LMFDB - Embedded newform 225.4.b.h.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$13.2754297513$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.4.b.h.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.35890i q^{2} -3.28220 q^{4} +30.4356i q^{7} -15.8466i q^{8} -31.4356 q^{11} +60.7424i q^{13} +102.230 q^{14} -79.4848 q^{16} +121.178i q^{17} +14.4356 q^{19} +105.589i q^{22} +13.6932i q^{23} +204.028 q^{26} -99.8958i q^{28} -76.0492 q^{29} +183.049 q^{31} +140.208i q^{32} +407.025 q^{34} -37.3864i q^{37} -48.4877i q^{38} +30.6627 q^{41} -327.564i q^{43} +103.178 q^{44} +45.9941 q^{46} +449.485i q^{47} -583.325 q^{49} -199.369i q^{52} -301.951i q^{53} +482.301 q^{56} +255.441i q^{58} +340.970 q^{59} +619.098 q^{61} -614.844i q^{62} -164.932 q^{64} +256.890i q^{67} -397.731i q^{68} -499.178 q^{71} +19.1288i q^{73} -125.577 q^{74} -47.3805 q^{76} -956.761i q^{77} -257.424 q^{79} -102.993i q^{82} +914.909i q^{83} -1100.26 q^{86} +498.148i q^{88} -1059.68 q^{89} -1848.73 q^{91} -44.9439i q^{92} +1509.77 q^{94} +521.000i q^{97} +1959.33i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 48 q^{4} - 56 q^{11} + 252 q^{14} + 240 q^{16} - 12 q^{19} + 1252 q^{26} + 184 q^{29} + 244 q^{31} + 1384 q^{34} - 784 q^{41} + 64 q^{44} - 1176 q^{46} - 520 q^{49} + 360 q^{56} + 248 q^{59} + 1500 q^{61}+ \cdots + 1192 q^{94}+O(q^{100}) $ 4 * q - 48 * q^4 - 56 * q^11 + 252 * q^14 + 240 * q^16 - 12 * q^19 + 1252 * q^26 + 184 * q^29 + 244 * q^31 + 1384 * q^34 - 784 * q^41 + 64 * q^44 - 1176 * q^46 - 520 * q^49 + 360 * q^56 + 248 * q^59 + 1500 * q^61 - 2752 * q^64 - 1648 * q^71 + 2392 * q^74 + 752 * q^76 + 1760 * q^79 + 1684 * q^86 - 1728 * q^89 - 4396 * q^91 + 1192 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 3.35890i	− 1.18755i	−0.804631	−	0.593775i	$-0.797637\pi$
$2$	− 3.35890i	− 1.18755i	0.804631	−	0.593775i	$-0.202363\pi$
$3$	0	0
$3$	0	0
$4$	−3.28220	−0.410275
$5$	0	0
$5$	0	0
$6$	0	0
$7$	30.4356i	1.64337i	0.569944	+	0.821684i	$0.306965\pi$
$7$	30.4356i	1.64337i	−0.569944	+	0.821684i	$0.693035\pi$
$8$	− 15.8466i	− 0.700328i
$9$	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
$13$	60.7424i	1.29592i	0.761676	+	0.647958i	$0.224377\pi$
$13$	60.7424i	1.29592i	−0.761676	+	0.647958i	$0.775623\pi$
$14$	102.230	1.95158
$15$	0	0
$16$	−79.4848	−1.24195
$17$	121.178i	1.72882i	0.502786	+	0.864411i	$0.332308\pi$
$17$	121.178i	1.72882i	−0.502786	+	0.864411i	$0.667692\pi$
$18$	0	0
$19$	14.4356	0.174303	0.0871514	−	0.996195i	$-0.472224\pi$
$19$	14.4356	0.174303	0.0871514	+	0.996195i	$0.472224\pi$
$20$	0	0
$21$	0	0
$22$	105.589i	1.02326i
$23$	13.6932i	0.124141i	0.998072	+	0.0620703i	$0.0197703\pi$
$23$	13.6932i	0.124141i	−0.998072	+	0.0620703i	$0.980230\pi$
$24$	0	0
$25$	0	0
$26$	204.028	1.53896
$27$	0	0
$28$	− 99.8958i	− 0.674233i
$29$	−76.0492	−0.486965	−0.243482	−	0.969905i	$-0.578290\pi$
$29$	−76.0492	−0.486965	−0.243482	+	0.969905i	$0.578290\pi$
$30$	0	0
$31$	183.049	1.06054	0.530268	−	0.847830i	$-0.322091\pi$
$31$	183.049	1.06054	0.530268	+	0.847830i	$0.322091\pi$
$32$	140.208i	0.774550i
$33$	0	0
$34$	407.025	2.05306
$35$	0	0
$36$	0	0
$37$	− 37.3864i	− 0.166116i	−0.996545	−	0.0830580i	$-0.973531\pi$
$37$	− 37.3864i	− 0.166116i	0.996545	−	0.0830580i	$-0.0264687\pi$
$38$	− 48.4877i	− 0.206993i
$39$	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
$43$	− 327.564i	− 1.16170i	−0.814011	−	0.580850i	$-0.802720\pi$
$43$	− 327.564i	− 1.16170i	0.814011	−	0.580850i	$-0.197280\pi$
$44$	103.178	0.353515
$45$	0	0
$46$	45.9941	0.147423
$47$	449.485i	1.39498i	0.716594	+	0.697490i	$0.245700\pi$
$47$	449.485i	1.39498i	−0.716594	+	0.697490i	$0.754300\pi$
$48$	0	0
$49$	−583.325	−1.70066
$50$	0	0
$51$	0	0
$52$	− 199.369i	− 0.531682i
$53$	− 301.951i	− 0.782569i	−0.920270	−	0.391284i	$-0.872031\pi$
$53$	− 301.951i	− 0.782569i	0.920270	−	0.391284i	$-0.127969\pi$
$54$	0	0
$55$	0	0
$56$	482.301	1.15090
$57$	0	0
$58$	255.441i	0.578295i
$59$	340.970	0.752381	0.376190	−	0.926542i	$-0.377234\pi$
$59$	340.970	0.752381	0.376190	+	0.926542i	$0.377234\pi$
$60$	0	0
$61$	619.098	1.29947	0.649733	−	0.760163i	$-0.274881\pi$
$61$	619.098	1.29947	0.649733	+	0.760163i	$0.274881\pi$
$62$	− 614.844i	− 1.25944i
$63$	0	0
$64$	−164.932	−0.322133
$65$	0	0
$66$	0	0
$67$	256.890i	0.468419i	0.972186	+	0.234210i	$0.0752502\pi$
$67$	256.890i	0.468419i	−0.972186	+	0.234210i	$0.924750\pi$
$68$	− 397.731i	− 0.709293i
$69$	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
$73$	19.1288i	0.0306693i	0.999882	+	0.0153346i	$0.00488136\pi$
$73$	19.1288i	0.0306693i	−0.999882	+	0.0153346i	$0.995119\pi$
$74$	−125.577	−0.197271
$75$	0	0
$76$	−47.3805	−0.0715121
$77$	− 956.761i	− 1.41601i
$78$	0	0
$79$	−257.424	−0.366613	−0.183307	−	0.983056i	$-0.558680\pi$
$79$	−257.424	−0.366613	−0.183307	+	0.983056i	$0.558680\pi$
$80$	0	0
$81$	0	0
$82$	− 102.993i	− 0.138703i
$83$	914.909i	1.20993i	0.796252	+	0.604965i	$0.206813\pi$
$83$	914.909i	1.20993i	−0.796252	+	0.604965i	$0.793187\pi$
$84$	0	0
$85$	0	0
$86$	−1100.26	−1.37958
$87$	0	0
$88$	498.148i	0.603440i
$89$	−1059.68	−1.26209	−0.631045	−	0.775746i	$-0.717374\pi$
$89$	−1059.68	−1.26209	−0.631045	+	0.775746i	$0.717374\pi$
$90$	0	0
$91$	−1848.73	−2.12967
$92$	− 44.9439i	− 0.0509318i
$93$	0	0
$94$	1509.77	1.65661
$95$	0	0
$96$	0	0
$97$	521.000i	0.545356i	0.962105	+	0.272678i	$0.0879094\pi$
$97$	521.000i	0.545356i	−0.962105	+	0.272678i	$0.912091\pi$
$98$	1959.33i	2.01962i
$99$	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
$103$	770.749i	0.737322i	0.929564	+	0.368661i	$0.120184\pi$
$103$	770.749i	0.737322i	−0.929564	+	0.368661i	$0.879816\pi$
$104$	962.561	0.907566
$105$	0	0
$106$	−1014.22	−0.929339
$107$	1415.37i	1.27878i	0.768883	+	0.639390i	$0.220813\pi$
$107$	1415.37i	1.27878i	−0.768883	+	0.639390i	$0.779187\pi$
$108$	0	0
$109$	−908.386	−0.798235	−0.399118	−	0.916900i	$-0.630683\pi$
$109$	−908.386	−0.798235	−0.399118	+	0.916900i	$0.630683\pi$
$110$	0	0
$111$	0	0
$112$	− 2419.17i	− 2.04098i
$113$	− 2049.94i	− 1.70657i	−0.521447	−	0.853283i	$-0.674608\pi$
$113$	− 2049.94i	− 1.70657i	0.521447	−	0.853283i	$-0.325392\pi$
$114$	0	0
$115$	0	0
$116$	249.609	0.199790
$117$	0	0
$118$	− 1145.28i	− 0.893490i
$119$	−3688.12	−2.84109
$120$	0	0
$121$	−342.803	−0.257553
$122$	− 2079.49i	− 1.54318i
$123$	0	0
$124$	−600.804	−0.435111
$125$	0	0
$126$	0	0
$127$	281.644i	0.196786i	0.995148	+	0.0983932i	$0.0313703\pi$
$127$	281.644i	0.196786i	−0.995148	+	0.0983932i	$0.968630\pi$
$128$	1675.66i	1.15710i
$129$	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
$133$	439.356i	0.286444i
$134$	862.867	0.556271
$135$	0	0
$136$	1920.26	1.21074
$137$	− 909.386i	− 0.567110i	−0.958956	−	0.283555i	$-0.908486\pi$
$137$	− 909.386i	− 0.567110i	0.958956	−	0.283555i	$-0.0915139\pi$
$138$	0	0
$139$	2049.52	1.25063	0.625317	−	0.780371i	$-0.284970\pi$
$139$	2049.52	1.25063	0.625317	+	0.780371i	$0.284970\pi$
$140$	0	0
$141$	0	0
$142$	1676.69i	0.990877i
$143$	− 1909.47i	− 1.11663i
$144$	0	0
$145$	0	0
$146$	64.2517	0.0364213
$147$	0	0
$148$	122.710i	0.0681533i
$149$	3601.14	1.97998	0.989990	−	0.141136i	$-0.0450753\pi$
$149$	3601.14	1.97998	0.989990	+	0.141136i	$0.0450753\pi$
$150$	0	0
$151$	1383.38	0.745550	0.372775	−	0.927922i	$-0.378406\pi$
$151$	1383.38	0.745550	0.372775	+	0.927922i	$0.378406\pi$
$152$	− 228.755i	− 0.122069i
$153$	0	0
$154$	−3213.66	−1.68159
$155$	0	0
$156$	0	0
$157$	− 131.749i	− 0.0669729i	−0.999439	−	0.0334864i	$-0.989339\pi$
$157$	− 131.749i	− 0.0669729i	0.999439	−	0.0334864i	$-0.0106611\pi$
$158$	864.661i	0.435372i
$159$	0	0
$160$	0	0
$161$	−416.761	−0.204009
$162$	0	0
$163$	− 2897.74i	− 1.39244i	−0.717827	−	0.696222i	$-0.754863\pi$
$163$	− 2897.74i	− 1.39244i	0.717827	−	0.696222i	$-0.245137\pi$
$164$	−100.641	−0.0479193
$165$	0	0
$166$	3073.09	1.43685
$167$	− 260.283i	− 0.120607i	−0.998180	−	0.0603034i	$-0.980793\pi$
$167$	− 260.283i	− 0.120607i	0.998180	−	0.0603034i	$-0.0192068\pi$
$168$	0	0
$169$	−1492.64	−0.679398
$170$	0	0
$171$	0	0
$172$	1075.13i	0.476617i
$173$	− 1935.83i	− 0.850742i	−0.905019	−	0.425371i	$-0.860144\pi$
$173$	− 1935.83i	− 0.850742i	0.905019	−	0.425371i	$-0.139856\pi$
$174$	0	0
$175$	0	0
$176$	2498.65	1.07013
$177$	0	0
$178$	3559.36i	1.49880i
$179$	576.627	0.240777	0.120389	−	0.992727i	$-0.461586\pi$
$179$	576.627	0.240777	0.120389	+	0.992727i	$0.461586\pi$
$180$	0	0
$181$	−1962.04	−0.805733	−0.402866	−	0.915259i	$-0.631986\pi$
$181$	−1962.04	−0.805733	−0.402866	+	0.915259i	$0.631986\pi$
$182$	6209.70i	2.52909i
$183$	0	0
$184$	216.991	0.0869390
$185$	0	0
$186$	0	0
$187$	− 3809.30i	− 1.48965i
$188$	− 1475.30i	− 0.572326i
$189$	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
$193$	− 2.97647i	− 0.00111011i	−1.00000		0.000555054i	$-0.999823\pi$
$193$	− 2.97647i	− 0.00111011i	1.00000		0.000555054i	$-0.000176679\pi$
$194$	1749.99	0.647638
$195$	0	0
$196$	1914.59	0.697738
$197$	− 569.705i	− 0.206040i	−0.994679	−	0.103020i	$-0.967149\pi$
$197$	− 569.705i	− 0.206040i	0.994679	−	0.103020i	$-0.0328505\pi$
$198$	0	0
$199$	3050.73	1.08674	0.543368	−	0.839494i	$-0.317149\pi$
$199$	3050.73	1.08674	0.543368	+	0.839494i	$0.317149\pi$
$200$	0	0
$201$	0	0
$202$	1165.81i	0.406068i
$203$	− 2314.60i	− 0.800262i
$204$	0	0
$205$	0	0
$206$	2588.87	0.875607
$207$	0	0
$208$	− 4828.09i	− 1.60946i
$209$	−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	$0.502623\pi$
$212$	991.064i	0.321069i
$213$	0	0
$214$	4754.10	1.51862
$215$	0	0
$216$	0	0
$217$	5571.21i	1.74285i
$218$	3051.18i	0.947944i
$219$	0	0
$220$	0	0
$221$	−7360.64	−2.24041
$222$	0	0
$223$	− 5453.55i	− 1.63765i	−0.574040	−	0.818827i	$-0.694625\pi$
$223$	− 5453.55i	− 1.63765i	0.574040	−	0.818827i	$-0.305375\pi$
$224$	−4267.33	−1.27287
$225$	0	0
$226$	−6885.54	−2.02663
$227$	4777.14i	1.39678i	0.715715	+	0.698392i	$0.246101\pi$
$227$	4777.14i	1.39678i	−0.715715	+	0.698392i	$0.753899\pi$
$228$	0	0
$229$	2085.51	0.601808	0.300904	−	0.953654i	$-0.402712\pi$
$229$	2085.51	0.601808	0.300904	+	0.953654i	$0.402712\pi$
$230$	0	0
$231$	0	0
$232$	1205.12i	0.341035i
$233$	6484.53i	1.82324i	0.411030	+	0.911622i	$0.365169\pi$
$233$	6484.53i	1.82324i	−0.411030	+	0.911622i	$0.634831\pi$
$234$	0	0
$235$	0	0
$236$	−1119.13	−0.308683
$237$	0	0
$238$	12388.0i	3.37394i
$239$	2234.62	0.604792	0.302396	−	0.953182i	$-0.402214\pi$
$239$	2234.62	0.604792	0.302396	+	0.953182i	$0.402214\pi$
$240$	0	0
$241$	−2393.01	−0.639616	−0.319808	−	0.947482i	$-0.603618\pi$
$241$	−2393.01	−0.639616	−0.319808	+	0.947482i	$0.603618\pi$
$242$	1151.44i	0.305857i
$243$	0	0
$244$	−2032.01	−0.533139
$245$	0	0
$246$	0	0
$247$	876.852i	0.225882i
$248$	− 2900.71i	− 0.742722i
$249$	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
$253$	− 430.454i	− 0.106966i
$254$	946.014	0.233694
$255$	0	0
$256$	4308.91	1.05198
$257$	306.112i	0.0742987i	0.999310	+	0.0371493i	$0.0118277\pi$
$257$	306.112i	0.0742987i	−0.999310	+	0.0371493i	$0.988172\pi$
$258$	0	0
$259$	1137.88	0.272990
$260$	0	0
$261$	0	0
$262$	816.401i	0.192509i
$263$	283.839i	0.0665484i	0.999446	+	0.0332742i	$0.0105935\pi$
$263$	283.839i	0.0665484i	−0.999446	+	0.0332742i	$0.989407\pi$
$264$	0	0
$265$	0	0
$266$	1475.75	0.340166
$267$	0	0
$268$	− 843.165i	− 0.192181i
$269$	2426.21	0.549920	0.274960	−	0.961456i	$-0.411335\pi$
$269$	2426.21	0.549920	0.274960	+	0.961456i	$0.411335\pi$
$270$	0	0
$271$	−174.946	−0.0392148	−0.0196074	−	0.999808i	$-0.506242\pi$
$271$	−174.946	−0.0392148	−0.0196074	+	0.999808i	$0.506242\pi$
$272$	− 9631.80i	− 2.14711i
$273$	0	0
$274$	−3054.54	−0.673472
$275$	0	0
$276$	0	0
$277$	− 7807.07i	− 1.69344i	−0.532042	−	0.846718i	$-0.678575\pi$
$277$	− 7807.07i	− 1.69344i	0.532042	−	0.846718i	$-0.321425\pi$
$278$	− 6884.14i	− 1.48519i
$279$	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
$283$	5897.31i	1.23872i	0.785106	+	0.619362i	$0.212609\pi$
$283$	5897.31i	1.23872i	−0.785106	+	0.619362i	$0.787391\pi$
$284$	1638.40	0.342329
$285$	0	0
$286$	−6413.73	−1.32605
$287$	933.239i	0.191942i
$288$	0	0
$289$	−9771.10	−1.98883
$290$	0	0
$291$	0	0
$292$	− 62.7846i	− 0.0125828i
$293$	1609.73i	0.320960i	0.987039	+	0.160480i	$0.0513042\pi$
$293$	1609.73i	0.320960i	−0.987039	+	0.160480i	$0.948696\pi$
$294$	0	0
$295$	0	0
$296$	−592.448	−0.116336
$297$	0	0
$298$	− 12095.9i	− 2.35133i
$299$	−831.758	−0.160876
$300$	0	0
$301$	9969.62	1.90910
$302$	− 4646.64i	− 0.885378i
$303$	0	0
$304$	−1147.41	−0.216475
$305$	0	0
$306$	0	0
$307$	234.473i	0.0435898i	0.999762	+	0.0217949i	$0.00693809\pi$
$307$	234.473i	0.0435898i	−0.999762	+	0.0217949i	$0.993062\pi$
$308$	3140.28i	0.580955i
$309$	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
$313$	8440.61i	1.52425i	0.647427	+	0.762127i	$0.275845\pi$
$313$	8440.61i	1.52425i	−0.647427	+	0.762127i	$0.724155\pi$
$314$	−442.533	−0.0795336
$315$	0	0
$316$	844.917	0.150412
$317$	− 10551.7i	− 1.86953i	−0.355264	−	0.934766i	$-0.615609\pi$
$317$	− 10551.7i	− 1.86953i	0.355264	−	0.934766i	$-0.384391\pi$
$318$	0	0
$319$	2390.65	0.419595
$320$	0	0
$321$	0	0
$322$	1399.86i	0.242270i
$323$	1749.28i	0.301339i
$324$	0	0
$325$	0	0
$326$	−9733.21	−1.65360
$327$	0	0
$328$	− 485.900i	− 0.0817968i
$329$	−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	$0.310849\pi$
$332$	− 3002.91i	− 0.496405i
$333$	0	0
$334$	−874.265	−0.143227
$335$	0	0
$336$	0	0
$337$	− 8437.26i	− 1.36382i	−0.731437	−	0.681909i	$-0.761150\pi$
$337$	− 8437.26i	− 1.36382i	0.731437	−	0.681909i	$-0.238850\pi$
$338$	5013.62i	0.806819i
$339$	0	0
$340$	0	0
$341$	−5754.26	−0.913814
$342$	0	0
$343$	− 7314.45i	− 1.15144i
$344$	−5190.78	−0.813571
$345$	0	0
$346$	−6502.25	−1.01030
$347$	1848.85i	0.286027i	0.989721	+	0.143013i	$0.0456792\pi$
$347$	1848.85i	0.286027i	−0.989721	+	0.143013i	$0.954321\pi$
$348$	0	0
$349$	1148.38	0.176136	0.0880678	−	0.996114i	$-0.471931\pi$
$349$	1148.38	0.176136	0.0880678	+	0.996114i	$0.471931\pi$
$350$	0	0
$351$	0	0
$352$	− 4407.54i	− 0.667393i
$353$	− 5753.60i	− 0.867516i	−0.901029	−	0.433758i	$-0.857187\pi$
$353$	− 5753.60i	− 0.867516i	0.901029	−	0.433758i	$-0.142813\pi$
$354$	0	0
$355$	0	0
$356$	3478.09	0.517804
$357$	0	0
$358$	− 1936.83i	− 0.285935i
$359$	5452.01	0.801521	0.400761	−	0.916183i	$-0.368746\pi$
$359$	5452.01	0.801521	0.400761	+	0.916183i	$0.368746\pi$
$360$	0	0
$361$	−6650.61	−0.969619
$362$	6590.31i	0.956848i
$363$	0	0
$364$	6067.91	0.873749
$365$	0	0
$366$	0	0
$367$	8385.93i	1.19276i	0.802703	+	0.596379i	$0.203394\pi$
$367$	8385.93i	1.19276i	−0.802703	+	0.596379i	$0.796606\pi$
$368$	− 1088.40i	− 0.154176i
$369$	0	0
$370$	0	0
$371$	9190.05	1.28605
$372$	0	0
$373$	− 2728.30i	− 0.378730i	−0.981907	−	0.189365i	$-0.939357\pi$
$373$	− 2728.30i	− 0.378730i	0.981907	−	0.189365i	$-0.0606429\pi$
$374$	−12795.1	−1.76903
$375$	0	0
$376$	7122.81	0.976944
$377$	− 4619.41i	− 0.631065i
$378$	0	0
$379$	−3348.99	−0.453895	−0.226947	−	0.973907i	$-0.572875\pi$
$379$	−3348.99	−0.453895	−0.226947	+	0.973907i	$0.572875\pi$
$380$	0	0
$381$	0	0
$382$	− 14506.2i	− 1.94294i
$383$	10430.3i	1.39155i	0.718261	+	0.695774i	$0.244938\pi$
$383$	10430.3i	1.39155i	−0.718261	+	0.695774i	$0.755062\pi$
$384$	0	0
$385$	0	0
$386$	−9.99766	−0.00131831
$387$	0	0
$388$	− 1710.03i	− 0.223746i
$389$	9827.23	1.28088	0.640438	−	0.768010i	$-0.278753\pi$
$389$	9827.23	1.28088	0.640438	+	0.768010i	$0.278753\pi$
$390$	0	0
$391$	−1659.32	−0.214617
$392$	9243.73i	1.19102i
$393$	0	0
$394$	−1913.58	−0.244682
$395$	0	0
$396$	0	0
$397$	− 436.382i	− 0.0551672i	−0.999619	−	0.0275836i	$-0.991219\pi$
$397$	− 436.382i	− 0.0551672i	0.999619	−	0.0275836i	$-0.00878125\pi$
$398$	− 10247.1i	− 1.29055i
$399$	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
$403$	11118.8i	1.37436i
$404$	1139.19	0.140289
$405$	0	0
$406$	−7774.51	−0.950351
$407$	1175.26i	0.143134i
$408$	0	0
$409$	12058.4	1.45782	0.728911	−	0.684609i	$-0.240027\pi$
$409$	12058.4	1.45782	0.728911	+	0.684609i	$0.240027\pi$
$410$	0	0
$411$	0	0
$412$	− 2529.76i	− 0.302505i
$413$	10377.6i	1.23644i
$414$	0	0
$415$	0	0
$416$	−8516.60	−1.00375
$417$	0	0
$418$	1524.24i	0.178356i
$419$	6042.95	0.704577	0.352288	−	0.935892i	$-0.385404\pi$
$419$	6042.95	0.704577	0.352288	+	0.935892i	$0.385404\pi$
$420$	0	0
$421$	−9994.67	−1.15703	−0.578516	−	0.815671i	$-0.696368\pi$
$421$	−9994.67	−1.15703	−0.578516	+	0.815671i	$0.696368\pi$
$422$	169.659i	0.0195708i
$423$	0	0
$424$	−4784.90	−0.548054
$425$	0	0
$426$	0	0
$427$	18842.6i	2.13550i
$428$	− 4645.55i	− 0.524652i
$429$	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
$433$	7861.22i	0.872485i	0.899829	+	0.436243i	$0.143691\pi$
$433$	7861.22i	0.872485i	−0.899829	+	0.436243i	$0.856309\pi$
$434$	18713.1	2.06972
$435$	0	0
$436$	2981.51	0.327496
$437$	197.670i	0.0216380i
$438$	0	0
$439$	−7412.06	−0.805828	−0.402914	−	0.915238i	$-0.632003\pi$
$439$	−7412.06	−0.805828	−0.402914	+	0.915238i	$0.632003\pi$
$440$	0	0
$441$	0	0
$442$	24723.6i	2.66060i
$443$	− 3043.66i	− 0.326430i	−0.986591	−	0.163215i	$-0.947814\pi$
$443$	− 3043.66i	− 0.326430i	0.986591	−	0.163215i	$-0.0521864\pi$
$444$	0	0
$445$	0	0
$446$	−18317.9	−1.94480
$447$	0	0
$448$	− 5019.81i	− 0.529383i
$449$	−9547.87	−1.00355	−0.501773	−	0.864999i	$-0.667319\pi$
$449$	−9547.87	−1.00355	−0.501773	+	0.864999i	$0.667319\pi$
$450$	0	0
$451$	−963.902	−0.100639
$452$	6728.31i	0.700162i
$453$	0	0
$454$	16045.9	1.65875
$455$	0	0
$456$	0	0
$457$	13401.9i	1.37180i	0.727695	+	0.685901i	$0.240592\pi$
$457$	13401.9i	1.37180i	−0.727695	+	0.685901i	$0.759408\pi$
$458$	− 7005.00i	− 0.714677i
$459$	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
$463$	13976.3i	1.40288i	0.712729	+	0.701439i	$0.247459\pi$
$463$	13976.3i	1.40288i	−0.712729	+	0.701439i	$0.752541\pi$
$464$	6044.75	0.604786
$465$	0	0
$466$	21780.9	2.16519
$467$	10796.5i	1.06982i	0.844910	+	0.534908i	$0.179654\pi$
$467$	10796.5i	1.06982i	−0.844910	+	0.534908i	$0.820346\pi$
$468$	0	0
$469$	−7818.60	−0.769785
$470$	0	0
$471$	0	0
$472$	− 5403.21i	− 0.526913i
$473$	10297.2i	1.00098i
$474$	0	0
$475$	0	0
$476$	12105.2	1.16563
$477$	0	0
$478$	− 7505.85i	− 0.718221i
$479$	14568.4	1.38966	0.694830	−	0.719174i	$-0.255479\pi$
$479$	14568.4	1.38966	0.694830	+	0.719174i	$0.255479\pi$
$480$	0	0
$481$	2270.94	0.215272
$482$	8037.89i	0.759577i
$483$	0	0
$484$	1125.15	0.105668
$485$	0	0
$486$	0	0
$487$	− 11456.6i	− 1.06601i	−0.846113	−	0.533004i	$-0.821063\pi$
$487$	− 11456.6i	− 1.06601i	0.846113	−	0.533004i	$-0.178937\pi$
$488$	− 9810.61i	− 0.910052i
$489$	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
$493$	− 9215.48i	− 0.841875i
$494$	2945.26	0.268246
$495$	0	0
$496$	−14549.6	−1.31713
$497$	− 15192.8i	− 1.37121i
$498$	0	0
$499$	−8379.31	−0.751722	−0.375861	−	0.926676i	$-0.622653\pi$
$499$	−8379.31	−0.751722	−0.375861	+	0.926676i	$0.622653\pi$
$500$	0	0
$501$	0	0
$502$	− 2057.87i	− 0.182962i
$503$	− 15678.1i	− 1.38976i	−0.719124	−	0.694881i	$-0.755457\pi$
$503$	− 15678.1i	− 1.38976i	0.719124	−	0.694881i	$-0.244543\pi$
$504$	0	0
$505$	0	0
$506$	−1445.85	−0.127028
$507$	0	0
$508$	− 924.413i	− 0.0807366i
$509$	−17037.3	−1.48363	−0.741813	−	0.670606i	$-0.766034\pi$
$509$	−17037.3	−1.48363	−0.741813	+	0.670606i	$0.766034\pi$
$510$	0	0
$511$	−582.197	−0.0504009
$512$	− 1067.93i	− 0.0921798i
$513$	0	0
$514$	1028.20	0.0882334
$515$	0	0
$516$	0	0
$517$	− 14129.8i	− 1.20199i
$518$	− 3822.02i	− 0.324189i
$519$	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
$523$	− 11120.4i	− 0.929753i	−0.885375	−	0.464877i	$-0.846099\pi$
$523$	− 11120.4i	− 0.929753i	0.885375	−	0.464877i	$-0.153901\pi$
$524$	797.759	0.0665082
$525$	0	0
$526$	953.385	0.0790296
$527$	22181.5i	1.83348i
$528$	0	0
$529$	11979.5	0.984589
$530$	0	0
$531$	0	0
$532$	− 1442.06i	− 0.117521i
$533$	1862.53i	0.151360i
$534$	0	0
$535$	0	0
$536$	4070.83	0.328047
$537$	0	0
$538$	− 8149.39i	− 0.653058i
$539$	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	$0.168288\pi$
$542$	587.626i	0.0465695i
$543$	0	0
$544$	−16990.2	−1.33906
$545$	0	0
$546$	0	0
$547$	− 6926.17i	− 0.541392i	−0.962665	−	0.270696i	$-0.912746\pi$
$547$	− 6926.17i	− 0.541392i	0.962665	−	0.270696i	$-0.0872539\pi$
$548$	2984.79i	0.232671i
$549$	0	0
$550$	0	0
$551$	−1097.82	−0.0848793
$552$	0	0
$553$	− 7834.85i	− 0.602480i
$554$	−26223.2	−2.01104
$555$	0	0
$556$	−6726.95	−0.513104
$557$	6589.22i	0.501246i	0.968085	+	0.250623i	$0.0806355\pi$
$557$	6589.22i	0.501246i	−0.968085	+	0.250623i	$0.919364\pi$
$558$	0	0
$559$	19897.0	1.50547
$560$	0	0
$561$	0	0
$562$	1962.17i	0.147276i
$563$	3839.63i	0.287426i	0.989619	+	0.143713i	$0.0459043\pi$
$563$	3839.63i	0.287426i	−0.989619	+	0.143713i	$0.954096\pi$
$564$	0	0
$565$	0	0
$566$	19808.5	1.47105
$567$	0	0
$568$	7910.28i	0.584345i
$569$	−20874.0	−1.53794	−0.768968	−	0.639287i	$-0.779229\pi$
$569$	−20874.0	−1.53794	−0.768968	+	0.639287i	$0.779229\pi$
$570$	0	0
$571$	21175.9	1.55199	0.775994	−	0.630740i	$-0.217248\pi$
$571$	21175.9	1.55199	0.775994	+	0.630740i	$0.217248\pi$
$572$	6267.28i	0.458126i
$573$	0	0
$574$	3134.66	0.227941
$575$	0	0
$576$	0	0
$577$	− 14924.2i	− 1.07678i	−0.842695	−	0.538391i	$-0.819032\pi$
$577$	− 14924.2i	− 1.07678i	0.842695	−	0.538391i	$-0.180968\pi$
$578$	32820.1i	2.36183i
$579$	0	0
$580$	0	0
$581$	−27845.8	−1.98836
$582$	0	0
$583$	9492.00i	0.674303i
$584$	303.127	0.0214785
$585$	0	0
$586$	5406.92	0.381156
$587$	− 25218.0i	− 1.77318i	−0.462552	−	0.886592i	$-0.653066\pi$
$587$	− 25218.0i	− 1.77318i	0.462552	−	0.886592i	$-0.346934\pi$
$588$	0	0
$589$	2642.42	0.184854
$590$	0	0
$591$	0	0
$592$	2971.65i	0.206308i
$593$	5011.77i	0.347063i	0.984828	+	0.173532i	$0.0555179\pi$
$593$	5011.77i	0.347063i	−0.984828	+	0.173532i	$0.944482\pi$
$594$	0	0
$595$	0	0
$596$	−11819.7	−0.812337
$597$	0	0
$598$	2793.79i	0.191048i
$599$	4943.41	0.337199	0.168600	−	0.985685i	$-0.446075\pi$
$599$	4943.41	0.337199	0.168600	+	0.985685i	$0.446075\pi$
$600$	0	0
$601$	−24334.8	−1.65164	−0.825821	−	0.563932i	$-0.809288\pi$
$601$	−24334.8	−1.65164	−0.825821	+	0.563932i	$0.809288\pi$
$602$	− 33486.9i	− 2.26715i
$603$	0	0
$604$	−4540.54	−0.305881
$605$	0	0
$606$	0	0
$607$	28973.8i	1.93741i	0.248207	+	0.968707i	$0.420159\pi$
$607$	28973.8i	1.93741i	−0.248207	+	0.968707i	$0.579841\pi$
$608$	2023.99i	0.135006i
$609$	0	0
$610$	0	0
$611$	−27302.8	−1.80778
$612$	0	0
$613$	− 15139.1i	− 0.997490i	−0.866749	−	0.498745i	$-0.833794\pi$
$613$	− 15139.1i	− 0.997490i	0.866749	−	0.498745i	$-0.166206\pi$
$614$	787.571	0.0517651
$615$	0	0
$616$	−15161.4	−0.991673
$617$	− 13894.8i	− 0.906617i	−0.891354	−	0.453309i	$-0.850244\pi$
$617$	− 13894.8i	− 0.906617i	0.891354	−	0.453309i	$-0.149756\pi$
$618$	0	0
$619$	4589.69	0.298021	0.149011	−	0.988836i	$-0.452391\pi$
$619$	4589.69	0.298021	0.149011	+	0.988836i	$0.452391\pi$
$620$	0	0
$621$	0	0
$622$	− 6030.07i	− 0.388720i
$623$	− 32252.0i	− 2.07408i
$624$	0	0
$625$	0	0
$626$	28351.2	1.81013
$627$	0	0
$628$	432.428i	0.0274773i
$629$	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	$0.469774\pi$
$632$	4079.29i	0.256749i
$633$	0	0
$634$	−35442.0	−2.22016
$635$	0	0
$636$	0	0
$637$	− 35432.6i	− 2.20391i
$638$	− 8029.96i	− 0.498290i
$639$	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
$643$	− 11305.1i	− 0.693358i	−0.937984	−	0.346679i	$-0.887309\pi$
$643$	− 11305.1i	− 0.693358i	0.937984	−	0.346679i	$-0.112691\pi$
$644$	1367.89	0.0836997
$645$	0	0
$646$	5875.64	0.357855
$647$	8614.30i	0.523436i	0.965144	+	0.261718i	$0.0842891\pi$
$647$	8614.30i	0.523436i	−0.965144	+	0.261718i	$0.915711\pi$
$648$	0	0
$649$	−10718.6	−0.648291
$650$	0	0
$651$	0	0
$652$	9510.96i	0.571285i
$653$	12639.8i	0.757479i	0.925503	+	0.378739i	$0.123642\pi$
$653$	12639.8i	0.757479i	−0.925503	+	0.378739i	$0.876358\pi$
$654$	0	0
$655$	0	0
$656$	−2437.22	−0.145057
$657$	0	0
$658$	45950.9i	2.72242i
$659$	13640.4	0.806306	0.403153	−	0.915133i	$-0.367914\pi$
$659$	13640.4	0.806306	0.403153	+	0.915133i	$0.367914\pi$
$660$	0	0
$661$	−17052.0	−1.00340	−0.501699	−	0.865042i	$-0.667291\pi$
$661$	−17052.0	−1.00340	−0.501699	+	0.865042i	$0.667291\pi$
$662$	− 22649.6i	− 1.32976i
$663$	0	0
$664$	14498.2	0.847348
$665$	0	0
$666$	0	0
$667$	− 1041.36i	− 0.0604521i
$668$	854.302i	0.0494820i
$669$	0	0
$670$	0	0
$671$	−19461.7	−1.11969
$672$	0	0
$673$	16419.7i	0.940467i	0.882542	+	0.470234i	$0.155830\pi$
$673$	16419.7i	0.940467i	−0.882542	+	0.470234i	$0.844170\pi$
$674$	−28339.9	−1.61960
$675$	0	0
$676$	4899.14	0.278740
$677$	8670.47i	0.492221i	0.969242	+	0.246110i	$0.0791525\pi$
$677$	8670.47i	0.492221i	−0.969242	+	0.246110i	$0.920847\pi$
$678$	0	0
$679$	−15856.9	−0.896220
$680$	0	0
$681$	0	0
$682$	19328.0i	1.08520i
$683$	5973.36i	0.334647i	0.985902	+	0.167324i	$0.0535125\pi$
$683$	5973.36i	0.334647i	−0.985902	+	0.167324i	$0.946488\pi$
$684$	0	0
$685$	0	0
$686$	−24568.5	−1.36739
$687$	0	0
$688$	26036.4i	1.44277i
$689$	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	$0.638533\pi$
$692$	6353.78i	0.349038i
$693$	0	0
$694$	6210.09	0.339671
$695$	0	0
$696$	0	0
$697$	3715.65i	0.201923i
$698$	− 3857.29i	− 0.209170i
$699$	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
$703$	− 539.695i	− 0.0289545i
$704$	5184.74	0.277567
$705$	0	0
$706$	−19325.8	−1.03022
$707$	− 10563.6i	− 0.561929i
$708$	0	0
$709$	−11194.1	−0.592955	−0.296477	−	0.955040i	$-0.595812\pi$
$709$	−11194.1	−0.592955	−0.296477	+	0.955040i	$0.595812\pi$
$710$	0	0
$711$	0	0
$712$	16792.4i	0.883877i
$713$	2506.53i	0.131655i
$714$	0	0
$715$	0	0
$716$	−1892.61	−0.0987850
$717$	0	0
$718$	− 18312.8i	− 0.951847i
$719$	−15491.9	−0.803549	−0.401774	−	0.915739i	$-0.631606\pi$
$719$	−15491.9	−0.803549	−0.401774	+	0.915739i	$0.631606\pi$
$720$	0	0
$721$	−23458.2	−1.21169
$722$	22338.7i	1.15147i
$723$	0	0
$724$	6439.83	0.330572
$725$	0	0
$726$	0	0
$727$	− 6272.72i	− 0.320003i	−0.987117	−	0.160002i	$-0.948850\pi$
$727$	− 6272.72i	− 0.320003i	0.987117	−	0.160002i	$-0.0511500\pi$
$728$	29296.1i	1.49146i
$729$	0	0
$730$	0	0
$731$	39693.6	2.00837
$732$	0	0
$733$	− 24980.5i	− 1.25877i	−0.777095	−	0.629383i	$-0.783308\pi$
$733$	− 24980.5i	− 1.25877i	0.777095	−	0.629383i	$-0.216692\pi$
$734$	28167.5	1.41646
$735$	0	0
$736$	−1919.90	−0.0961530
$737$	− 8075.49i	− 0.403615i
$738$	0	0
$739$	30660.7	1.52621	0.763107	−	0.646272i	$-0.223673\pi$
$739$	30660.7	1.52621	0.763107	+	0.646272i	$0.223673\pi$
$740$	0	0
$741$	0	0
$742$	− 30868.5i	− 1.52725i
$743$	− 17205.7i	− 0.849551i	−0.905299	−	0.424776i	$-0.860353\pi$
$743$	− 17205.7i	− 0.849551i	0.905299	−	0.424776i	$-0.139647\pi$
$744$	0	0
$745$	0	0
$746$	−9164.09	−0.449760
$747$	0	0
$748$	12502.9i	0.611165i
$749$	−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	$0.352510\pi$
$752$	− 35727.2i	− 1.73250i
$753$	0	0
$754$	−15516.1	−0.749422
$755$	0	0
$756$	0	0
$757$	22305.1i	1.07093i	0.844557	+	0.535465i	$0.179864\pi$
$757$	22305.1i	1.07093i	−0.844557	+	0.535465i	$0.820136\pi$
$758$	11248.9i	0.539023i
$759$	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
$763$	− 27647.3i	− 1.31179i
$764$	−14175.0	−0.671249
$765$	0	0
$766$	35034.3	1.65253
$767$	20711.3i	0.975022i
$768$	0	0
$769$	39897.6	1.87093	0.935463	−	0.353424i	$-0.114983\pi$
$769$	39897.6	1.87093	0.935463	+	0.353424i	$0.114983\pi$
$770$	0	0
$771$	0	0
$772$	9.76938i	0 0.000455450i
$773$	20070.2i	0.933863i	0.884294	+	0.466931i	$0.154641\pi$
$773$	20070.2i	0.933863i	−0.884294	+	0.466931i	$0.845359\pi$
$774$	0	0
$775$	0	0
$776$	8256.08	0.381928
$777$	0	0
$778$	− 33008.7i	− 1.52110i
$779$	442.635	0.0203582
$780$	0	0
$781$	15692.0	0.718953
$782$	5573.47i	0.254868i
$783$	0	0
$784$	46365.5	2.11213
$785$	0	0
$786$	0	0
$787$	10733.0i	0.486137i	0.970009	+	0.243069i	$0.0781540\pi$
$787$	10733.0i	0.486137i	−0.970009	+	0.243069i	$0.921846\pi$
$788$	1869.89i	0.0845329i
$789$	0	0
$790$	0	0
$791$	62391.1	2.80452
$792$	0	0
$793$	37605.5i	1.68400i
$794$	−1465.76	−0.0655139
$795$	0	0
$796$	−10013.1	−0.445861
$797$	14335.8i	0.637140i	0.947899	+	0.318570i	$0.103203\pi$
$797$	14335.8i	0.637140i	−0.947899	+	0.318570i	$0.896797\pi$
$798$	0	0
$799$	−54467.7	−2.41167
$800$	0	0
$801$	0	0
$802$	− 48709.0i	− 2.14461i
$803$	− 601.325i	− 0.0264263i
$804$	0	0
$805$	0	0
$806$	37347.1	1.63213
$807$	0	0
$808$	5500.03i	0.239468i
$809$	20920.7	0.909187	0.454593	−	0.890699i	$-0.349785\pi$
$809$	20920.7	0.909187	0.454593	+	0.890699i	$0.349785\pi$
$810$	0	0
$811$	12816.5	0.554931	0.277465	−	0.960736i	$-0.410506\pi$
$811$	12816.5	0.554931	0.277465	+	0.960736i	$0.410506\pi$
$812$	7596.99i	0.328328i
$813$	0	0
$814$	3947.59	0.169979
$815$	0	0
$816$	0	0
$817$	− 4728.59i	− 0.202488i
$818$	− 40502.9i	− 1.73124i
$819$	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
$823$	− 35288.1i	− 1.49461i	−0.664479	−	0.747307i	$-0.731346\pi$
$823$	− 35288.1i	− 1.49461i	0.664479	−	0.747307i	$-0.268654\pi$
$824$	12213.8	0.516367
$825$	0	0
$826$	34857.3	1.46833
$827$	32205.9i	1.35418i	0.735899	+	0.677092i	$0.236760\pi$
$827$	32205.9i	1.35418i	−0.735899	+	0.677092i	$0.763240\pi$
$828$	0	0
$829$	−29993.3	−1.25659	−0.628294	−	0.777976i	$-0.716246\pi$
$829$	−29993.3	−1.25659	−0.628294	+	0.777976i	$0.716246\pi$
$830$	0	0
$831$	0	0
$832$	− 10018.4i	− 0.417457i
$833$	− 70686.2i	− 2.94013i
$834$	0	0
$835$	0	0
$836$	1489.44	0.0616187
$837$	0	0
$838$	− 20297.7i	− 0.836720i
$839$	−35608.7	−1.46526	−0.732628	−	0.680629i	$-0.761706\pi$
$839$	−35608.7	−1.46526	−0.732628	+	0.680629i	$0.761706\pi$
$840$	0	0
$841$	−18605.5	−0.762865
$842$	33571.1i	1.37403i
$843$	0	0
$844$	165.785	0.00676134
$845$	0	0
$846$	0	0
$847$	− 10433.4i	− 0.423255i
$848$	24000.5i	0.971911i
$849$	0	0
$850$	0	0
$851$	511.940	0.0206217
$852$	0	0
$853$	11229.3i	0.450744i	0.974273	+	0.225372i	$0.0723598\pi$
$853$	11229.3i	0.450744i	−0.974273	+	0.225372i	$0.927640\pi$
$854$	63290.5	2.53601
$855$	0	0
$856$	22428.9	0.895565
$857$	− 22136.1i	− 0.882327i	−0.897427	−	0.441164i	$-0.854566\pi$
$857$	− 22136.1i	− 0.882327i	0.897427	−	0.441164i	$-0.145434\pi$
$858$	0	0
$859$	820.727	0.0325994	0.0162997	−	0.999867i	$-0.494811\pi$
$859$	820.727	0.0325994	0.0162997	+	0.999867i	$0.494811\pi$
$860$	0	0
$861$	0	0
$862$	31329.5i	1.23792i
$863$	− 245.223i	− 0.00967264i	−0.999988	−	0.00483632i	$-0.998461\pi$
$863$	− 245.223i	− 0.00967264i	0.999988	−	0.00483632i	$-0.00153945\pi$
$864$	0	0
$865$	0	0
$866$	26405.0	1.03612
$867$	0	0
$868$	− 18285.8i	− 0.715048i
$869$	8092.27	0.315894
$870$	0	0
$871$	−15604.1	−0.607032
$872$	14394.8i	0.559026i
$873$	0	0
$874$	663.952	0.0256963
$875$	0	0
$876$	0	0
$877$	37727.8i	1.45265i	0.687350	+	0.726326i	$0.258774\pi$
$877$	37727.8i	1.45265i	−0.687350	+	0.726326i	$0.741226\pi$
$878$	24896.4i	0.956961i
$879$	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
$883$	44340.4i	1.68989i	0.534852	+	0.844946i	$0.320368\pi$
$883$	44340.4i	1.68989i	−0.534852	+	0.844946i	$0.679632\pi$
$884$	24159.1	0.919184
$885$	0	0
$886$	−10223.3	−0.387652
$887$	681.008i	0.0257790i	0.999917	+	0.0128895i	$0.00410298\pi$
$887$	681.008i	0.0257790i	−0.999917	+	0.0128895i	$0.995897\pi$
$888$	0	0
$889$	−8572.00	−0.323392
$890$	0	0
$891$	0	0
$892$	17899.7i	0.671889i
$893$	6488.58i	0.243149i
$894$	0	0
$895$	0	0
$896$	−50999.6	−1.90154
$897$	0	0
$898$	32070.3i	1.19176i
$899$	−13920.7	−0.516443
$900$	0	0
$901$	36589.8	1.35292
$902$	3237.65i	0.119514i
$903$	0	0
$904$	−32484.6	−1.19516
$905$	0	0
$906$	0	0
$907$	5348.01i	0.195786i	0.995197	+	0.0978929i	$0.0312103\pi$
$907$	5348.01i	0.195786i	−0.995197	+	0.0978929i	$0.968790\pi$
$908$	− 15679.5i	− 0.573066i
$909$	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
$913$	− 28760.7i	− 1.04254i
$914$	45015.6	1.62908
$915$	0	0
$916$	−6845.05	−0.246907
$917$	− 7397.56i	− 0.266400i
$918$	0	0
$919$	22546.9	0.809308	0.404654	−	0.914470i	$-0.367392\pi$
$919$	22546.9	0.809308	0.404654	+	0.914470i	$0.367392\pi$
$920$	0	0
$921$	0	0
$922$	− 13895.9i	− 0.496351i
$923$	− 30321.3i	− 1.08130i
$924$	0	0
$925$	0	0
$926$	46944.9	1.66599
$927$	0	0
$928$	− 10662.7i	− 0.377178i
$929$	−13980.7	−0.493747	−0.246874	−	0.969048i	$-0.579403\pi$
$929$	−13980.7	−0.493747	−0.246874	+	0.969048i	$0.579403\pi$
$930$	0	0
$931$	−8420.65	−0.296429
$932$	− 21283.5i	− 0.748032i
$933$	0	0
$934$	36264.5	1.27046
$935$	0	0
$936$	0	0
$937$	26362.6i	0.919133i	0.888143	+	0.459567i	$0.151995\pi$
$937$	26362.6i	0.919133i	−0.888143	+	0.459567i	$0.848005\pi$
$938$	26261.9i	0.914159i
$939$	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
$943$	419.871i	0.0144994i
$944$	−27101.9	−0.934419
$945$	0	0
$946$	34587.2	1.18872
$947$	11818.2i	0.405532i	0.979227	+	0.202766i	$0.0649931\pi$
$947$	11818.2i	0.405532i	−0.979227	+	0.202766i	$0.935007\pi$
$948$	0	0
$949$	−1161.93	−0.0397448
$950$	0	0
$951$	0	0
$952$	58444.2i	1.98969i
$953$	− 43832.3i	− 1.48989i	−0.667125	−	0.744945i	$-0.732476\pi$
$953$	− 43832.3i	− 1.48989i	0.667125	−	0.744945i	$-0.267524\pi$
$954$	0	0
$955$	0	0
$956$	−7334.46	−0.248131
$957$	0	0
$958$	− 48933.8i	− 1.65029i
$959$	27677.7	0.931971
$960$	0	0
$961$	3716.00	0.124736
$962$	− 7627.86i	− 0.255647i
$963$	0	0
$964$	7854.36	0.262419
$965$	0	0
$966$	0	0
$967$	10696.2i	0.355706i	0.984057	+	0.177853i	$0.0569152\pi$
$967$	10696.2i	0.355706i	−0.984057	+	0.177853i	$0.943085\pi$
$968$	5432.27i	0.180372i
$969$	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
$973$	62378.4i	2.05525i
$974$	−38481.4	−1.26594
$975$	0	0
$976$	−49208.9	−1.61387
$977$	− 24341.7i	− 0.797094i	−0.917148	−	0.398547i	$-0.869515\pi$
$977$	− 24341.7i	− 0.797094i	0.917148	−	0.398547i	$-0.130485\pi$
$978$	0	0
$979$	33311.7	1.08748
$980$	0	0
$981$	0	0
$982$	− 66057.7i	− 2.14662i
$983$	12553.0i	0.407301i	0.979044	+	0.203651i	$0.0652807\pi$
$983$	12553.0i	0.407301i	−0.979044	+	0.203651i	$0.934719\pi$
$984$	0	0
$985$	0	0
$986$	−30953.9	−0.999769
$987$	0	0
$988$	− 2878.01i	− 0.0926737i
$989$	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	$0.761111\pi$
$992$	25665.0i	0.821437i
$993$	0	0
$994$	−51031.0	−1.62838
$995$	0	0
$996$	0	0
$997$	− 34499.0i	− 1.09588i	−0.836517	−	0.547941i	$-0.815412\pi$
$997$	− 34499.0i	− 1.09588i	0.836517	−	0.547941i	$-0.184588\pi$
$998$	28145.3i	0.892708i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.2	✓	2	15.8	even	4
75.4.a.e.1.1	yes	2	15.2	even	4
75.4.b.c.49.2		4	15.14	odd	2
75.4.b.c.49.3		4	3.2	odd	2
225.4.a.j.1.2		2	5.2	odd	4
225.4.a.n.1.1		2	5.3	odd	4
225.4.b.h.199.2		4	1.1	even	1	trivial
225.4.b.h.199.3		4	5.4	even	2	inner
1200.4.a.bl.1.1		2	60.23	odd	4
1200.4.a.bu.1.2		2	60.47	odd	4
1200.4.f.v.49.2		4	60.59	even	2
1200.4.f.v.49.3		4	12.11	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.35890i q^{2} -3.28220 q^{4} +30.4356i q^{7} -15.8466i q^{8} -31.4356 q^{11} +60.7424i q^{13} +102.230 q^{14} -79.4848 q^{16} +121.178i q^{17} +14.4356 q^{19} +105.589i q^{22} +13.6932i q^{23} +204.028 q^{26} -99.8958i q^{28} -76.0492 q^{29} +183.049 q^{31} +140.208i q^{32} +407.025 q^{34} -37.3864i q^{37} -48.4877i q^{38} +30.6627 q^{41} -327.564i q^{43} +103.178 q^{44} +45.9941 q^{46} +449.485i q^{47} -583.325 q^{49} -199.369i q^{52} -301.951i q^{53} +482.301 q^{56} +255.441i q^{58} +340.970 q^{59} +619.098 q^{61} -614.844i q^{62} -164.932 q^{64} +256.890i q^{67} -397.731i q^{68} -499.178 q^{71} +19.1288i q^{73} -125.577 q^{74} -47.3805 q^{76} -956.761i q^{77} -257.424 q^{79} -102.993i q^{82} +914.909i q^{83} -1100.26 q^{86} +498.148i q^{88} -1059.68 q^{89} -1848.73 q^{91} -44.9439i q^{92} +1509.77 q^{94} +521.000i q^{97} +1959.33i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 48 q^{4} - 56 q^{11} + 252 q^{14} + 240 q^{16} - 12 q^{19} + 1252 q^{26} + 184 q^{29} + 244 q^{31} + 1384 q^{34} - 784 q^{41} + 64 q^{44} - 1176 q^{46} - 520 q^{49} + 360 q^{56} + 248 q^{59} + 1500 q^{61}+ \cdots + 1192 q^{94}+O(q^{100}) \) 4 * q - 48 * q^4 - 56 * q^11 + 252 * q^14 + 240 * q^16 - 12 * q^19 + 1252 * q^26 + 184 * q^29 + 244 * q^31 + 1384 * q^34 - 784 * q^41 + 64 * q^44 - 1176 * q^46 - 520 * q^49 + 360 * q^56 + 248 * q^59 + 1500 * q^61 - 2752 * q^64 - 1648 * q^71 + 2392 * q^74 + 752 * q^76 + 1760 * q^79 + 1684 * q^86 - 1728 * q^89 - 4396 * q^91 + 1192 * q^94

Introduction

L-functions

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