q^{64} -967.611 q^{67} +9.10487 q^{68} +401.080 q^{71} -97.6223 q^{73} -1312.27 q^{74} +1411.76 q^{76} -1312.38 q^{77} +1132.32 q^{79} -1309.70 q^{82} -573.219 q^{83} +887.663 q^{86} +1420.60 q^{88} +90.6612 q^{89} -1639.05 q^{91} -2281.14 q^{92} +2097.56 q^{94} +9.24640 q^{97} -2861.98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) \) 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

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