LMFDB - Embedded newform 2025.4.a.be.1.10

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			1.10
Root			$-3.81662$ 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)

$f(q)$	$=$	$q+3.81662 q^{2} +6.56661 q^{4} -9.98804 q^{7} -5.47071 q^{8} +O(q^{10})$	$q+3.81662 q^{2} +6.56661 q^{4} -9.98804 q^{7} -5.47071 q^{8} +27.2172 q^{11} -11.9334 q^{13} -38.1206 q^{14} -73.4125 q^{16} +69.0588 q^{17} -99.8945 q^{19} +103.878 q^{22} +177.764 q^{23} -45.5454 q^{26} -65.5876 q^{28} +125.681 q^{29} -147.125 q^{31} -236.422 q^{32} +263.571 q^{34} +129.338 q^{37} -381.260 q^{38} +303.254 q^{41} -520.876 q^{43} +178.724 q^{44} +678.460 q^{46} -442.532 q^{47} -243.239 q^{49} -78.3623 q^{52} -209.930 q^{53} +54.6417 q^{56} +479.676 q^{58} -668.960 q^{59} -444.592 q^{61} -561.522 q^{62} -315.034 q^{64} -439.927 q^{67} +453.482 q^{68} +284.335 q^{71} +80.3311 q^{73} +493.635 q^{74} -655.968 q^{76} -271.846 q^{77} -1348.61 q^{79} +1157.41 q^{82} -1194.14 q^{83} -1987.99 q^{86} -148.897 q^{88} +107.486 q^{89} +119.192 q^{91} +1167.31 q^{92} -1688.98 q^{94} +451.652 q^{97} -928.352 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.81662	1.34938	0.674690	−	0.738101i	$-0.264277\pi$
$2$	3.81662	1.34938	0.674690	+	0.738101i	$0.264277\pi$
$3$	0	0
$3$	0	0
$4$	6.56661	0.820826
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−9.98804	−0.539304	−0.269652	−	0.962958i	$-0.586909\pi$
$7$	−9.98804	−0.539304	−0.269652	+	0.962958i	$0.586909\pi$
$8$	−5.47071	−0.241773
$9$	0	0
$10$	0	0
$11$	27.2172	0.746025	0.373013	−	0.927826i	$-0.378325\pi$
$11$	27.2172	0.746025	0.373013	+	0.927826i	$0.378325\pi$
$12$	0	0
$13$	−11.9334	−0.254595	−0.127298	−	0.991865i	$-0.540630\pi$
$13$	−11.9334	−0.254595	−0.127298	+	0.991865i	$0.540630\pi$
$14$	−38.1206	−0.727725
$15$	0	0
$16$	−73.4125	−1.14707
$17$	69.0588	0.985248	0.492624	−	0.870242i	$-0.336038\pi$
$17$	69.0588	0.985248	0.492624	+	0.870242i	$0.336038\pi$
$18$	0	0
$19$	−99.8945	−1.20618	−0.603089	−	0.797674i	$-0.706063\pi$
$19$	−99.8945	−1.20618	−0.603089	+	0.797674i	$0.706063\pi$
$20$	0	0
$21$	0	0
$22$	103.878	1.00667
$23$	177.764	1.61158	0.805792	−	0.592198i	$-0.201740\pi$
$23$	177.764	1.61158	0.805792	+	0.592198i	$0.201740\pi$
$24$	0	0
$25$	0	0
$26$	−45.5454	−0.343546
$27$	0	0
$28$	−65.5876	−0.442675
$29$	125.681	0.804769	0.402385	−	0.915471i	$-0.368181\pi$
$29$	125.681	0.804769	0.402385	+	0.915471i	$0.368181\pi$
$30$	0	0
$31$	−147.125	−0.852402	−0.426201	−	0.904628i	$-0.640148\pi$
$31$	−147.125	−0.852402	−0.426201	+	0.904628i	$0.640148\pi$
$32$	−236.422	−1.30606
$33$	0	0
$34$	263.571	1.32947
$35$	0	0
$36$	0	0
$37$	129.338	0.574677	0.287339	−	0.957829i	$-0.407230\pi$
$37$	129.338	0.574677	0.287339	+	0.957829i	$0.407230\pi$
$38$	−381.260	−1.62759
$39$	0	0
$40$	0	0
$41$	303.254	1.15513	0.577566	−	0.816344i	$-0.304003\pi$
$41$	303.254	1.15513	0.577566	+	0.816344i	$0.304003\pi$
$42$	0	0
$43$	−520.876	−1.84727	−0.923637	−	0.383267i	$-0.874799\pi$
$43$	−520.876	−1.84727	−0.923637	+	0.383267i	$0.874799\pi$
$44$	178.724	0.612357
$45$	0	0
$46$	678.460	2.17464
$47$	−442.532	−1.37340	−0.686702	−	0.726939i	$-0.740942\pi$
$47$	−442.532	−1.37340	−0.686702	+	0.726939i	$0.740942\pi$
$48$	0	0
$49$	−243.239	−0.709152
$50$	0	0
$51$	0	0
$52$	−78.3623	−0.208979
$53$	−209.930	−0.544078	−0.272039	−	0.962286i	$-0.587698\pi$
$53$	−209.930	−0.544078	−0.272039	+	0.962286i	$0.587698\pi$
$54$	0	0
$55$	0	0
$56$	54.6417	0.130389
$57$	0	0
$58$	479.676	1.08594
$59$	−668.960	−1.47612	−0.738060	−	0.674735i	$-0.764258\pi$
$59$	−668.960	−1.47612	−0.738060	+	0.674735i	$0.764258\pi$
$60$	0	0
$61$	−444.592	−0.933183	−0.466591	−	0.884473i	$-0.654518\pi$
$61$	−444.592	−0.933183	−0.466591	+	0.884473i	$0.654518\pi$
$62$	−561.522	−1.15021
$63$	0	0
$64$	−315.034	−0.615301
$65$	0	0
$66$	0	0
$67$	−439.927	−0.802174	−0.401087	−	0.916040i	$-0.631368\pi$
$67$	−439.927	−0.802174	−0.401087	+	0.916040i	$0.631368\pi$
$68$	453.482	0.808718
$69$	0	0
$70$	0	0
$71$	284.335	0.475272	0.237636	−	0.971354i	$-0.423627\pi$
$71$	284.335	0.475272	0.237636	+	0.971354i	$0.423627\pi$
$72$	0	0
$73$	80.3311	0.128795	0.0643976	−	0.997924i	$-0.479487\pi$
$73$	80.3311	0.128795	0.0643976	+	0.997924i	$0.479487\pi$
$74$	493.635	0.775458
$75$	0	0
$76$	−655.968	−0.990062
$77$	−271.846	−0.402334
$78$	0	0
$79$	−1348.61	−1.92063	−0.960316	−	0.278913i	$-0.910026\pi$
$79$	−1348.61	−1.92063	−0.960316	+	0.278913i	$0.910026\pi$
$80$	0	0
$81$	0	0
$82$	1157.41	1.55871
$83$	−1194.14	−1.57920	−0.789601	−	0.613621i	$-0.789712\pi$
$83$	−1194.14	−1.57920	−0.789601	+	0.613621i	$0.789712\pi$
$84$	0	0
$85$	0	0
$86$	−1987.99	−2.49268
$87$	0	0
$88$	−148.897	−0.180369
$89$	107.486	0.128017	0.0640086	−	0.997949i	$-0.479611\pi$
$89$	107.486	0.128017	0.0640086	+	0.997949i	$0.479611\pi$
$90$	0	0
$91$	119.192	0.137304
$92$	1167.31	1.32283
$93$	0	0
$94$	−1688.98	−1.85324
$95$	0	0
$96$	0	0
$97$	451.652	0.472766	0.236383	−	0.971660i	$-0.424038\pi$
$97$	451.652	0.472766	0.236383	+	0.971660i	$0.424038\pi$
$98$	−928.352	−0.956915
$99$	0	0
$100$	0	0
$101$	1366.93	1.34668	0.673342	−	0.739331i	$-0.264858\pi$
$101$	1366.93	1.34668	0.673342	+	0.739331i	$0.264858\pi$
$102$	0	0
$103$	−1177.86	−1.12678	−0.563390	−	0.826191i	$-0.690503\pi$
$103$	−1177.86	−1.12678	−0.563390	+	0.826191i	$0.690503\pi$
$104$	65.2844	0.0615544
$105$	0	0
$106$	−801.225	−0.734169
$107$	−375.335	−0.339112	−0.169556	−	0.985521i	$-0.554233\pi$
$107$	−375.335	−0.339112	−0.169556	+	0.985521i	$0.554233\pi$
$108$	0	0
$109$	319.719	0.280950	0.140475	−	0.990084i	$-0.455137\pi$
$109$	319.719	0.280950	0.140475	+	0.990084i	$0.455137\pi$
$110$	0	0
$111$	0	0
$112$	733.247	0.618619
$113$	265.108	0.220701	0.110351	−	0.993893i	$-0.464803\pi$
$113$	265.108	0.220701	0.110351	+	0.993893i	$0.464803\pi$
$114$	0	0
$115$	0	0
$116$	825.296	0.660576
$117$	0	0
$118$	−2553.17	−1.99185
$119$	−689.762	−0.531348
$120$	0	0
$121$	−590.227	−0.443446
$122$	−1696.84	−1.25922
$123$	0	0
$124$	−966.114	−0.699674
$125$	0	0
$126$	0	0
$127$	2720.86	1.90108	0.950541	−	0.310599i	$-0.100530\pi$
$127$	2720.86	1.90108	0.950541	+	0.310599i	$0.100530\pi$
$128$	689.010	0.475785
$129$	0	0
$130$	0	0
$131$	−1201.71	−0.801480	−0.400740	−	0.916192i	$-0.631247\pi$
$131$	−1201.71	−0.801480	−0.400740	+	0.916192i	$0.631247\pi$
$132$	0	0
$133$	997.750	0.650496
$134$	−1679.04	−1.08244
$135$	0	0
$136$	−377.800	−0.238207
$137$	−518.131	−0.323116	−0.161558	−	0.986863i	$-0.551652\pi$
$137$	−518.131	−0.323116	−0.161558	+	0.986863i	$0.551652\pi$
$138$	0	0
$139$	−844.475	−0.515305	−0.257653	−	0.966238i	$-0.582949\pi$
$139$	−844.475	−0.515305	−0.257653	+	0.966238i	$0.582949\pi$
$140$	0	0
$141$	0	0
$142$	1085.20	0.641323
$143$	−324.794	−0.189935
$144$	0	0
$145$	0	0
$146$	306.594	0.173794
$147$	0	0
$148$	849.313	0.471710
$149$	2129.43	1.17080	0.585401	−	0.810744i	$-0.300937\pi$
$149$	2129.43	1.17080	0.585401	+	0.810744i	$0.300937\pi$
$150$	0	0
$151$	1950.03	1.05094	0.525468	−	0.850813i	$-0.323890\pi$
$151$	1950.03	1.05094	0.525468	+	0.850813i	$0.323890\pi$
$152$	546.493	0.291622
$153$	0	0
$154$	−1037.53	−0.542902
$155$	0	0
$156$	0	0
$157$	−1610.76	−0.818805	−0.409402	−	0.912354i	$-0.634263\pi$
$157$	−1610.76	−0.818805	−0.409402	+	0.912354i	$0.634263\pi$
$158$	−5147.12	−2.59166
$159$	0	0
$160$	0	0
$161$	−1775.52	−0.869133
$162$	0	0
$163$	−3785.13	−1.81886	−0.909430	−	0.415857i	$-0.863482\pi$
$163$	−3785.13	−1.81886	−0.909430	+	0.415857i	$0.863482\pi$
$164$	1991.35	0.948162
$165$	0	0
$166$	−4557.58	−2.13094
$167$	−575.673	−0.266748	−0.133374	−	0.991066i	$-0.542581\pi$
$167$	−575.673	−0.266748	−0.133374	+	0.991066i	$0.542581\pi$
$168$	0	0
$169$	−2054.59	−0.935181
$170$	0	0
$171$	0	0
$172$	−3420.39	−1.51629
$173$	−1962.51	−0.862469	−0.431235	−	0.902240i	$-0.641922\pi$
$173$	−1962.51	−0.862469	−0.431235	+	0.902240i	$0.641922\pi$
$174$	0	0
$175$	0	0
$176$	−1998.08	−0.855744
$177$	0	0
$178$	410.235	0.172744
$179$	3890.89	1.62468	0.812342	−	0.583181i	$-0.198192\pi$
$179$	3890.89	1.62468	0.812342	+	0.583181i	$0.198192\pi$
$180$	0	0
$181$	−1314.02	−0.539614	−0.269807	−	0.962914i	$-0.586960\pi$
$181$	−1314.02	−0.539614	−0.269807	+	0.962914i	$0.586960\pi$
$182$	454.910	0.185276
$183$	0	0
$184$	−972.497	−0.389638
$185$	0	0
$186$	0	0
$187$	1879.58	0.735020
$188$	−2905.94	−1.12733
$189$	0	0
$190$	0	0
$191$	−1946.58	−0.737432	−0.368716	−	0.929542i	$-0.620203\pi$
$191$	−1946.58	−0.737432	−0.368716	+	0.929542i	$0.620203\pi$
$192$	0	0
$193$	−2678.90	−0.999125	−0.499563	−	0.866278i	$-0.666506\pi$
$193$	−2678.90	−0.999125	−0.499563	+	0.866278i	$0.666506\pi$
$194$	1723.79	0.637942
$195$	0	0
$196$	−1597.26	−0.582090
$197$	2421.67	0.875822	0.437911	−	0.899018i	$-0.355719\pi$
$197$	2421.67	0.875822	0.437911	+	0.899018i	$0.355719\pi$
$198$	0	0
$199$	−2192.17	−0.780897	−0.390449	−	0.920625i	$-0.627680\pi$
$199$	−2192.17	−0.780897	−0.390449	+	0.920625i	$0.627680\pi$
$200$	0	0
$201$	0	0
$202$	5217.07	1.81719
$203$	−1255.30	−0.434015
$204$	0	0
$205$	0	0
$206$	−4495.46	−1.52045
$207$	0	0
$208$	876.064	0.292039
$209$	−2718.84	−0.899839
$210$	0	0
$211$	3512.52	1.14603	0.573014	−	0.819546i	$-0.305774\pi$
$211$	3512.52	1.14603	0.573014	+	0.819546i	$0.305774\pi$
$212$	−1378.53	−0.446594
$213$	0	0
$214$	−1432.51	−0.457591
$215$	0	0
$216$	0	0
$217$	1469.49	0.459704
$218$	1220.25	0.379108
$219$	0	0
$220$	0	0
$221$	−824.109	−0.250840
$222$	0	0
$223$	−2459.81	−0.738661	−0.369330	−	0.929298i	$-0.620413\pi$
$223$	−2459.81	−0.738661	−0.369330	+	0.929298i	$0.620413\pi$
$224$	2361.39	0.704363
$225$	0	0
$226$	1011.82	0.297810
$227$	133.072	0.0389087	0.0194544	−	0.999811i	$-0.493807\pi$
$227$	133.072	0.0389087	0.0194544	+	0.999811i	$0.493807\pi$
$228$	0	0
$229$	983.754	0.283879	0.141940	−	0.989875i	$-0.454666\pi$
$229$	983.754	0.283879	0.141940	+	0.989875i	$0.454666\pi$
$230$	0	0
$231$	0	0
$232$	−687.562	−0.194572
$233$	2922.38	0.821679	0.410840	−	0.911708i	$-0.365236\pi$
$233$	2922.38	0.821679	0.410840	+	0.911708i	$0.365236\pi$
$234$	0	0
$235$	0	0
$236$	−4392.80	−1.21164
$237$	0	0
$238$	−2632.56	−0.716990
$239$	386.386	0.104574	0.0522872	−	0.998632i	$-0.483349\pi$
$239$	386.386	0.104574	0.0522872	+	0.998632i	$0.483349\pi$
$240$	0	0
$241$	−1239.78	−0.331374	−0.165687	−	0.986178i	$-0.552984\pi$
$241$	−1239.78	−0.331374	−0.165687	+	0.986178i	$0.552984\pi$
$242$	−2252.67	−0.598377
$243$	0	0
$244$	−2919.46	−0.765981
$245$	0	0
$246$	0	0
$247$	1192.08	0.307087
$248$	804.879	0.206088
$249$	0	0
$250$	0	0
$251$	352.122	0.0885487	0.0442743	−	0.999019i	$-0.485902\pi$
$251$	352.122	0.0885487	0.0442743	+	0.999019i	$0.485902\pi$
$252$	0	0
$253$	4838.24	1.20228
$254$	10384.5	2.56528
$255$	0	0
$256$	5149.97	1.25732
$257$	4927.96	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$257$	4927.96	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$258$	0	0
$259$	−1291.83	−0.309925
$260$	0	0
$261$	0	0
$262$	−4586.47	−1.08150
$263$	−5955.35	−1.39628	−0.698141	−	0.715960i	$-0.745989\pi$
$263$	−5955.35	−1.39628	−0.698141	+	0.715960i	$0.745989\pi$
$264$	0	0
$265$	0	0
$266$	3808.04	0.877766
$267$	0	0
$268$	−2888.83	−0.658446
$269$	6539.68	1.48227	0.741136	−	0.671355i	$-0.234287\pi$
$269$	6539.68	1.48227	0.741136	+	0.671355i	$0.234287\pi$
$270$	0	0
$271$	−1978.70	−0.443533	−0.221767	−	0.975100i	$-0.571182\pi$
$271$	−1978.70	−0.443533	−0.221767	+	0.975100i	$0.571182\pi$
$272$	−5069.78	−1.13015
$273$	0	0
$274$	−1977.51	−0.436007
$275$	0	0
$276$	0	0
$277$	3063.18	0.664435	0.332217	−	0.943203i	$-0.392203\pi$
$277$	3063.18	0.664435	0.332217	+	0.943203i	$0.392203\pi$
$278$	−3223.04	−0.695343
$279$	0	0
$280$	0	0
$281$	−4060.67	−0.862061	−0.431030	−	0.902337i	$-0.641850\pi$
$281$	−4060.67	−0.862061	−0.431030	+	0.902337i	$0.641850\pi$
$282$	0	0
$283$	3703.82	0.777984	0.388992	−	0.921241i	$-0.372823\pi$
$283$	3703.82	0.777984	0.388992	+	0.921241i	$0.372823\pi$
$284$	1867.12	0.390116
$285$	0	0
$286$	−1239.62	−0.256294
$287$	−3028.92	−0.622966
$288$	0	0
$289$	−143.882	−0.0292859
$290$	0	0
$291$	0	0
$292$	527.503	0.105718
$293$	−114.691	−0.0228679	−0.0114340	−	0.999935i	$-0.503640\pi$
$293$	−114.691	−0.0228679	−0.0114340	+	0.999935i	$0.503640\pi$
$294$	0	0
$295$	0	0
$296$	−707.571	−0.138942
$297$	0	0
$298$	8127.23	1.57986
$299$	−2121.34	−0.410302
$300$	0	0
$301$	5202.53	0.996242
$302$	7442.54	1.41811
$303$	0	0
$304$	7333.50	1.38357
$305$	0	0
$306$	0	0
$307$	9171.12	1.70496	0.852481	−	0.522758i	$-0.175097\pi$
$307$	9171.12	1.70496	0.852481	+	0.522758i	$0.175097\pi$
$308$	−1785.11	−0.330246
$309$	0	0
$310$	0	0
$311$	7718.42	1.40730	0.703652	−	0.710545i	$-0.251551\pi$
$311$	7718.42	1.40730	0.703652	+	0.710545i	$0.251551\pi$
$312$	0	0
$313$	−10381.9	−1.87483	−0.937416	−	0.348212i	$-0.886789\pi$
$313$	−10381.9	−1.87483	−0.937416	+	0.348212i	$0.886789\pi$
$314$	−6147.65	−1.10488
$315$	0	0
$316$	−8855.77	−1.57651
$317$	−4514.98	−0.799957	−0.399979	−	0.916524i	$-0.630982\pi$
$317$	−4514.98	−0.799957	−0.399979	+	0.916524i	$0.630982\pi$
$318$	0	0
$319$	3420.67	0.600378
$320$	0	0
$321$	0	0
$322$	−6776.48	−1.17279
$323$	−6898.59	−1.18838
$324$	0	0
$325$	0	0
$326$	−14446.4	−2.45433
$327$	0	0
$328$	−1659.02	−0.279280
$329$	4420.03	0.740682
$330$	0	0
$331$	2930.94	0.486704	0.243352	−	0.969938i	$-0.421753\pi$
$331$	2930.94	0.486704	0.243352	+	0.969938i	$0.421753\pi$
$332$	−7841.44	−1.29625
$333$	0	0
$334$	−2197.13	−0.359944
$335$	0	0
$336$	0	0
$337$	−2664.65	−0.430721	−0.215360	−	0.976535i	$-0.569093\pi$
$337$	−2664.65	−0.430721	−0.215360	+	0.976535i	$0.569093\pi$
$338$	−7841.61	−1.26191
$339$	0	0
$340$	0	0
$341$	−4004.33	−0.635914
$342$	0	0
$343$	5855.38	0.921752
$344$	2849.56	0.446622
$345$	0	0
$346$	−7490.18	−1.16380
$347$	−3379.56	−0.522837	−0.261418	−	0.965226i	$-0.584190\pi$
$347$	−3379.56	−0.522837	−0.261418	+	0.965226i	$0.584190\pi$
$348$	0	0
$349$	7712.27	1.18289	0.591445	−	0.806346i	$-0.298558\pi$
$349$	7712.27	1.18289	0.591445	+	0.806346i	$0.298558\pi$
$350$	0	0
$351$	0	0
$352$	−6434.74	−0.974354
$353$	−1723.79	−0.259909	−0.129955	−	0.991520i	$-0.541483\pi$
$353$	−1723.79	−0.259909	−0.129955	+	0.991520i	$0.541483\pi$
$354$	0	0
$355$	0	0
$356$	705.821	0.105080
$357$	0	0
$358$	14850.0	2.19232
$359$	−5090.68	−0.748401	−0.374200	−	0.927348i	$-0.622083\pi$
$359$	−5090.68	−0.748401	−0.374200	+	0.927348i	$0.622083\pi$
$360$	0	0
$361$	3119.91	0.454863
$362$	−5015.11	−0.728144
$363$	0	0
$364$	782.686	0.112703
$365$	0	0
$366$	0	0
$367$	−3736.58	−0.531466	−0.265733	−	0.964047i	$-0.585614\pi$
$367$	−3736.58	−0.531466	−0.265733	+	0.964047i	$0.585614\pi$
$368$	−13050.1	−1.84860
$369$	0	0
$370$	0	0
$371$	2096.79	0.293423
$372$	0	0
$373$	3417.47	0.474396	0.237198	−	0.971461i	$-0.423771\pi$
$373$	3417.47	0.474396	0.237198	+	0.971461i	$0.423771\pi$
$374$	7173.66	0.991822
$375$	0	0
$376$	2420.97	0.332053
$377$	−1499.80	−0.204891
$378$	0	0
$379$	10243.4	1.38830	0.694152	−	0.719829i	$-0.255780\pi$
$379$	10243.4	1.38830	0.694152	+	0.719829i	$0.255780\pi$
$380$	0	0
$381$	0	0
$382$	−7429.36	−0.995076
$383$	−6091.12	−0.812642	−0.406321	−	0.913730i	$-0.633189\pi$
$383$	−6091.12	−0.812642	−0.406321	+	0.913730i	$0.633189\pi$
$384$	0	0
$385$	0	0
$386$	−10224.3	−1.34820
$387$	0	0
$388$	2965.82	0.388059
$389$	−5791.92	−0.754916	−0.377458	−	0.926027i	$-0.623202\pi$
$389$	−5791.92	−0.754916	−0.377458	+	0.926027i	$0.623202\pi$
$390$	0	0
$391$	12276.2	1.58781
$392$	1330.69	0.171454
$393$	0	0
$394$	9242.60	1.18182
$395$	0	0
$396$	0	0
$397$	−11206.7	−1.41675	−0.708374	−	0.705838i	$-0.750571\pi$
$397$	−11206.7	−1.41675	−0.708374	+	0.705838i	$0.750571\pi$
$398$	−8366.67	−1.05373
$399$	0	0
$400$	0	0
$401$	−5230.78	−0.651404	−0.325702	−	0.945473i	$-0.605601\pi$
$401$	−5230.78	−0.651404	−0.325702	+	0.945473i	$0.605601\pi$
$402$	0	0
$403$	1755.71	0.217018
$404$	8976.13	1.10539
$405$	0	0
$406$	−4791.02	−0.585651
$407$	3520.21	0.428724
$408$	0	0
$409$	13469.7	1.62845	0.814224	−	0.580551i	$-0.197163\pi$
$409$	13469.7	1.62845	0.814224	+	0.580551i	$0.197163\pi$
$410$	0	0
$411$	0	0
$412$	−7734.57	−0.924890
$413$	6681.60	0.796077
$414$	0	0
$415$	0	0
$416$	2821.33	0.332517
$417$	0	0
$418$	−10376.8	−1.21422
$419$	−2716.10	−0.316683	−0.158342	−	0.987384i	$-0.550615\pi$
$419$	−2716.10	−0.316683	−0.158342	+	0.987384i	$0.550615\pi$
$420$	0	0
$421$	−892.409	−0.103310	−0.0516548	−	0.998665i	$-0.516450\pi$
$421$	−892.409	−0.103310	−0.0516548	+	0.998665i	$0.516450\pi$
$422$	13406.0	1.54643
$423$	0	0
$424$	1148.47	0.131544
$425$	0	0
$426$	0	0
$427$	4440.60	0.503269
$428$	−2464.68	−0.278352
$429$	0	0
$430$	0	0
$431$	14017.4	1.56658	0.783288	−	0.621659i	$-0.213541\pi$
$431$	14017.4	1.56658	0.783288	+	0.621659i	$0.213541\pi$
$432$	0	0
$433$	−16826.3	−1.86749	−0.933744	−	0.357941i	$-0.883479\pi$
$433$	−16826.3	−1.86749	−0.933744	+	0.357941i	$0.883479\pi$
$434$	5608.50	0.620315
$435$	0	0
$436$	2099.47	0.230611
$437$	−17757.7	−1.94386
$438$	0	0
$439$	−15751.8	−1.71252	−0.856258	−	0.516549i	$-0.827216\pi$
$439$	−15751.8	−1.71252	−0.856258	+	0.516549i	$0.827216\pi$
$440$	0	0
$441$	0	0
$442$	−3145.31	−0.338478
$443$	5988.11	0.642220	0.321110	−	0.947042i	$-0.395944\pi$
$443$	5988.11	0.642220	0.321110	+	0.947042i	$0.395944\pi$
$444$	0	0
$445$	0	0
$446$	−9388.18	−0.996734
$447$	0	0
$448$	3146.58	0.331834
$449$	114.851	0.0120716	0.00603580	−	0.999982i	$-0.498079\pi$
$449$	114.851	0.0120716	0.00603580	+	0.999982i	$0.498079\pi$
$450$	0	0
$451$	8253.72	0.861757
$452$	1740.86	0.181157
$453$	0	0
$454$	507.885	0.0525027
$455$	0	0
$456$	0	0
$457$	13669.8	1.39922	0.699611	−	0.714524i	$-0.253357\pi$
$457$	13669.8	1.39922	0.699611	+	0.714524i	$0.253357\pi$
$458$	3754.62	0.383061
$459$	0	0
$460$	0	0
$461$	8784.67	0.887512	0.443756	−	0.896148i	$-0.353646\pi$
$461$	8784.67	0.887512	0.443756	+	0.896148i	$0.353646\pi$
$462$	0	0
$463$	9101.55	0.913574	0.456787	−	0.889576i	$-0.349000\pi$
$463$	9101.55	0.913574	0.456787	+	0.889576i	$0.349000\pi$
$464$	−9226.53	−0.923127
$465$	0	0
$466$	11153.6	1.10876
$467$	−14817.7	−1.46827	−0.734136	−	0.679003i	$-0.762412\pi$
$467$	−14817.7	−1.46827	−0.734136	+	0.679003i	$0.762412\pi$
$468$	0	0
$469$	4394.01	0.432615
$470$	0	0
$471$	0	0
$472$	3659.68	0.356887
$473$	−14176.8	−1.37811
$474$	0	0
$475$	0	0
$476$	−4529.40	−0.436144
$477$	0	0
$478$	1474.69	0.141110
$479$	−7517.89	−0.717121	−0.358561	−	0.933506i	$-0.616732\pi$
$479$	−7517.89	−0.717121	−0.358561	+	0.933506i	$0.616732\pi$
$480$	0	0
$481$	−1543.45	−0.146310
$482$	−4731.77	−0.447150
$483$	0	0
$484$	−3875.79	−0.363992
$485$	0	0
$486$	0	0
$487$	−9323.21	−0.867505	−0.433752	−	0.901032i	$-0.642811\pi$
$487$	−9323.21	−0.867505	−0.433752	+	0.901032i	$0.642811\pi$
$488$	2432.23	0.225619
$489$	0	0
$490$	0	0
$491$	−11234.9	−1.03263	−0.516316	−	0.856398i	$-0.672697\pi$
$491$	−11234.9	−1.03263	−0.516316	+	0.856398i	$0.672697\pi$
$492$	0	0
$493$	8679.36	0.792898
$494$	4549.74	0.414377
$495$	0	0
$496$	10800.8	0.977766
$497$	−2839.95	−0.256316
$498$	0	0
$499$	14435.1	1.29500	0.647498	−	0.762067i	$-0.275815\pi$
$499$	14435.1	1.29500	0.647498	+	0.762067i	$0.275815\pi$
$500$	0	0
$501$	0	0
$502$	1343.92	0.119486
$503$	3906.55	0.346292	0.173146	−	0.984896i	$-0.444607\pi$
$503$	3906.55	0.346292	0.173146	+	0.984896i	$0.444607\pi$
$504$	0	0
$505$	0	0
$506$	18465.7	1.62234
$507$	0	0
$508$	17866.8	1.56046
$509$	−3568.51	−0.310749	−0.155375	−	0.987856i	$-0.549658\pi$
$509$	−3568.51	−0.310749	−0.155375	+	0.987856i	$0.549658\pi$
$510$	0	0
$511$	−802.351	−0.0694597
$512$	14143.4	1.22081
$513$	0	0
$514$	18808.2	1.61399
$515$	0	0
$516$	0	0
$517$	−12044.5	−1.02459
$518$	−4930.44	−0.418207
$519$	0	0
$520$	0	0
$521$	−3076.09	−0.258668	−0.129334	−	0.991601i	$-0.541284\pi$
$521$	−3076.09	−0.258668	−0.129334	+	0.991601i	$0.541284\pi$
$522$	0	0
$523$	1084.19	0.0906472	0.0453236	−	0.998972i	$-0.485568\pi$
$523$	1084.19	0.0906472	0.0453236	+	0.998972i	$0.485568\pi$
$524$	−7891.16	−0.657876
$525$	0	0
$526$	−22729.3	−1.88412
$527$	−10160.3	−0.839828
$528$	0	0
$529$	19433.2	1.59720
$530$	0	0
$531$	0	0
$532$	6551.84	0.533944
$533$	−3618.87	−0.294091
$534$	0	0
$535$	0	0
$536$	2406.71	0.193944
$537$	0	0
$538$	24959.5	2.00015
$539$	−6620.27	−0.529045
$540$	0	0
$541$	9699.97	0.770858	0.385429	−	0.922737i	$-0.374053\pi$
$541$	9699.97	0.770858	0.385429	+	0.922737i	$0.374053\pi$
$542$	−7551.96	−0.598495
$543$	0	0
$544$	−16327.0	−1.28679
$545$	0	0
$546$	0	0
$547$	−5573.10	−0.435628	−0.217814	−	0.975990i	$-0.569893\pi$
$547$	−5573.10	−0.435628	−0.217814	+	0.975990i	$0.569893\pi$
$548$	−3402.36	−0.265222
$549$	0	0
$550$	0	0
$551$	−12554.8	−0.970694
$552$	0	0
$553$	13469.9	1.03580
$554$	11691.0	0.896575
$555$	0	0
$556$	−5545.34	−0.422976
$557$	−22526.6	−1.71361	−0.856806	−	0.515639i	$-0.827555\pi$
$557$	−22526.6	−1.71361	−0.856806	+	0.515639i	$0.827555\pi$
$558$	0	0
$559$	6215.84	0.470308
$560$	0	0
$561$	0	0
$562$	−15498.0	−1.16325
$563$	4469.38	0.334568	0.167284	−	0.985909i	$-0.446500\pi$
$563$	4469.38	0.334568	0.167284	+	0.985909i	$0.446500\pi$
$564$	0	0
$565$	0	0
$566$	14136.1	1.04980
$567$	0	0
$568$	−1555.51	−0.114908
$569$	−9063.69	−0.667785	−0.333893	−	0.942611i	$-0.608362\pi$
$569$	−9063.69	−0.667785	−0.333893	+	0.942611i	$0.608362\pi$
$570$	0	0
$571$	−7121.92	−0.521967	−0.260983	−	0.965343i	$-0.584047\pi$
$571$	−7121.92	−0.521967	−0.260983	+	0.965343i	$0.584047\pi$
$572$	−2132.80	−0.155903
$573$	0	0
$574$	−11560.2	−0.840618
$575$	0	0
$576$	0	0
$577$	−1890.44	−0.136396	−0.0681978	−	0.997672i	$-0.521725\pi$
$577$	−1890.44	−0.136396	−0.0681978	+	0.997672i	$0.521725\pi$
$578$	−549.143	−0.0395179
$579$	0	0
$580$	0	0
$581$	11927.1	0.851669
$582$	0	0
$583$	−5713.71	−0.405896
$584$	−439.468	−0.0311392
$585$	0	0
$586$	−437.731	−0.0308575
$587$	25330.3	1.78108	0.890539	−	0.454906i	$-0.150327\pi$
$587$	25330.3	1.78108	0.890539	+	0.454906i	$0.150327\pi$
$588$	0	0
$589$	14697.0	1.02815
$590$	0	0
$591$	0	0
$592$	−9495.03	−0.659195
$593$	−11140.5	−0.771476	−0.385738	−	0.922608i	$-0.626053\pi$
$593$	−11140.5	−0.771476	−0.385738	+	0.922608i	$0.626053\pi$
$594$	0	0
$595$	0	0
$596$	13983.1	0.961026
$597$	0	0
$598$	−8096.36	−0.553654
$599$	2229.45	0.152075	0.0760375	−	0.997105i	$-0.475773\pi$
$599$	2229.45	0.152075	0.0760375	+	0.997105i	$0.475773\pi$
$600$	0	0
$601$	25347.4	1.72037	0.860185	−	0.509982i	$-0.170348\pi$
$601$	25347.4	1.72037	0.860185	+	0.509982i	$0.170348\pi$
$602$	19856.1	1.34431
$603$	0	0
$604$	12805.1	0.862636
$605$	0	0
$606$	0	0
$607$	6774.24	0.452979	0.226489	−	0.974014i	$-0.427275\pi$
$607$	6774.24	0.452979	0.226489	+	0.974014i	$0.427275\pi$
$608$	23617.3	1.57534
$609$	0	0
$610$	0	0
$611$	5280.94	0.349663
$612$	0	0
$613$	−12357.6	−0.814225	−0.407113	−	0.913378i	$-0.633464\pi$
$613$	−12357.6	−0.814225	−0.407113	+	0.913378i	$0.633464\pi$
$614$	35002.7	2.30064
$615$	0	0
$616$	1487.19	0.0972737
$617$	10045.8	0.655474	0.327737	−	0.944769i	$-0.393714\pi$
$617$	10045.8	0.655474	0.327737	+	0.944769i	$0.393714\pi$
$618$	0	0
$619$	15622.7	1.01443	0.507213	−	0.861821i	$-0.330676\pi$
$619$	15622.7	1.01443	0.507213	+	0.861821i	$0.330676\pi$
$620$	0	0
$621$	0	0
$622$	29458.3	1.89899
$623$	−1073.58	−0.0690402
$624$	0	0
$625$	0	0
$626$	−39624.0	−2.52986
$627$	0	0
$628$	−10577.2	−0.672097
$629$	8931.93	0.566200
$630$	0	0
$631$	23403.3	1.47650	0.738249	−	0.674529i	$-0.235653\pi$
$631$	23403.3	1.47650	0.738249	+	0.674529i	$0.235653\pi$
$632$	7377.83	0.464358
$633$	0	0
$634$	−17232.0	−1.07945
$635$	0	0
$636$	0	0
$637$	2902.68	0.180547
$638$	13055.4	0.810139
$639$	0	0
$640$	0	0
$641$	−402.056	−0.0247742	−0.0123871	−	0.999923i	$-0.503943\pi$
$641$	−402.056	−0.0247742	−0.0123871	+	0.999923i	$0.503943\pi$
$642$	0	0
$643$	−21698.8	−1.33082	−0.665409	−	0.746479i	$-0.731743\pi$
$643$	−21698.8	−1.33082	−0.665409	+	0.746479i	$0.731743\pi$
$644$	−11659.1	−0.713407
$645$	0	0
$646$	−26329.3	−1.60358
$647$	9372.54	0.569509	0.284755	−	0.958600i	$-0.408088\pi$
$647$	9372.54	0.569509	0.284755	+	0.958600i	$0.408088\pi$
$648$	0	0
$649$	−18207.2	−1.10122
$650$	0	0
$651$	0	0
$652$	−24855.5	−1.49297
$653$	30489.5	1.82718	0.913588	−	0.406641i	$-0.133300\pi$
$653$	30489.5	1.82718	0.913588	+	0.406641i	$0.133300\pi$
$654$	0	0
$655$	0	0
$656$	−22262.7	−1.32502
$657$	0	0
$658$	16869.6	0.999461
$659$	21969.0	1.29862	0.649310	−	0.760524i	$-0.275058\pi$
$659$	21969.0	1.29862	0.649310	+	0.760524i	$0.275058\pi$
$660$	0	0
$661$	25474.1	1.49898	0.749491	−	0.662015i	$-0.230298\pi$
$661$	25474.1	1.49898	0.749491	+	0.662015i	$0.230298\pi$
$662$	11186.3	0.656749
$663$	0	0
$664$	6532.78	0.381809
$665$	0	0
$666$	0	0
$667$	22341.6	1.29695
$668$	−3780.22	−0.218954
$669$	0	0
$670$	0	0
$671$	−12100.5	−0.696178
$672$	0	0
$673$	−20461.8	−1.17198	−0.585992	−	0.810317i	$-0.699295\pi$
$673$	−20461.8	−1.17198	−0.585992	+	0.810317i	$0.699295\pi$
$674$	−10170.0	−0.581206
$675$	0	0
$676$	−13491.7	−0.767621
$677$	−7333.59	−0.416326	−0.208163	−	0.978094i	$-0.566748\pi$
$677$	−7333.59	−0.416326	−0.208163	+	0.978094i	$0.566748\pi$
$678$	0	0
$679$	−4511.12	−0.254965
$680$	0	0
$681$	0	0
$682$	−15283.0	−0.858089
$683$	−11120.6	−0.623011	−0.311505	−	0.950244i	$-0.600833\pi$
$683$	−11120.6	−0.623011	−0.311505	+	0.950244i	$0.600833\pi$
$684$	0	0
$685$	0	0
$686$	22347.8	1.24379
$687$	0	0
$688$	38238.8	2.11895
$689$	2505.19	0.138520
$690$	0	0
$691$	−8445.77	−0.464967	−0.232483	−	0.972600i	$-0.574685\pi$
$691$	−8445.77	−0.464967	−0.232483	+	0.972600i	$0.574685\pi$
$692$	−12887.1	−0.707938
$693$	0	0
$694$	−12898.5	−0.705505
$695$	0	0
$696$	0	0
$697$	20942.4	1.13809
$698$	29434.8	1.59617
$699$	0	0
$700$	0	0
$701$	23726.6	1.27838	0.639188	−	0.769050i	$-0.279271\pi$
$701$	23726.6	1.27838	0.639188	+	0.769050i	$0.279271\pi$
$702$	0	0
$703$	−12920.2	−0.693162
$704$	−8574.34	−0.459031
$705$	0	0
$706$	−6579.05	−0.350716
$707$	−13653.0	−0.726271
$708$	0	0
$709$	−7055.05	−0.373706	−0.186853	−	0.982388i	$-0.559829\pi$
$709$	−7055.05	−0.373706	−0.186853	+	0.982388i	$0.559829\pi$
$710$	0	0
$711$	0	0
$712$	−588.027	−0.0309512
$713$	−26153.6	−1.37372
$714$	0	0
$715$	0	0
$716$	25549.9	1.33358
$717$	0	0
$718$	−19429.2	−1.00988
$719$	−402.558	−0.0208802	−0.0104401	−	0.999946i	$-0.503323\pi$
$719$	−402.558	−0.0208802	−0.0104401	+	0.999946i	$0.503323\pi$
$720$	0	0
$721$	11764.5	0.607676
$722$	11907.5	0.613783
$723$	0	0
$724$	−8628.64	−0.442929
$725$	0	0
$726$	0	0
$727$	−5896.34	−0.300802	−0.150401	−	0.988625i	$-0.548056\pi$
$727$	−5896.34	−0.300802	−0.150401	+	0.988625i	$0.548056\pi$
$728$	−652.063	−0.0331965
$729$	0	0
$730$	0	0
$731$	−35971.1	−1.82002
$732$	0	0
$733$	−26561.0	−1.33841	−0.669205	−	0.743078i	$-0.733365\pi$
$733$	−26561.0	−1.33841	−0.669205	+	0.743078i	$0.733365\pi$
$734$	−14261.1	−0.717150
$735$	0	0
$736$	−42027.5	−2.10483
$737$	−11973.6	−0.598442
$738$	0	0
$739$	−22157.8	−1.10296	−0.551481	−	0.834187i	$-0.685937\pi$
$739$	−22157.8	−1.10296	−0.551481	+	0.834187i	$0.685937\pi$
$740$	0	0
$741$	0	0
$742$	8002.67	0.395940
$743$	14102.1	0.696305	0.348152	−	0.937438i	$-0.386809\pi$
$743$	14102.1	0.696305	0.348152	+	0.937438i	$0.386809\pi$
$744$	0	0
$745$	0	0
$746$	13043.2	0.640141
$747$	0	0
$748$	12342.5	0.603324
$749$	3748.86	0.182884
$750$	0	0
$751$	−20538.1	−0.997931	−0.498965	−	0.866622i	$-0.666287\pi$
$751$	−20538.1	−0.997931	−0.498965	+	0.866622i	$0.666287\pi$
$752$	32487.4	1.57539
$753$	0	0
$754$	−5724.18	−0.276475
$755$	0	0
$756$	0	0
$757$	−6851.03	−0.328937	−0.164468	−	0.986382i	$-0.552591\pi$
$757$	−6851.03	−0.328937	−0.164468	+	0.986382i	$0.552591\pi$
$758$	39095.1	1.87335
$759$	0	0
$760$	0	0
$761$	33783.9	1.60928	0.804642	−	0.593760i	$-0.202357\pi$
$761$	33783.9	1.60928	0.804642	+	0.593760i	$0.202357\pi$
$762$	0	0
$763$	−3193.36	−0.151517
$764$	−12782.4	−0.605304
$765$	0	0
$766$	−23247.5	−1.09656
$767$	7982.99	0.375814
$768$	0	0
$769$	−32314.1	−1.51531	−0.757656	−	0.652654i	$-0.773656\pi$
$769$	−32314.1	−1.51531	−0.757656	+	0.652654i	$0.773656\pi$
$770$	0	0
$771$	0	0
$772$	−17591.3	−0.820108
$773$	35408.5	1.64755	0.823774	−	0.566918i	$-0.191865\pi$
$773$	35408.5	1.64755	0.823774	+	0.566918i	$0.191865\pi$
$774$	0	0
$775$	0	0
$776$	−2470.86	−0.114302
$777$	0	0
$778$	−22105.6	−1.01867
$779$	−30293.4	−1.39329
$780$	0	0
$781$	7738.78	0.354565
$782$	46853.6	2.14256
$783$	0	0
$784$	17856.8	0.813447
$785$	0	0
$786$	0	0
$787$	1861.73	0.0843247	0.0421624	−	0.999111i	$-0.486575\pi$
$787$	1861.73	0.0843247	0.0421624	+	0.999111i	$0.486575\pi$
$788$	15902.2	0.718897
$789$	0	0
$790$	0	0
$791$	−2647.91	−0.119025
$792$	0	0
$793$	5305.51	0.237584
$794$	−42771.8	−1.91173
$795$	0	0
$796$	−14395.1	−0.640981
$797$	21126.6	0.938951	0.469475	−	0.882946i	$-0.344443\pi$
$797$	21126.6	0.938951	0.469475	+	0.882946i	$0.344443\pi$
$798$	0	0
$799$	−30560.8	−1.35314
$800$	0	0
$801$	0	0
$802$	−19963.9	−0.878991
$803$	2186.38	0.0960845
$804$	0	0
$805$	0	0
$806$	6700.88	0.292839
$807$	0	0
$808$	−7478.10	−0.325592
$809$	−4486.43	−0.194975	−0.0974873	−	0.995237i	$-0.531081\pi$
$809$	−4486.43	−0.194975	−0.0974873	+	0.995237i	$0.531081\pi$
$810$	0	0
$811$	5352.19	0.231740	0.115870	−	0.993264i	$-0.463034\pi$
$811$	5352.19	0.231740	0.115870	+	0.993264i	$0.463034\pi$
$812$	−8243.09	−0.356251
$813$	0	0
$814$	13435.3	0.578511
$815$	0	0
$816$	0	0
$817$	52032.6	2.22814
$818$	51408.9	2.19740
$819$	0	0
$820$	0	0
$821$	35371.0	1.50360	0.751801	−	0.659390i	$-0.229185\pi$
$821$	35371.0	1.50360	0.751801	+	0.659390i	$0.229185\pi$
$822$	0	0
$823$	18829.7	0.797524	0.398762	−	0.917055i	$-0.369440\pi$
$823$	18829.7	0.797524	0.398762	+	0.917055i	$0.369440\pi$
$824$	6443.74	0.272425
$825$	0	0
$826$	25501.1	1.07421
$827$	1000.09	0.0420516	0.0210258	−	0.999779i	$-0.493307\pi$
$827$	1000.09	0.0420516	0.0210258	+	0.999779i	$0.493307\pi$
$828$	0	0
$829$	−13900.5	−0.582369	−0.291184	−	0.956667i	$-0.594049\pi$
$829$	−13900.5	−0.582369	−0.291184	+	0.956667i	$0.594049\pi$
$830$	0	0
$831$	0	0
$832$	3759.44	0.156653
$833$	−16797.8	−0.698690
$834$	0	0
$835$	0	0
$836$	−17853.6	−0.738611
$837$	0	0
$838$	−10366.3	−0.427326
$839$	−22012.2	−0.905774	−0.452887	−	0.891568i	$-0.649606\pi$
$839$	−22012.2	−0.905774	−0.452887	+	0.891568i	$0.649606\pi$
$840$	0	0
$841$	−8593.37	−0.352346
$842$	−3405.99	−0.139404
$843$	0	0
$844$	23065.4	0.940690
$845$	0	0
$846$	0	0
$847$	5895.21	0.239152
$848$	15411.5	0.624096
$849$	0	0
$850$	0	0
$851$	22991.7	0.926141
$852$	0	0
$853$	−19666.5	−0.789411	−0.394705	−	0.918808i	$-0.629153\pi$
$853$	−19666.5	−0.789411	−0.394705	+	0.918808i	$0.629153\pi$
$854$	16948.1	0.679101
$855$	0	0
$856$	2053.35	0.0819882
$857$	−9497.62	−0.378568	−0.189284	−	0.981922i	$-0.560617\pi$
$857$	−9497.62	−0.378568	−0.189284	+	0.981922i	$0.560617\pi$
$858$	0	0
$859$	10030.4	0.398409	0.199205	−	0.979958i	$-0.436164\pi$
$859$	10030.4	0.398409	0.199205	+	0.979958i	$0.436164\pi$
$860$	0	0
$861$	0	0
$862$	53499.1	2.11391
$863$	−35966.9	−1.41869	−0.709343	−	0.704863i	$-0.751008\pi$
$863$	−35966.9	−1.41869	−0.709343	+	0.704863i	$0.751008\pi$
$864$	0	0
$865$	0	0
$866$	−64219.8	−2.51995
$867$	0	0
$868$	9649.59	0.377337
$869$	−36705.2	−1.43284
$870$	0	0
$871$	5249.85	0.204230
$872$	−1749.09	−0.0679261
$873$	0	0
$874$	−67774.4	−2.62300
$875$	0	0
$876$	0	0
$877$	31264.0	1.20378	0.601888	−	0.798581i	$-0.294415\pi$
$877$	31264.0	1.20378	0.601888	+	0.798581i	$0.294415\pi$
$878$	−60118.8	−2.31083
$879$	0	0
$880$	0	0
$881$	−33061.6	−1.26433	−0.632164	−	0.774835i	$-0.717833\pi$
$881$	−33061.6	−1.26433	−0.632164	+	0.774835i	$0.717833\pi$
$882$	0	0
$883$	4173.39	0.159055	0.0795275	−	0.996833i	$-0.474659\pi$
$883$	4173.39	0.159055	0.0795275	+	0.996833i	$0.474659\pi$
$884$	−5411.60	−0.205896
$885$	0	0
$886$	22854.3	0.866599
$887$	910.349	0.0344606	0.0172303	−	0.999852i	$-0.494515\pi$
$887$	910.349	0.0344606	0.0172303	+	0.999852i	$0.494515\pi$
$888$	0	0
$889$	−27176.1	−1.02526
$890$	0	0
$891$	0	0
$892$	−16152.6	−0.606312
$893$	44206.6	1.65657
$894$	0	0
$895$	0	0
$896$	−6881.86	−0.256593
$897$	0	0
$898$	438.343	0.0162892
$899$	−18490.8	−0.685987
$900$	0	0
$901$	−14497.5	−0.536052
$902$	31501.3	1.16284
$903$	0	0
$904$	−1450.33	−0.0533597
$905$	0	0
$906$	0	0
$907$	−13904.3	−0.509023	−0.254511	−	0.967070i	$-0.581915\pi$
$907$	−13904.3	−0.509023	−0.254511	+	0.967070i	$0.581915\pi$
$908$	873.830	0.0319373
$909$	0	0
$910$	0	0
$911$	53888.3	1.95982	0.979912	−	0.199429i	$-0.0639089\pi$
$911$	53888.3	1.95982	0.979912	+	0.199429i	$0.0639089\pi$
$912$	0	0
$913$	−32501.1	−1.17812
$914$	52172.3	1.88808
$915$	0	0
$916$	6459.93	0.233015
$917$	12002.7	0.432241
$918$	0	0
$919$	29460.3	1.05746	0.528729	−	0.848790i	$-0.322669\pi$
$919$	29460.3	1.05746	0.528729	+	0.848790i	$0.322669\pi$
$920$	0	0
$921$	0	0
$922$	33527.8	1.19759
$923$	−3393.09	−0.121002
$924$	0	0
$925$	0	0
$926$	34737.2	1.23276
$927$	0	0
$928$	−29713.7	−1.05108
$929$	9706.33	0.342792	0.171396	−	0.985202i	$-0.445172\pi$
$929$	9706.33	0.342792	0.171396	+	0.985202i	$0.445172\pi$
$930$	0	0
$931$	24298.2	0.855363
$932$	19190.1	0.674456
$933$	0	0
$934$	−56553.7	−1.98126
$935$	0	0
$936$	0	0
$937$	17097.4	0.596104	0.298052	−	0.954550i	$-0.403663\pi$
$937$	17097.4	0.596104	0.298052	+	0.954550i	$0.403663\pi$
$938$	16770.3	0.583762
$939$	0	0
$940$	0	0
$941$	31845.4	1.10322	0.551610	−	0.834102i	$-0.314014\pi$
$941$	31845.4	1.10322	0.551610	+	0.834102i	$0.314014\pi$
$942$	0	0
$943$	53907.8	1.86159
$944$	49110.0	1.69321
$945$	0	0
$946$	−54107.3	−1.85960
$947$	−51569.9	−1.76958	−0.884791	−	0.465987i	$-0.845699\pi$
$947$	−51569.9	−1.76958	−0.884791	+	0.465987i	$0.845699\pi$
$948$	0	0
$949$	−958.627	−0.0327907
$950$	0	0
$951$	0	0
$952$	3773.49	0.128466
$953$	29747.0	1.01112	0.505562	−	0.862790i	$-0.331285\pi$
$953$	29747.0	1.01112	0.505562	+	0.862790i	$0.331285\pi$
$954$	0	0
$955$	0	0
$956$	2537.25	0.0858374
$957$	0	0
$958$	−28692.9	−0.967669
$959$	5175.11	0.174258
$960$	0	0
$961$	−8145.17	−0.273410
$962$	−5890.76	−0.197428
$963$	0	0
$964$	−8141.15	−0.272001
$965$	0	0
$966$	0	0
$967$	9178.41	0.305230	0.152615	−	0.988286i	$-0.451230\pi$
$967$	9178.41	0.305230	0.152615	+	0.988286i	$0.451230\pi$
$968$	3228.96	0.107213
$969$	0	0
$970$	0	0
$971$	56195.2	1.85725	0.928625	−	0.371019i	$-0.120991\pi$
$971$	56195.2	1.85725	0.928625	+	0.371019i	$0.120991\pi$
$972$	0	0
$973$	8434.65	0.277906
$974$	−35583.2	−1.17059
$975$	0	0
$976$	32638.6	1.07043
$977$	−32107.1	−1.05138	−0.525690	−	0.850676i	$-0.676193\pi$
$977$	−32107.1	−1.05138	−0.525690	+	0.850676i	$0.676193\pi$
$978$	0	0
$979$	2925.47	0.0955041
$980$	0	0
$981$	0	0
$982$	−42879.2	−1.39341
$983$	14108.4	0.457771	0.228885	−	0.973453i	$-0.426492\pi$
$983$	14108.4	0.457771	0.228885	+	0.973453i	$0.426492\pi$
$984$	0	0
$985$	0	0
$986$	33125.8	1.06992
$987$	0	0
$988$	7827.96	0.252065
$989$	−92593.2	−2.97704
$990$	0	0
$991$	−30909.9	−0.990802	−0.495401	−	0.868664i	$-0.664979\pi$
$991$	−30909.9	−0.990802	−0.495401	+	0.868664i	$0.664979\pi$
$992$	34783.7	1.11329
$993$	0	0
$994$	−10839.0	−0.345868
$995$	0	0
$996$	0	0
$997$	12253.6	0.389243	0.194622	−	0.980878i	$-0.437652\pi$
$997$	12253.6	0.389243	0.194622	+	0.980878i	$0.437652\pi$
$998$	55093.3	1.74744
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.be.1.10		12
3.2	odd	2		2025.4.a.bi.1.3		12
5.4	even	2		2025.4.a.bj.1.3		12
9.2	odd	6		225.4.e.e.76.10	✓	24
9.5	odd	6		225.4.e.e.151.10	yes	24
15.14	odd	2		2025.4.a.bf.1.10		12
45.2	even	12		225.4.k.e.49.5		48
45.14	odd	6		225.4.e.f.151.3	yes	24
45.23	even	12		225.4.k.e.124.5		48
45.29	odd	6		225.4.e.f.76.3	yes	24
45.32	even	12		225.4.k.e.124.20		48
45.38	even	12		225.4.k.e.49.20		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.e.e.76.10	✓	24	9.2	odd	6
225.4.e.e.151.10	yes	24	9.5	odd	6
225.4.e.f.76.3	yes	24	45.29	odd	6
225.4.e.f.151.3	yes	24	45.14	odd	6
225.4.k.e.49.5		48	45.2	even	12
225.4.k.e.49.20		48	45.38	even	12
225.4.k.e.124.5		48	45.23	even	12
225.4.k.e.124.20		48	45.32	even	12
2025.4.a.be.1.10		12	1.1	even	1	trivial
2025.4.a.bf.1.10		12	15.14	odd	2
2025.4.a.bi.1.3		12	3.2	odd	2
2025.4.a.bj.1.3		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)

\(f(q)\)	\(=\)	\(q+3.81662 q^{2} +6.56661 q^{4} -9.98804 q^{7} -5.47071 q^{8} +O(q^{10})\)	\(q+3.81662 q^{2} +6.56661 q^{4} -9.98804 q^{7} -5.47071 q^{8} +27.2172 q^{11} -11.9334 q^{13} -38.1206 q^{14} -73.4125 q^{16} +69.0588 q^{17} -99.8945 q^{19} +103.878 q^{22} +177.764 q^{23} -45.5454 q^{26} -65.5876 q^{28} +125.681 q^{29} -147.125 q^{31} -236.422 q^{32} +263.571 q^{34} +129.338 q^{37} -381.260 q^{38} +303.254 q^{41} -520.876 q^{43} +178.724 q^{44} +678.460 q^{46} -442.532 q^{47} -243.239 q^{49} -78.3623 q^{52} -209.930 q^{53} +54.6417 q^{56} +479.676 q^{58} -668.960 q^{59} -444.592 q^{61} -561.522 q^{62} -315.034 q^{64} -439.927 q^{67} +453.482 q^{68} +284.335 q^{71} +80.3311 q^{73} +493.635 q^{74} -655.968 q^{76} -271.846 q^{77} -1348.61 q^{79} +1157.41 q^{82} -1194.14 q^{83} -1987.99 q^{86} -148.897 q^{88} +107.486 q^{89} +119.192 q^{91} +1167.31 q^{92} -1688.98 q^{94} +451.652 q^{97} -928.352 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) \) 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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Representations

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Database

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