LMFDB - Embedded newform 225.4.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.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:	$13.2754297513$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.35890 q^{2} +3.28220 q^{4} -30.4356 q^{7} -15.8466 q^{8} -31.4356 q^{11} +60.7424 q^{13} -102.230 q^{14} -79.4848 q^{16} -121.178 q^{17} -14.4356 q^{19} -105.589 q^{22} +13.6932 q^{23} +204.028 q^{26} -99.8958 q^{28} +76.0492 q^{29} +183.049 q^{31} -140.208 q^{32} -407.025 q^{34} +37.3864 q^{37} -48.4877 q^{38} +30.6627 q^{41} -327.564 q^{43} -103.178 q^{44} +45.9941 q^{46} -449.485 q^{47} +583.325 q^{49} +199.369 q^{52} -301.951 q^{53} +482.301 q^{56} +255.441 q^{58} -340.970 q^{59} +619.098 q^{61} +614.844 q^{62} +164.932 q^{64} -256.890 q^{67} -397.731 q^{68} -499.178 q^{71} +19.1288 q^{73} +125.577 q^{74} -47.3805 q^{76} +956.761 q^{77} +257.424 q^{79} +102.993 q^{82} +914.909 q^{83} -1100.26 q^{86} +498.148 q^{88} +1059.68 q^{89} -1848.73 q^{91} +44.9439 q^{92} -1509.77 q^{94} -521.000 q^{97} +1959.33 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8} - 28 q^{11} - 18 q^{13} - 126 q^{14} + 120 q^{16} - 68 q^{17} + 6 q^{19} - 124 q^{22} + 132 q^{23} + 626 q^{26} - 8 q^{28} - 92 q^{29} + 122 q^{31} - 664 q^{32}+ \cdots + 3692 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 24 * q^4 - 26 * q^7 - 84 * q^8 - 28 * q^11 - 18 * q^13 - 126 * q^14 + 120 * q^16 - 68 * q^17 + 6 * q^19 - 124 * q^22 + 132 * q^23 + 626 * q^26 - 8 * q^28 - 92 * q^29 + 122 * q^31 - 664 * q^32 - 692 * q^34 + 284 * q^37 - 158 * q^38 - 392 * q^41 - 690 * q^43 - 32 * q^44 - 588 * q^46 - 620 * q^47 + 260 * q^49 - 1432 * q^52 - 848 * q^53 + 180 * q^56 + 1156 * q^58 - 124 * q^59 + 750 * q^61 + 942 * q^62 + 1376 * q^64 + 358 * q^67 + 704 * q^68 - 824 * q^71 + 108 * q^73 - 1196 * q^74 + 376 * q^76 + 972 * q^77 - 880 * q^79 + 2368 * q^82 + 156 * q^83 + 842 * q^86 + 264 * q^88 + 864 * q^89 - 2198 * q^91 + 2496 * q^92 - 596 * q^94 - 1042 * q^97 + 3692 * q^98

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 $

$2$	3.35890	1.18755	0.593775	−	0.804631i	$-0.297637\pi$
$2$	3.35890	1.18755	0.593775	+	0.804631i	$0.297637\pi$
$3$	0	0
$3$	0	0
$4$	3.28220	0.410275
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−30.4356	−1.64337	−0.821684	−	0.569944i	$-0.806965\pi$
$7$	−30.4356	−1.64337	−0.821684	+	0.569944i	$0.806965\pi$
$8$	−15.8466	−0.700328
$9$	0	0
$10$	0	0
$11$	−31.4356	−0.861654	−0.430827	−	0.902435i	$-0.641778\pi$
$11$	−31.4356	−0.861654	−0.430827	+	0.902435i	$0.641778\pi$
$12$	0	0
$13$	60.7424	1.29592	0.647958	−	0.761676i	$-0.275623\pi$
$13$	60.7424	1.29592	0.647958	+	0.761676i	$0.275623\pi$
$14$	−102.230	−1.95158
$15$	0	0
$16$	−79.4848	−1.24195
$17$	−121.178	−1.72882	−0.864411	−	0.502786i	$-0.832308\pi$
$17$	−121.178	−1.72882	−0.864411	+	0.502786i	$0.832308\pi$
$18$	0	0
$19$	−14.4356	−0.174303	−0.0871514	−	0.996195i	$-0.527776\pi$
$19$	−14.4356	−0.174303	−0.0871514	+	0.996195i	$0.527776\pi$
$20$	0	0
$21$	0	0
$22$	−105.589	−1.02326
$23$	13.6932	0.124141	0.0620703	−	0.998072i	$-0.480230\pi$
$23$	13.6932	0.124141	0.0620703	+	0.998072i	$0.480230\pi$
$24$	0	0
$25$	0	0
$26$	204.028	1.53896
$27$	0	0
$28$	−99.8958	−0.674233
$29$	76.0492	0.486965	0.243482	−	0.969905i	$-0.421710\pi$
$29$	76.0492	0.486965	0.243482	+	0.969905i	$0.421710\pi$
$30$	0	0
$31$	183.049	1.06054	0.530268	−	0.847830i	$-0.322091\pi$
$31$	183.049	1.06054	0.530268	+	0.847830i	$0.322091\pi$
$32$	−140.208	−0.774550
$33$	0	0
$34$	−407.025	−2.05306
$35$	0	0
$36$	0	0
$37$	37.3864	0.166116	0.0830580	−	0.996545i	$-0.473531\pi$
$37$	37.3864	0.166116	0.0830580	+	0.996545i	$0.473531\pi$
$38$	−48.4877	−0.206993
$39$	0	0
$40$	0	0
$41$	30.6627	0.116798	0.0583990	−	0.998293i	$-0.481400\pi$
$41$	30.6627	0.116798	0.0583990	+	0.998293i	$0.481400\pi$
$42$	0	0
$43$	−327.564	−1.16170	−0.580850	−	0.814011i	$-0.697280\pi$
$43$	−327.564	−1.16170	−0.580850	+	0.814011i	$0.697280\pi$
$44$	−103.178	−0.353515
$45$	0	0
$46$	45.9941	0.147423
$47$	−449.485	−1.39498	−0.697490	−	0.716594i	$-0.745700\pi$
$47$	−449.485	−1.39498	−0.697490	+	0.716594i	$0.745700\pi$
$48$	0	0
$49$	583.325	1.70066
$50$	0	0
$51$	0	0
$52$	199.369	0.531682
$53$	−301.951	−0.782569	−0.391284	−	0.920270i	$-0.627969\pi$
$53$	−301.951	−0.782569	−0.391284	+	0.920270i	$0.627969\pi$
$54$	0	0
$55$	0	0
$56$	482.301	1.15090
$57$	0	0
$58$	255.441	0.578295
$59$	−340.970	−0.752381	−0.376190	−	0.926542i	$-0.622766\pi$
$59$	−340.970	−0.752381	−0.376190	+	0.926542i	$0.622766\pi$
$60$	0	0
$61$	619.098	1.29947	0.649733	−	0.760163i	$-0.274881\pi$
$61$	619.098	1.29947	0.649733	+	0.760163i	$0.274881\pi$
$62$	614.844	1.25944
$63$	0	0
$64$	164.932	0.322133
$65$	0	0
$66$	0	0
$67$	−256.890	−0.468419	−0.234210	−	0.972186i	$-0.575250\pi$
$67$	−256.890	−0.468419	−0.234210	+	0.972186i	$0.575250\pi$
$68$	−397.731	−0.709293
$69$	0	0
$70$	0	0
$71$	−499.178	−0.834388	−0.417194	−	0.908818i	$-0.636986\pi$
$71$	−499.178	−0.834388	−0.417194	+	0.908818i	$0.636986\pi$
$72$	0	0
$73$	19.1288	0.0306693	0.0153346	−	0.999882i	$-0.495119\pi$
$73$	19.1288	0.0306693	0.0153346	+	0.999882i	$0.495119\pi$
$74$	125.577	0.197271
$75$	0	0
$76$	−47.3805	−0.0715121
$77$	956.761	1.41601
$78$	0	0
$79$	257.424	0.366613	0.183307	−	0.983056i	$-0.441320\pi$
$79$	257.424	0.366613	0.183307	+	0.983056i	$0.441320\pi$
$80$	0	0
$81$	0	0
$82$	102.993	0.138703
$83$	914.909	1.20993	0.604965	−	0.796252i	$-0.293187\pi$
$83$	914.909	1.20993	0.604965	+	0.796252i	$0.293187\pi$
$84$	0	0
$85$	0	0
$86$	−1100.26	−1.37958
$87$	0	0
$88$	498.148	0.603440
$89$	1059.68	1.26209	0.631045	−	0.775746i	$-0.282626\pi$
$89$	1059.68	1.26209	0.631045	+	0.775746i	$0.282626\pi$
$90$	0	0
$91$	−1848.73	−2.12967
$92$	44.9439	0.0509318
$93$	0	0
$94$	−1509.77	−1.65661
$95$	0	0
$96$	0	0
$97$	−521.000	−0.545356	−0.272678	−	0.962105i	$-0.587909\pi$
$97$	−521.000	−0.545356	−0.272678	+	0.962105i	$0.587909\pi$
$98$	1959.33	2.01962
$99$	0	0
$100$	0	0
$101$	−347.080	−0.341938	−0.170969	−	0.985276i	$-0.554690\pi$
$101$	−347.080	−0.341938	−0.170969	+	0.985276i	$0.554690\pi$
$102$	0	0
$103$	770.749	0.737322	0.368661	−	0.929564i	$-0.379816\pi$
$103$	770.749	0.737322	0.368661	+	0.929564i	$0.379816\pi$
$104$	−962.561	−0.907566
$105$	0	0
$106$	−1014.22	−0.929339
$107$	−1415.37	−1.27878	−0.639390	−	0.768883i	$-0.720813\pi$
$107$	−1415.37	−1.27878	−0.639390	+	0.768883i	$0.720813\pi$
$108$	0	0
$109$	908.386	0.798235	0.399118	−	0.916900i	$-0.369317\pi$
$109$	908.386	0.798235	0.399118	+	0.916900i	$0.369317\pi$
$110$	0	0
$111$	0	0
$112$	2419.17	2.04098
$113$	−2049.94	−1.70657	−0.853283	−	0.521447i	$-0.825392\pi$
$113$	−2049.94	−1.70657	−0.853283	+	0.521447i	$0.825392\pi$
$114$	0	0
$115$	0	0
$116$	249.609	0.199790
$117$	0	0
$118$	−1145.28	−0.893490
$119$	3688.12	2.84109
$120$	0	0
$121$	−342.803	−0.257553
$122$	2079.49	1.54318
$123$	0	0
$124$	600.804	0.435111
$125$	0	0
$126$	0	0
$127$	−281.644	−0.196786	−0.0983932	−	0.995148i	$-0.531370\pi$
$127$	−281.644	−0.196786	−0.0983932	+	0.995148i	$0.531370\pi$
$128$	1675.66	1.15710
$129$	0	0
$130$	0	0
$131$	−243.056	−0.162106	−0.0810531	−	0.996710i	$-0.525828\pi$
$131$	−243.056	−0.162106	−0.0810531	+	0.996710i	$0.525828\pi$
$132$	0	0
$133$	439.356	0.286444
$134$	−862.867	−0.556271
$135$	0	0
$136$	1920.26	1.21074
$137$	909.386	0.567110	0.283555	−	0.958956i	$-0.408486\pi$
$137$	909.386	0.567110	0.283555	+	0.958956i	$0.408486\pi$
$138$	0	0
$139$	−2049.52	−1.25063	−0.625317	−	0.780371i	$-0.715030\pi$
$139$	−2049.52	−1.25063	−0.625317	+	0.780371i	$0.715030\pi$
$140$	0	0
$141$	0	0
$142$	−1676.69	−0.990877
$143$	−1909.47	−1.11663
$144$	0	0
$145$	0	0
$146$	64.2517	0.0364213
$147$	0	0
$148$	122.710	0.0681533
$149$	−3601.14	−1.97998	−0.989990	−	0.141136i	$-0.954925\pi$
$149$	−3601.14	−1.97998	−0.989990	+	0.141136i	$0.954925\pi$
$150$	0	0
$151$	1383.38	0.745550	0.372775	−	0.927922i	$-0.378406\pi$
$151$	1383.38	0.745550	0.372775	+	0.927922i	$0.378406\pi$
$152$	228.755	0.122069
$153$	0	0
$154$	3213.66	1.68159
$155$	0	0
$156$	0	0
$157$	131.749	0.0669729	0.0334864	−	0.999439i	$-0.489339\pi$
$157$	131.749	0.0669729	0.0334864	+	0.999439i	$0.489339\pi$
$158$	864.661	0.435372
$159$	0	0
$160$	0	0
$161$	−416.761	−0.204009
$162$	0	0
$163$	−2897.74	−1.39244	−0.696222	−	0.717827i	$-0.745137\pi$
$163$	−2897.74	−1.39244	−0.696222	+	0.717827i	$0.745137\pi$
$164$	100.641	0.0479193
$165$	0	0
$166$	3073.09	1.43685
$167$	260.283	0.120607	0.0603034	−	0.998180i	$-0.480793\pi$
$167$	260.283	0.120607	0.0603034	+	0.998180i	$0.480793\pi$
$168$	0	0
$169$	1492.64	0.679398
$170$	0	0
$171$	0	0
$172$	−1075.13	−0.476617
$173$	−1935.83	−0.850742	−0.425371	−	0.905019i	$-0.639856\pi$
$173$	−1935.83	−0.850742	−0.425371	+	0.905019i	$0.639856\pi$
$174$	0	0
$175$	0	0
$176$	2498.65	1.07013
$177$	0	0
$178$	3559.36	1.49880
$179$	−576.627	−0.240777	−0.120389	−	0.992727i	$-0.538414\pi$
$179$	−576.627	−0.240777	−0.120389	+	0.992727i	$0.538414\pi$
$180$	0	0
$181$	−1962.04	−0.805733	−0.402866	−	0.915259i	$-0.631986\pi$
$181$	−1962.04	−0.805733	−0.402866	+	0.915259i	$0.631986\pi$
$182$	−6209.70	−2.52909
$183$	0	0
$184$	−216.991	−0.0869390
$185$	0	0
$186$	0	0
$187$	3809.30	1.48965
$188$	−1475.30	−0.572326
$189$	0	0
$190$	0	0
$191$	4318.75	1.63609	0.818047	−	0.575152i	$-0.195057\pi$
$191$	4318.75	1.63609	0.818047	+	0.575152i	$0.195057\pi$
$192$	0	0
$193$	−2.97647	−0.00111011	−0.000555054	−	1.00000i	$-0.500177\pi$
$193$	−2.97647	−0.00111011	−0.000555054		1.00000i	$0.500177\pi$
$194$	−1749.99	−0.647638
$195$	0	0
$196$	1914.59	0.697738
$197$	569.705	0.206040	0.103020	−	0.994679i	$-0.467149\pi$
$197$	569.705	0.206040	0.103020	+	0.994679i	$0.467149\pi$
$198$	0	0
$199$	−3050.73	−1.08674	−0.543368	−	0.839494i	$-0.682851\pi$
$199$	−3050.73	−1.08674	−0.543368	+	0.839494i	$0.682851\pi$
$200$	0	0
$201$	0	0
$202$	−1165.81	−0.406068
$203$	−2314.60	−0.800262
$204$	0	0
$205$	0	0
$206$	2588.87	0.875607
$207$	0	0
$208$	−4828.09	−1.60946
$209$	453.792	0.150189
$210$	0	0
$211$	−50.5104	−0.0164800	−0.00824000	−	0.999966i	$-0.502623\pi$
$211$	−50.5104	−0.0164800	−0.00824000	+	0.999966i	$0.502623\pi$
$212$	−991.064	−0.321069
$213$	0	0
$214$	−4754.10	−1.51862
$215$	0	0
$216$	0	0
$217$	−5571.21	−1.74285
$218$	3051.18	0.947944
$219$	0	0
$220$	0	0
$221$	−7360.64	−2.24041
$222$	0	0
$223$	−5453.55	−1.63765	−0.818827	−	0.574040i	$-0.805375\pi$
$223$	−5453.55	−1.63765	−0.818827	+	0.574040i	$0.805375\pi$
$224$	4267.33	1.27287
$225$	0	0
$226$	−6885.54	−2.02663
$227$	−4777.14	−1.39678	−0.698392	−	0.715715i	$-0.746101\pi$
$227$	−4777.14	−1.39678	−0.698392	+	0.715715i	$0.746101\pi$
$228$	0	0
$229$	−2085.51	−0.601808	−0.300904	−	0.953654i	$-0.597288\pi$
$229$	−2085.51	−0.601808	−0.300904	+	0.953654i	$0.597288\pi$
$230$	0	0
$231$	0	0
$232$	−1205.12	−0.341035
$233$	6484.53	1.82324	0.911622	−	0.411030i	$-0.134831\pi$
$233$	6484.53	1.82324	0.911622	+	0.411030i	$0.134831\pi$
$234$	0	0
$235$	0	0
$236$	−1119.13	−0.308683
$237$	0	0
$238$	12388.0	3.37394
$239$	−2234.62	−0.604792	−0.302396	−	0.953182i	$-0.597786\pi$
$239$	−2234.62	−0.604792	−0.302396	+	0.953182i	$0.597786\pi$
$240$	0	0
$241$	−2393.01	−0.639616	−0.319808	−	0.947482i	$-0.603618\pi$
$241$	−2393.01	−0.639616	−0.319808	+	0.947482i	$0.603618\pi$
$242$	−1151.44	−0.305857
$243$	0	0
$244$	2032.01	0.533139
$245$	0	0
$246$	0	0
$247$	−876.852	−0.225882
$248$	−2900.71	−0.742722
$249$	0	0
$250$	0	0
$251$	612.661	0.154067	0.0770335	−	0.997029i	$-0.475455\pi$
$251$	612.661	0.154067	0.0770335	+	0.997029i	$0.475455\pi$
$252$	0	0
$253$	−430.454	−0.106966
$254$	−946.014	−0.233694
$255$	0	0
$256$	4308.91	1.05198
$257$	−306.112	−0.0742987	−0.0371493	−	0.999310i	$-0.511828\pi$
$257$	−306.112	−0.0742987	−0.0371493	+	0.999310i	$0.511828\pi$
$258$	0	0
$259$	−1137.88	−0.272990
$260$	0	0
$261$	0	0
$262$	−816.401	−0.192509
$263$	283.839	0.0665484	0.0332742	−	0.999446i	$-0.489407\pi$
$263$	283.839	0.0665484	0.0332742	+	0.999446i	$0.489407\pi$
$264$	0	0
$265$	0	0
$266$	1475.75	0.340166
$267$	0	0
$268$	−843.165	−0.192181
$269$	−2426.21	−0.549920	−0.274960	−	0.961456i	$-0.588665\pi$
$269$	−2426.21	−0.549920	−0.274960	+	0.961456i	$0.588665\pi$
$270$	0	0
$271$	−174.946	−0.0392148	−0.0196074	−	0.999808i	$-0.506242\pi$
$271$	−174.946	−0.0392148	−0.0196074	+	0.999808i	$0.506242\pi$
$272$	9631.80	2.14711
$273$	0	0
$274$	3054.54	0.673472
$275$	0	0
$276$	0	0
$277$	7807.07	1.69344	0.846718	−	0.532042i	$-0.178575\pi$
$277$	7807.07	1.69344	0.846718	+	0.532042i	$0.178575\pi$
$278$	−6884.14	−1.48519
$279$	0	0
$280$	0	0
$281$	−584.171	−0.124017	−0.0620084	−	0.998076i	$-0.519751\pi$
$281$	−584.171	−0.124017	−0.0620084	+	0.998076i	$0.519751\pi$
$282$	0	0
$283$	5897.31	1.23872	0.619362	−	0.785106i	$-0.287391\pi$
$283$	5897.31	1.23872	0.619362	+	0.785106i	$0.287391\pi$
$284$	−1638.40	−0.342329
$285$	0	0
$286$	−6413.73	−1.32605
$287$	−933.239	−0.191942
$288$	0	0
$289$	9771.10	1.98883
$290$	0	0
$291$	0	0
$292$	62.7846	0.0125828
$293$	1609.73	0.320960	0.160480	−	0.987039i	$-0.448696\pi$
$293$	1609.73	0.320960	0.160480	+	0.987039i	$0.448696\pi$
$294$	0	0
$295$	0	0
$296$	−592.448	−0.116336
$297$	0	0
$298$	−12095.9	−2.35133
$299$	831.758	0.160876
$300$	0	0
$301$	9969.62	1.90910
$302$	4646.64	0.885378
$303$	0	0
$304$	1147.41	0.216475
$305$	0	0
$306$	0	0
$307$	−234.473	−0.0435898	−0.0217949	−	0.999762i	$-0.506938\pi$
$307$	−234.473	−0.0435898	−0.0217949	+	0.999762i	$0.506938\pi$
$308$	3140.28	0.580955
$309$	0	0
$310$	0	0
$311$	1795.25	0.327329	0.163665	−	0.986516i	$-0.447669\pi$
$311$	1795.25	0.327329	0.163665	+	0.986516i	$0.447669\pi$
$312$	0	0
$313$	8440.61	1.52425	0.762127	−	0.647427i	$-0.224155\pi$
$313$	8440.61	1.52425	0.762127	+	0.647427i	$0.224155\pi$
$314$	442.533	0.0795336
$315$	0	0
$316$	844.917	0.150412
$317$	10551.7	1.86953	0.934766	−	0.355264i	$-0.115609\pi$
$317$	10551.7	1.86953	0.934766	+	0.355264i	$0.115609\pi$
$318$	0	0
$319$	−2390.65	−0.419595
$320$	0	0
$321$	0	0
$322$	−1399.86	−0.242270
$323$	1749.28	0.301339
$324$	0	0
$325$	0	0
$326$	−9733.21	−1.65360
$327$	0	0
$328$	−485.900	−0.0817968
$329$	13680.3	2.29247
$330$	0	0
$331$	6743.17	1.11975	0.559876	−	0.828576i	$-0.310849\pi$
$331$	6743.17	1.11975	0.559876	+	0.828576i	$0.310849\pi$
$332$	3002.91	0.496405
$333$	0	0
$334$	874.265	0.143227
$335$	0	0
$336$	0	0
$337$	8437.26	1.36382	0.681909	−	0.731437i	$-0.261150\pi$
$337$	8437.26	1.36382	0.681909	+	0.731437i	$0.261150\pi$
$338$	5013.62	0.806819
$339$	0	0
$340$	0	0
$341$	−5754.26	−0.913814
$342$	0	0
$343$	−7314.45	−1.15144
$344$	5190.78	0.813571
$345$	0	0
$346$	−6502.25	−1.01030
$347$	−1848.85	−0.286027	−0.143013	−	0.989721i	$-0.545679\pi$
$347$	−1848.85	−0.286027	−0.143013	+	0.989721i	$0.545679\pi$
$348$	0	0
$349$	−1148.38	−0.176136	−0.0880678	−	0.996114i	$-0.528069\pi$
$349$	−1148.38	−0.176136	−0.0880678	+	0.996114i	$0.528069\pi$
$350$	0	0
$351$	0	0
$352$	4407.54	0.667393
$353$	−5753.60	−0.867516	−0.433758	−	0.901029i	$-0.642813\pi$
$353$	−5753.60	−0.867516	−0.433758	+	0.901029i	$0.642813\pi$
$354$	0	0
$355$	0	0
$356$	3478.09	0.517804
$357$	0	0
$358$	−1936.83	−0.285935
$359$	−5452.01	−0.801521	−0.400761	−	0.916183i	$-0.631254\pi$
$359$	−5452.01	−0.801521	−0.400761	+	0.916183i	$0.631254\pi$
$360$	0	0
$361$	−6650.61	−0.969619
$362$	−6590.31	−0.956848
$363$	0	0
$364$	−6067.91	−0.873749
$365$	0	0
$366$	0	0
$367$	−8385.93	−1.19276	−0.596379	−	0.802703i	$-0.703394\pi$
$367$	−8385.93	−1.19276	−0.596379	+	0.802703i	$0.703394\pi$
$368$	−1088.40	−0.154176
$369$	0	0
$370$	0	0
$371$	9190.05	1.28605
$372$	0	0
$373$	−2728.30	−0.378730	−0.189365	−	0.981907i	$-0.560643\pi$
$373$	−2728.30	−0.378730	−0.189365	+	0.981907i	$0.560643\pi$
$374$	12795.1	1.76903
$375$	0	0
$376$	7122.81	0.976944
$377$	4619.41	0.631065
$378$	0	0
$379$	3348.99	0.453895	0.226947	−	0.973907i	$-0.427125\pi$
$379$	3348.99	0.453895	0.226947	+	0.973907i	$0.427125\pi$
$380$	0	0
$381$	0	0
$382$	14506.2	1.94294
$383$	10430.3	1.39155	0.695774	−	0.718261i	$-0.255062\pi$
$383$	10430.3	1.39155	0.695774	+	0.718261i	$0.255062\pi$
$384$	0	0
$385$	0	0
$386$	−9.99766	−0.00131831
$387$	0	0
$388$	−1710.03	−0.223746
$389$	−9827.23	−1.28088	−0.640438	−	0.768010i	$-0.721247\pi$
$389$	−9827.23	−1.28088	−0.640438	+	0.768010i	$0.721247\pi$
$390$	0	0
$391$	−1659.32	−0.214617
$392$	−9243.73	−1.19102
$393$	0	0
$394$	1913.58	0.244682
$395$	0	0
$396$	0	0
$397$	436.382	0.0551672	0.0275836	−	0.999619i	$-0.491219\pi$
$397$	436.382	0.0551672	0.0275836	+	0.999619i	$0.491219\pi$
$398$	−10247.1	−1.29055
$399$	0	0
$400$	0	0
$401$	14501.5	1.80591	0.902955	−	0.429736i	$-0.141393\pi$
$401$	14501.5	1.80591	0.902955	+	0.429736i	$0.141393\pi$
$402$	0	0
$403$	11118.8	1.37436
$404$	−1139.19	−0.140289
$405$	0	0
$406$	−7774.51	−0.950351
$407$	−1175.26	−0.143134
$408$	0	0
$409$	−12058.4	−1.45782	−0.728911	−	0.684609i	$-0.759973\pi$
$409$	−12058.4	−1.45782	−0.728911	+	0.684609i	$0.759973\pi$
$410$	0	0
$411$	0	0
$412$	2529.76	0.302505
$413$	10377.6	1.23644
$414$	0	0
$415$	0	0
$416$	−8516.60	−1.00375
$417$	0	0
$418$	1524.24	0.178356
$419$	−6042.95	−0.704577	−0.352288	−	0.935892i	$-0.614596\pi$
$419$	−6042.95	−0.704577	−0.352288	+	0.935892i	$0.614596\pi$
$420$	0	0
$421$	−9994.67	−1.15703	−0.578516	−	0.815671i	$-0.696368\pi$
$421$	−9994.67	−1.15703	−0.578516	+	0.815671i	$0.696368\pi$
$422$	−169.659	−0.0195708
$423$	0	0
$424$	4784.90	0.548054
$425$	0	0
$426$	0	0
$427$	−18842.6	−2.13550
$428$	−4645.55	−0.524652
$429$	0	0
$430$	0	0
$431$	−9327.32	−1.04242	−0.521208	−	0.853430i	$-0.674518\pi$
$431$	−9327.32	−1.04242	−0.521208	+	0.853430i	$0.674518\pi$
$432$	0	0
$433$	7861.22	0.872485	0.436243	−	0.899829i	$-0.356309\pi$
$433$	7861.22	0.872485	0.436243	+	0.899829i	$0.356309\pi$
$434$	−18713.1	−2.06972
$435$	0	0
$436$	2981.51	0.327496
$437$	−197.670	−0.0216380
$438$	0	0
$439$	7412.06	0.805828	0.402914	−	0.915238i	$-0.367997\pi$
$439$	7412.06	0.805828	0.402914	+	0.915238i	$0.367997\pi$
$440$	0	0
$441$	0	0
$442$	−24723.6	−2.66060
$443$	−3043.66	−0.326430	−0.163215	−	0.986591i	$-0.552186\pi$
$443$	−3043.66	−0.326430	−0.163215	+	0.986591i	$0.552186\pi$
$444$	0	0
$445$	0	0
$446$	−18317.9	−1.94480
$447$	0	0
$448$	−5019.81	−0.529383
$449$	9547.87	1.00355	0.501773	−	0.864999i	$-0.332681\pi$
$449$	9547.87	1.00355	0.501773	+	0.864999i	$0.332681\pi$
$450$	0	0
$451$	−963.902	−0.100639
$452$	−6728.31	−0.700162
$453$	0	0
$454$	−16045.9	−1.65875
$455$	0	0
$456$	0	0
$457$	−13401.9	−1.37180	−0.685901	−	0.727695i	$-0.740592\pi$
$457$	−13401.9	−1.37180	−0.685901	+	0.727695i	$0.740592\pi$
$458$	−7005.00	−0.714677
$459$	0	0
$460$	0	0
$461$	4137.03	0.417962	0.208981	−	0.977920i	$-0.432985\pi$
$461$	4137.03	0.417962	0.208981	+	0.977920i	$0.432985\pi$
$462$	0	0
$463$	13976.3	1.40288	0.701439	−	0.712729i	$-0.252541\pi$
$463$	13976.3	1.40288	0.701439	+	0.712729i	$0.252541\pi$
$464$	−6044.75	−0.604786
$465$	0	0
$466$	21780.9	2.16519
$467$	−10796.5	−1.06982	−0.534908	−	0.844910i	$-0.679654\pi$
$467$	−10796.5	−1.06982	−0.534908	+	0.844910i	$0.679654\pi$
$468$	0	0
$469$	7818.60	0.769785
$470$	0	0
$471$	0	0
$472$	5403.21	0.526913
$473$	10297.2	1.00098
$474$	0	0
$475$	0	0
$476$	12105.2	1.16563
$477$	0	0
$478$	−7505.85	−0.718221
$479$	−14568.4	−1.38966	−0.694830	−	0.719174i	$-0.744521\pi$
$479$	−14568.4	−1.38966	−0.694830	+	0.719174i	$0.744521\pi$
$480$	0	0
$481$	2270.94	0.215272
$482$	−8037.89	−0.759577
$483$	0	0
$484$	−1125.15	−0.105668
$485$	0	0
$486$	0	0
$487$	11456.6	1.06601	0.533004	−	0.846113i	$-0.321063\pi$
$487$	11456.6	1.06601	0.533004	+	0.846113i	$0.321063\pi$
$488$	−9810.61	−0.910052
$489$	0	0
$490$	0	0
$491$	19666.5	1.80761	0.903804	−	0.427948i	$-0.140763\pi$
$491$	19666.5	1.80761	0.903804	+	0.427948i	$0.140763\pi$
$492$	0	0
$493$	−9215.48	−0.841875
$494$	−2945.26	−0.268246
$495$	0	0
$496$	−14549.6	−1.31713
$497$	15192.8	1.37121
$498$	0	0
$499$	8379.31	0.751722	0.375861	−	0.926676i	$-0.377347\pi$
$499$	8379.31	0.751722	0.375861	+	0.926676i	$0.377347\pi$
$500$	0	0
$501$	0	0
$502$	2057.87	0.182962
$503$	−15678.1	−1.38976	−0.694881	−	0.719124i	$-0.744543\pi$
$503$	−15678.1	−1.38976	−0.694881	+	0.719124i	$0.744543\pi$
$504$	0	0
$505$	0	0
$506$	−1445.85	−0.127028
$507$	0	0
$508$	−924.413	−0.0807366
$509$	17037.3	1.48363	0.741813	−	0.670606i	$-0.233966\pi$
$509$	17037.3	1.48363	0.741813	+	0.670606i	$0.233966\pi$
$510$	0	0
$511$	−582.197	−0.0504009
$512$	1067.93	0.0921798
$513$	0	0
$514$	−1028.20	−0.0882334
$515$	0	0
$516$	0	0
$517$	14129.8	1.20199
$518$	−3822.02	−0.324189
$519$	0	0
$520$	0	0
$521$	8776.12	0.737982	0.368991	−	0.929433i	$-0.379703\pi$
$521$	8776.12	0.737982	0.368991	+	0.929433i	$0.379703\pi$
$522$	0	0
$523$	−11120.4	−0.929753	−0.464877	−	0.885375i	$-0.653901\pi$
$523$	−11120.4	−0.929753	−0.464877	+	0.885375i	$0.653901\pi$
$524$	−797.759	−0.0665082
$525$	0	0
$526$	953.385	0.0790296
$527$	−22181.5	−1.83348
$528$	0	0
$529$	−11979.5	−0.984589
$530$	0	0
$531$	0	0
$532$	1442.06	0.117521
$533$	1862.53	0.151360
$534$	0	0
$535$	0	0
$536$	4070.83	0.328047
$537$	0	0
$538$	−8149.39	−0.653058
$539$	−18337.2	−1.46538
$540$	0	0
$541$	21730.6	1.72693	0.863467	−	0.504405i	$-0.168288\pi$
$541$	21730.6	1.72693	0.863467	+	0.504405i	$0.168288\pi$
$542$	−587.626	−0.0465695
$543$	0	0
$544$	16990.2	1.33906
$545$	0	0
$546$	0	0
$547$	6926.17	0.541392	0.270696	−	0.962665i	$-0.412746\pi$
$547$	6926.17	0.541392	0.270696	+	0.962665i	$0.412746\pi$
$548$	2984.79	0.232671
$549$	0	0
$550$	0	0
$551$	−1097.82	−0.0848793
$552$	0	0
$553$	−7834.85	−0.602480
$554$	26223.2	2.01104
$555$	0	0
$556$	−6726.95	−0.513104
$557$	−6589.22	−0.501246	−0.250623	−	0.968085i	$-0.580636\pi$
$557$	−6589.22	−0.501246	−0.250623	+	0.968085i	$0.580636\pi$
$558$	0	0
$559$	−19897.0	−1.50547
$560$	0	0
$561$	0	0
$562$	−1962.17	−0.147276
$563$	3839.63	0.287426	0.143713	−	0.989619i	$-0.454096\pi$
$563$	3839.63	0.287426	0.143713	+	0.989619i	$0.454096\pi$
$564$	0	0
$565$	0	0
$566$	19808.5	1.47105
$567$	0	0
$568$	7910.28	0.584345
$569$	20874.0	1.53794	0.768968	−	0.639287i	$-0.220771\pi$
$569$	20874.0	1.53794	0.768968	+	0.639287i	$0.220771\pi$
$570$	0	0
$571$	21175.9	1.55199	0.775994	−	0.630740i	$-0.217248\pi$
$571$	21175.9	1.55199	0.775994	+	0.630740i	$0.217248\pi$
$572$	−6267.28	−0.458126
$573$	0	0
$574$	−3134.66	−0.227941
$575$	0	0
$576$	0	0
$577$	14924.2	1.07678	0.538391	−	0.842695i	$-0.319032\pi$
$577$	14924.2	1.07678	0.538391	+	0.842695i	$0.319032\pi$
$578$	32820.1	2.36183
$579$	0	0
$580$	0	0
$581$	−27845.8	−1.98836
$582$	0	0
$583$	9492.00	0.674303
$584$	−303.127	−0.0214785
$585$	0	0
$586$	5406.92	0.381156
$587$	25218.0	1.77318	0.886592	−	0.462552i	$-0.153066\pi$
$587$	25218.0	1.77318	0.886592	+	0.462552i	$0.153066\pi$
$588$	0	0
$589$	−2642.42	−0.184854
$590$	0	0
$591$	0	0
$592$	−2971.65	−0.206308
$593$	5011.77	0.347063	0.173532	−	0.984828i	$-0.444482\pi$
$593$	5011.77	0.347063	0.173532	+	0.984828i	$0.444482\pi$
$594$	0	0
$595$	0	0
$596$	−11819.7	−0.812337
$597$	0	0
$598$	2793.79	0.191048
$599$	−4943.41	−0.337199	−0.168600	−	0.985685i	$-0.553925\pi$
$599$	−4943.41	−0.337199	−0.168600	+	0.985685i	$0.553925\pi$
$600$	0	0
$601$	−24334.8	−1.65164	−0.825821	−	0.563932i	$-0.809288\pi$
$601$	−24334.8	−1.65164	−0.825821	+	0.563932i	$0.809288\pi$
$602$	33486.9	2.26715
$603$	0	0
$604$	4540.54	0.305881
$605$	0	0
$606$	0	0
$607$	−28973.8	−1.93741	−0.968707	−	0.248207i	$-0.920159\pi$
$607$	−28973.8	−1.93741	−0.968707	+	0.248207i	$0.920159\pi$
$608$	2023.99	0.135006
$609$	0	0
$610$	0	0
$611$	−27302.8	−1.80778
$612$	0	0
$613$	−15139.1	−0.997490	−0.498745	−	0.866749i	$-0.666206\pi$
$613$	−15139.1	−0.997490	−0.498745	+	0.866749i	$0.666206\pi$
$614$	−787.571	−0.0517651
$615$	0	0
$616$	−15161.4	−0.991673
$617$	13894.8	0.906617	0.453309	−	0.891354i	$-0.350244\pi$
$617$	13894.8	0.906617	0.453309	+	0.891354i	$0.350244\pi$
$618$	0	0
$619$	−4589.69	−0.298021	−0.149011	−	0.988836i	$-0.547609\pi$
$619$	−4589.69	−0.298021	−0.149011	+	0.988836i	$0.547609\pi$
$620$	0	0
$621$	0	0
$622$	6030.07	0.388720
$623$	−32252.0	−2.07408
$624$	0	0
$625$	0	0
$626$	28351.2	1.81013
$627$	0	0
$628$	432.428	0.0274773
$629$	−4530.41	−0.287185
$630$	0	0
$631$	3005.77	0.189632	0.0948160	−	0.995495i	$-0.469774\pi$
$631$	3005.77	0.189632	0.0948160	+	0.995495i	$0.469774\pi$
$632$	−4079.29	−0.256749
$633$	0	0
$634$	35442.0	2.22016
$635$	0	0
$636$	0	0
$637$	35432.6	2.20391
$638$	−8029.96	−0.498290
$639$	0	0
$640$	0	0
$641$	5631.47	0.347004	0.173502	−	0.984834i	$-0.444492\pi$
$641$	5631.47	0.347004	0.173502	+	0.984834i	$0.444492\pi$
$642$	0	0
$643$	−11305.1	−0.693358	−0.346679	−	0.937984i	$-0.612691\pi$
$643$	−11305.1	−0.693358	−0.346679	+	0.937984i	$0.612691\pi$
$644$	−1367.89	−0.0836997
$645$	0	0
$646$	5875.64	0.357855
$647$	−8614.30	−0.523436	−0.261718	−	0.965144i	$-0.584289\pi$
$647$	−8614.30	−0.523436	−0.261718	+	0.965144i	$0.584289\pi$
$648$	0	0
$649$	10718.6	0.648291
$650$	0	0
$651$	0	0
$652$	−9510.96	−0.571285
$653$	12639.8	0.757479	0.378739	−	0.925503i	$-0.376358\pi$
$653$	12639.8	0.757479	0.378739	+	0.925503i	$0.376358\pi$
$654$	0	0
$655$	0	0
$656$	−2437.22	−0.145057
$657$	0	0
$658$	45950.9	2.72242
$659$	−13640.4	−0.806306	−0.403153	−	0.915133i	$-0.632086\pi$
$659$	−13640.4	−0.806306	−0.403153	+	0.915133i	$0.632086\pi$
$660$	0	0
$661$	−17052.0	−1.00340	−0.501699	−	0.865042i	$-0.667291\pi$
$661$	−17052.0	−1.00340	−0.501699	+	0.865042i	$0.667291\pi$
$662$	22649.6	1.32976
$663$	0	0
$664$	−14498.2	−0.847348
$665$	0	0
$666$	0	0
$667$	1041.36	0.0604521
$668$	854.302	0.0494820
$669$	0	0
$670$	0	0
$671$	−19461.7	−1.11969
$672$	0	0
$673$	16419.7	0.940467	0.470234	−	0.882542i	$-0.344170\pi$
$673$	16419.7	0.940467	0.470234	+	0.882542i	$0.344170\pi$
$674$	28339.9	1.61960
$675$	0	0
$676$	4899.14	0.278740
$677$	−8670.47	−0.492221	−0.246110	−	0.969242i	$-0.579153\pi$
$677$	−8670.47	−0.492221	−0.246110	+	0.969242i	$0.579153\pi$
$678$	0	0
$679$	15856.9	0.896220
$680$	0	0
$681$	0	0
$682$	−19328.0	−1.08520
$683$	5973.36	0.334647	0.167324	−	0.985902i	$-0.446488\pi$
$683$	5973.36	0.334647	0.167324	+	0.985902i	$0.446488\pi$
$684$	0	0
$685$	0	0
$686$	−24568.5	−1.36739
$687$	0	0
$688$	26036.4	1.44277
$689$	−18341.2	−1.01414
$690$	0	0
$691$	−15316.3	−0.843212	−0.421606	−	0.906779i	$-0.638533\pi$
$691$	−15316.3	−0.843212	−0.421606	+	0.906779i	$0.638533\pi$
$692$	−6353.78	−0.349038
$693$	0	0
$694$	−6210.09	−0.339671
$695$	0	0
$696$	0	0
$697$	−3715.65	−0.201923
$698$	−3857.29	−0.209170
$699$	0	0
$700$	0	0
$701$	−34583.1	−1.86332	−0.931660	−	0.363333i	$-0.881639\pi$
$701$	−34583.1	−1.86332	−0.931660	+	0.363333i	$0.881639\pi$
$702$	0	0
$703$	−539.695	−0.0289545
$704$	−5184.74	−0.277567
$705$	0	0
$706$	−19325.8	−1.03022
$707$	10563.6	0.561929
$708$	0	0
$709$	11194.1	0.592955	0.296477	−	0.955040i	$-0.404188\pi$
$709$	11194.1	0.592955	0.296477	+	0.955040i	$0.404188\pi$
$710$	0	0
$711$	0	0
$712$	−16792.4	−0.883877
$713$	2506.53	0.131655
$714$	0	0
$715$	0	0
$716$	−1892.61	−0.0987850
$717$	0	0
$718$	−18312.8	−0.951847
$719$	15491.9	0.803549	0.401774	−	0.915739i	$-0.368394\pi$
$719$	15491.9	0.803549	0.401774	+	0.915739i	$0.368394\pi$
$720$	0	0
$721$	−23458.2	−1.21169
$722$	−22338.7	−1.15147
$723$	0	0
$724$	−6439.83	−0.330572
$725$	0	0
$726$	0	0
$727$	6272.72	0.320003	0.160002	−	0.987117i	$-0.448850\pi$
$727$	6272.72	0.320003	0.160002	+	0.987117i	$0.448850\pi$
$728$	29296.1	1.49146
$729$	0	0
$730$	0	0
$731$	39693.6	2.00837
$732$	0	0
$733$	−24980.5	−1.25877	−0.629383	−	0.777095i	$-0.716692\pi$
$733$	−24980.5	−1.25877	−0.629383	+	0.777095i	$0.716692\pi$
$734$	−28167.5	−1.41646
$735$	0	0
$736$	−1919.90	−0.0961530
$737$	8075.49	0.403615
$738$	0	0
$739$	−30660.7	−1.52621	−0.763107	−	0.646272i	$-0.776327\pi$
$739$	−30660.7	−1.52621	−0.763107	+	0.646272i	$0.776327\pi$
$740$	0	0
$741$	0	0
$742$	30868.5	1.52725
$743$	−17205.7	−0.849551	−0.424776	−	0.905299i	$-0.639647\pi$
$743$	−17205.7	−0.849551	−0.424776	+	0.905299i	$0.639647\pi$
$744$	0	0
$745$	0	0
$746$	−9164.09	−0.449760
$747$	0	0
$748$	12502.9	0.611165
$749$	43077.8	2.10151
$750$	0	0
$751$	18397.1	0.893901	0.446950	−	0.894559i	$-0.352510\pi$
$751$	18397.1	0.893901	0.446950	+	0.894559i	$0.352510\pi$
$752$	35727.2	1.73250
$753$	0	0
$754$	15516.1	0.749422
$755$	0	0
$756$	0	0
$757$	−22305.1	−1.07093	−0.535465	−	0.844557i	$-0.679864\pi$
$757$	−22305.1	−1.07093	−0.535465	+	0.844557i	$0.679864\pi$
$758$	11248.9	0.539023
$759$	0	0
$760$	0	0
$761$	−14458.5	−0.688727	−0.344364	−	0.938836i	$-0.611905\pi$
$761$	−14458.5	−0.688727	−0.344364	+	0.938836i	$0.611905\pi$
$762$	0	0
$763$	−27647.3	−1.31179
$764$	14175.0	0.671249
$765$	0	0
$766$	35034.3	1.65253
$767$	−20711.3	−0.975022
$768$	0	0
$769$	−39897.6	−1.87093	−0.935463	−	0.353424i	$-0.885017\pi$
$769$	−39897.6	−1.87093	−0.935463	+	0.353424i	$0.885017\pi$
$770$	0	0
$771$	0	0
$772$	−9.76938	−0.000455450 0
$773$	20070.2	0.933863	0.466931	−	0.884294i	$-0.345359\pi$
$773$	20070.2	0.933863	0.466931	+	0.884294i	$0.345359\pi$
$774$	0	0
$775$	0	0
$776$	8256.08	0.381928
$777$	0	0
$778$	−33008.7	−1.52110
$779$	−442.635	−0.0203582
$780$	0	0
$781$	15692.0	0.718953
$782$	−5573.47	−0.254868
$783$	0	0
$784$	−46365.5	−2.11213
$785$	0	0
$786$	0	0
$787$	−10733.0	−0.486137	−0.243069	−	0.970009i	$-0.578154\pi$
$787$	−10733.0	−0.486137	−0.243069	+	0.970009i	$0.578154\pi$
$788$	1869.89	0.0845329
$789$	0	0
$790$	0	0
$791$	62391.1	2.80452
$792$	0	0
$793$	37605.5	1.68400
$794$	1465.76	0.0655139
$795$	0	0
$796$	−10013.1	−0.445861
$797$	−14335.8	−0.637140	−0.318570	−	0.947899i	$-0.603203\pi$
$797$	−14335.8	−0.637140	−0.318570	+	0.947899i	$0.603203\pi$
$798$	0	0
$799$	54467.7	2.41167
$800$	0	0
$801$	0	0
$802$	48709.0	2.14461
$803$	−601.325	−0.0264263
$804$	0	0
$805$	0	0
$806$	37347.1	1.63213
$807$	0	0
$808$	5500.03	0.239468
$809$	−20920.7	−0.909187	−0.454593	−	0.890699i	$-0.650215\pi$
$809$	−20920.7	−0.909187	−0.454593	+	0.890699i	$0.650215\pi$
$810$	0	0
$811$	12816.5	0.554931	0.277465	−	0.960736i	$-0.410506\pi$
$811$	12816.5	0.554931	0.277465	+	0.960736i	$0.410506\pi$
$812$	−7596.99	−0.328328
$813$	0	0
$814$	−3947.59	−0.169979
$815$	0	0
$816$	0	0
$817$	4728.59	0.202488
$818$	−40502.9	−1.73124
$819$	0	0
$820$	0	0
$821$	−7253.55	−0.308344	−0.154172	−	0.988044i	$-0.549271\pi$
$821$	−7253.55	−0.308344	−0.154172	+	0.988044i	$0.549271\pi$
$822$	0	0
$823$	−35288.1	−1.49461	−0.747307	−	0.664479i	$-0.768654\pi$
$823$	−35288.1	−1.49461	−0.747307	+	0.664479i	$0.768654\pi$
$824$	−12213.8	−0.516367
$825$	0	0
$826$	34857.3	1.46833
$827$	−32205.9	−1.35418	−0.677092	−	0.735899i	$-0.736760\pi$
$827$	−32205.9	−1.35418	−0.677092	+	0.735899i	$0.736760\pi$
$828$	0	0
$829$	29993.3	1.25659	0.628294	−	0.777976i	$-0.283754\pi$
$829$	29993.3	1.25659	0.628294	+	0.777976i	$0.283754\pi$
$830$	0	0
$831$	0	0
$832$	10018.4	0.417457
$833$	−70686.2	−2.94013
$834$	0	0
$835$	0	0
$836$	1489.44	0.0616187
$837$	0	0
$838$	−20297.7	−0.836720
$839$	35608.7	1.46526	0.732628	−	0.680629i	$-0.238294\pi$
$839$	35608.7	1.46526	0.732628	+	0.680629i	$0.238294\pi$
$840$	0	0
$841$	−18605.5	−0.762865
$842$	−33571.1	−1.37403
$843$	0	0
$844$	−165.785	−0.00676134
$845$	0	0
$846$	0	0
$847$	10433.4	0.423255
$848$	24000.5	0.971911
$849$	0	0
$850$	0	0
$851$	511.940	0.0206217
$852$	0	0
$853$	11229.3	0.450744	0.225372	−	0.974273i	$-0.427640\pi$
$853$	11229.3	0.450744	0.225372	+	0.974273i	$0.427640\pi$
$854$	−63290.5	−2.53601
$855$	0	0
$856$	22428.9	0.895565
$857$	22136.1	0.882327	0.441164	−	0.897427i	$-0.354566\pi$
$857$	22136.1	0.882327	0.441164	+	0.897427i	$0.354566\pi$
$858$	0	0
$859$	−820.727	−0.0325994	−0.0162997	−	0.999867i	$-0.505189\pi$
$859$	−820.727	−0.0325994	−0.0162997	+	0.999867i	$0.505189\pi$
$860$	0	0
$861$	0	0
$862$	−31329.5	−1.23792
$863$	−245.223	−0.00967264	−0.00483632	−	0.999988i	$-0.501539\pi$
$863$	−245.223	−0.00967264	−0.00483632	+	0.999988i	$0.501539\pi$
$864$	0	0
$865$	0	0
$866$	26405.0	1.03612
$867$	0	0
$868$	−18285.8	−0.715048
$869$	−8092.27	−0.315894
$870$	0	0
$871$	−15604.1	−0.607032
$872$	−14394.8	−0.559026
$873$	0	0
$874$	−663.952	−0.0256963
$875$	0	0
$876$	0	0
$877$	−37727.8	−1.45265	−0.726326	−	0.687350i	$-0.758774\pi$
$877$	−37727.8	−1.45265	−0.726326	+	0.687350i	$0.758774\pi$
$878$	24896.4	0.956961
$879$	0	0
$880$	0	0
$881$	−21738.8	−0.831326	−0.415663	−	0.909519i	$-0.636450\pi$
$881$	−21738.8	−0.831326	−0.415663	+	0.909519i	$0.636450\pi$
$882$	0	0
$883$	44340.4	1.68989	0.844946	−	0.534852i	$-0.179632\pi$
$883$	44340.4	1.68989	0.844946	+	0.534852i	$0.179632\pi$
$884$	−24159.1	−0.919184
$885$	0	0
$886$	−10223.3	−0.387652
$887$	−681.008	−0.0257790	−0.0128895	−	0.999917i	$-0.504103\pi$
$887$	−681.008	−0.0257790	−0.0128895	+	0.999917i	$0.504103\pi$
$888$	0	0
$889$	8572.00	0.323392
$890$	0	0
$891$	0	0
$892$	−17899.7	−0.671889
$893$	6488.58	0.243149
$894$	0	0
$895$	0	0
$896$	−50999.6	−1.90154
$897$	0	0
$898$	32070.3	1.19176
$899$	13920.7	0.516443
$900$	0	0
$901$	36589.8	1.35292
$902$	−3237.65	−0.119514
$903$	0	0
$904$	32484.6	1.19516
$905$	0	0
$906$	0	0
$907$	−5348.01	−0.195786	−0.0978929	−	0.995197i	$-0.531210\pi$
$907$	−5348.01	−0.195786	−0.0978929	+	0.995197i	$0.531210\pi$
$908$	−15679.5	−0.573066
$909$	0	0
$910$	0	0
$911$	−14488.7	−0.526930	−0.263465	−	0.964669i	$-0.584865\pi$
$911$	−14488.7	−0.526930	−0.263465	+	0.964669i	$0.584865\pi$
$912$	0	0
$913$	−28760.7	−1.04254
$914$	−45015.6	−1.62908
$915$	0	0
$916$	−6845.05	−0.246907
$917$	7397.56	0.266400
$918$	0	0
$919$	−22546.9	−0.809308	−0.404654	−	0.914470i	$-0.632608\pi$
$919$	−22546.9	−0.809308	−0.404654	+	0.914470i	$0.632608\pi$
$920$	0	0
$921$	0	0
$922$	13895.9	0.496351
$923$	−30321.3	−1.08130
$924$	0	0
$925$	0	0
$926$	46944.9	1.66599
$927$	0	0
$928$	−10662.7	−0.377178
$929$	13980.7	0.493747	0.246874	−	0.969048i	$-0.420597\pi$
$929$	13980.7	0.493747	0.246874	+	0.969048i	$0.420597\pi$
$930$	0	0
$931$	−8420.65	−0.296429
$932$	21283.5	0.748032
$933$	0	0
$934$	−36264.5	−1.27046
$935$	0	0
$936$	0	0
$937$	−26362.6	−0.919133	−0.459567	−	0.888143i	$-0.651995\pi$
$937$	−26362.6	−0.919133	−0.459567	+	0.888143i	$0.651995\pi$
$938$	26261.9	0.914159
$939$	0	0
$940$	0	0
$941$	14715.6	0.509792	0.254896	−	0.966968i	$-0.417959\pi$
$941$	14715.6	0.509792	0.254896	+	0.966968i	$0.417959\pi$
$942$	0	0
$943$	419.871	0.0144994
$944$	27101.9	0.934419
$945$	0	0
$946$	34587.2	1.18872
$947$	−11818.2	−0.405532	−0.202766	−	0.979227i	$-0.564993\pi$
$947$	−11818.2	−0.405532	−0.202766	+	0.979227i	$0.564993\pi$
$948$	0	0
$949$	1161.93	0.0397448
$950$	0	0
$951$	0	0
$952$	−58444.2	−1.98969
$953$	−43832.3	−1.48989	−0.744945	−	0.667125i	$-0.767524\pi$
$953$	−43832.3	−1.48989	−0.744945	+	0.667125i	$0.767524\pi$
$954$	0	0
$955$	0	0
$956$	−7334.46	−0.248131
$957$	0	0
$958$	−48933.8	−1.65029
$959$	−27677.7	−0.931971
$960$	0	0
$961$	3716.00	0.124736
$962$	7627.86	0.255647
$963$	0	0
$964$	−7854.36	−0.262419
$965$	0	0
$966$	0	0
$967$	−10696.2	−0.355706	−0.177853	−	0.984057i	$-0.556915\pi$
$967$	−10696.2	−0.355706	−0.177853	+	0.984057i	$0.556915\pi$
$968$	5432.27	0.180372
$969$	0	0
$970$	0	0
$971$	−27933.0	−0.923187	−0.461593	−	0.887092i	$-0.652722\pi$
$971$	−27933.0	−0.923187	−0.461593	+	0.887092i	$0.652722\pi$
$972$	0	0
$973$	62378.4	2.05525
$974$	38481.4	1.26594
$975$	0	0
$976$	−49208.9	−1.61387
$977$	24341.7	0.797094	0.398547	−	0.917148i	$-0.369515\pi$
$977$	24341.7	0.797094	0.398547	+	0.917148i	$0.369515\pi$
$978$	0	0
$979$	−33311.7	−1.08748
$980$	0	0
$981$	0	0
$982$	66057.7	2.14662
$983$	12553.0	0.407301	0.203651	−	0.979044i	$-0.434719\pi$
$983$	12553.0	0.407301	0.203651	+	0.979044i	$0.434719\pi$
$984$	0	0
$985$	0	0
$986$	−30953.9	−0.999769
$987$	0	0
$988$	−2878.01	−0.0926737
$989$	−4485.41	−0.144214
$990$	0	0
$991$	−45631.8	−1.46271	−0.731353	−	0.681999i	$-0.761111\pi$
$991$	−45631.8	−1.46271	−0.731353	+	0.681999i	$0.761111\pi$
$992$	−25665.0	−0.821437
$993$	0	0
$994$	51031.0	1.62838
$995$	0	0
$996$	0	0
$997$	34499.0	1.09588	0.547941	−	0.836517i	$-0.315412\pi$
$997$	34499.0	1.09588	0.547941	+	0.836517i	$0.315412\pi$
$998$	28145.3	0.892708
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.a.j.1.2		2
3.2	odd	2		75.4.a.e.1.1	yes	2
5.2	odd	4		225.4.b.h.199.3		4
5.3	odd	4		225.4.b.h.199.2		4
5.4	even	2		225.4.a.n.1.1		2
12.11	even	2		1200.4.a.bu.1.2		2
15.2	even	4		75.4.b.c.49.2		4
15.8	even	4		75.4.b.c.49.3		4
15.14	odd	2		75.4.a.d.1.2	✓	2
60.23	odd	4		1200.4.f.v.49.3		4
60.47	odd	4		1200.4.f.v.49.2		4
60.59	even	2		1200.4.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.2	✓	2	15.14	odd	2
75.4.a.e.1.1	yes	2	3.2	odd	2
75.4.b.c.49.2		4	15.2	even	4
75.4.b.c.49.3		4	15.8	even	4
225.4.a.j.1.2		2	1.1	even	1	trivial
225.4.a.n.1.1		2	5.4	even	2
225.4.b.h.199.2		4	5.3	odd	4
225.4.b.h.199.3		4	5.2	odd	4
1200.4.a.bl.1.1		2	60.59	even	2
1200.4.a.bu.1.2		2	12.11	even	2
1200.4.f.v.49.2		4	60.47	odd	4
1200.4.f.v.49.3		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+3.35890 q^{2} +3.28220 q^{4} -30.4356 q^{7} -15.8466 q^{8} -31.4356 q^{11} +60.7424 q^{13} -102.230 q^{14} -79.4848 q^{16} -121.178 q^{17} -14.4356 q^{19} -105.589 q^{22} +13.6932 q^{23} +204.028 q^{26} -99.8958 q^{28} +76.0492 q^{29} +183.049 q^{31} -140.208 q^{32} -407.025 q^{34} +37.3864 q^{37} -48.4877 q^{38} +30.6627 q^{41} -327.564 q^{43} -103.178 q^{44} +45.9941 q^{46} -449.485 q^{47} +583.325 q^{49} +199.369 q^{52} -301.951 q^{53} +482.301 q^{56} +255.441 q^{58} -340.970 q^{59} +619.098 q^{61} +614.844 q^{62} +164.932 q^{64} -256.890 q^{67} -397.731 q^{68} -499.178 q^{71} +19.1288 q^{73} +125.577 q^{74} -47.3805 q^{76} +956.761 q^{77} +257.424 q^{79} +102.993 q^{82} +914.909 q^{83} -1100.26 q^{86} +498.148 q^{88} +1059.68 q^{89} -1848.73 q^{91} +44.9439 q^{92} -1509.77 q^{94} -521.000 q^{97} +1959.33 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8} - 28 q^{11} - 18 q^{13} - 126 q^{14} + 120 q^{16} - 68 q^{17} + 6 q^{19} - 124 q^{22} + 132 q^{23} + 626 q^{26} - 8 q^{28} - 92 q^{29} + 122 q^{31} - 664 q^{32}+ \cdots + 3692 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 24 * q^4 - 26 * q^7 - 84 * q^8 - 28 * q^11 - 18 * q^13 - 126 * q^14 + 120 * q^16 - 68 * q^17 + 6 * q^19 - 124 * q^22 + 132 * q^23 + 626 * q^26 - 8 * q^28 - 92 * q^29 + 122 * q^31 - 664 * q^32 - 692 * q^34 + 284 * q^37 - 158 * q^38 - 392 * q^41 - 690 * q^43 - 32 * q^44 - 588 * q^46 - 620 * q^47 + 260 * q^49 - 1432 * q^52 - 848 * q^53 + 180 * q^56 + 1156 * q^58 - 124 * q^59 + 750 * q^61 + 942 * q^62 + 1376 * q^64 + 358 * q^67 + 704 * q^68 - 824 * q^71 + 108 * q^73 - 1196 * q^74 + 376 * q^76 + 972 * q^77 - 880 * q^79 + 2368 * q^82 + 156 * q^83 + 842 * q^86 + 264 * q^88 + 864 * q^89 - 2198 * q^91 + 2496 * q^92 - 596 * q^94 - 1042 * q^97 + 3692 * q^98

Introduction

L-functions

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