LMFDB - Embedded newform 325.6.b.d.274.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 85x^{4} + 1668x^{2} + 2304 $ x^6 + 85x^4 + 1668x^2 + 2304
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.3
Root			$-1.22183i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.d.274.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.22183i q^{2} -16.8416i q^{3} +27.0635 q^{4} -37.4193 q^{6} -177.501i q^{7} -131.229i q^{8} -40.6407 q^{9} -353.532 q^{11} -455.793i q^{12} -169.000i q^{13} -394.376 q^{14} +574.462 q^{16} -1545.40i q^{17} +90.2967i q^{18} +132.374 q^{19} -2989.40 q^{21} +785.489i q^{22} +3897.96i q^{23} -2210.11 q^{24} -375.489 q^{26} -3408.06i q^{27} -4803.78i q^{28} -825.458 q^{29} -5360.62 q^{31} -5475.69i q^{32} +5954.06i q^{33} -3433.62 q^{34} -1099.88 q^{36} +5251.49i q^{37} -294.112i q^{38} -2846.24 q^{39} +17324.0 q^{41} +6641.94i q^{42} -11155.8i q^{43} -9567.80 q^{44} +8660.61 q^{46} +5169.48i q^{47} -9674.88i q^{48} -14699.4 q^{49} -26027.1 q^{51} -4573.73i q^{52} -37052.1i q^{53} -7572.14 q^{54} -23293.2 q^{56} -2229.39i q^{57} +1834.03i q^{58} +22894.2 q^{59} -56415.9 q^{61} +11910.4i q^{62} +7213.74i q^{63} +6216.74 q^{64} +13228.9 q^{66} +33700.6i q^{67} -41823.9i q^{68} +65648.0 q^{69} +19320.2 q^{71} +5333.24i q^{72} +47632.1i q^{73} +11667.9 q^{74} +3582.49 q^{76} +62752.1i q^{77} +6323.86i q^{78} -20458.1 q^{79} -67273.0 q^{81} -38491.0i q^{82} -16270.2i q^{83} -80903.5 q^{84} -24786.3 q^{86} +13902.1i q^{87} +46393.7i q^{88} +73643.6 q^{89} -29997.6 q^{91} +105492. i q^{92} +90281.5i q^{93} +11485.7 q^{94} -92219.5 q^{96} +140209. i q^{97} +32659.7i q^{98} +14367.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 20 q^{4} + 44 q^{6} - 286 q^{9} - 1232 q^{11} + 2568 q^{14} - 3740 q^{16} + 6720 q^{19} - 7584 q^{21} - 1044 q^{24} - 676 q^{26} - 4612 q^{29} - 12760 q^{31} + 5720 q^{34} - 1372 q^{36} + 5408 q^{39}+ \cdots + 138896 q^{99}+O(q^{100}) $ 6 * q + 20 * q^4 + 44 * q^6 - 286 * q^9 - 1232 * q^11 + 2568 * q^14 - 3740 * q^16 + 6720 * q^19 - 7584 * q^21 - 1044 * q^24 - 676 * q^26 - 4612 * q^29 - 12760 * q^31 + 5720 * q^34 - 1372 * q^36 + 5408 * q^39 + 16572 * q^41 - 15380 * q^44 + 28372 * q^46 - 32342 * q^49 - 86656 * q^51 - 60008 * q^54 + 31560 * q^56 - 9504 * q^59 - 173788 * q^61 - 67236 * q^64 - 42376 * q^66 + 17776 * q^69 - 246504 * q^71 - 187776 * q^74 - 51820 * q^76 + 221392 * q^79 - 304154 * q^81 - 159888 * q^84 - 237932 * q^86 + 224420 * q^89 - 70304 * q^91 - 255032 * q^94 - 212876 * q^96 + 138896 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.22183i	− 0.392768i	−0.980527	−	0.196384i	$-0.937080\pi$
$2$	− 2.22183i	− 0.392768i	0.980527	−	0.196384i	$-0.0629199\pi$
$3$	− 16.8416i	− 1.08039i	−0.841539	−	0.540196i	$-0.818350\pi$
$3$	− 16.8416i	− 1.08039i	0.841539	−	0.540196i	$-0.181650\pi$
$4$	27.0635	0.845733
$5$	0	0
$5$	0	0
$6$	−37.4193	−0.424343
$7$	− 177.501i	− 1.36916i	−0.728937	−	0.684580i	$-0.759985\pi$
$7$	− 177.501i	− 1.36916i	0.728937	−	0.684580i	$-0.240015\pi$
$8$	− 131.229i	− 0.724945i
$9$	−40.6407	−0.167246
$10$	0	0
$11$	−353.532	−0.880942	−0.440471	−	0.897767i	$-0.645188\pi$
$11$	−353.532	−0.880942	−0.440471	+	0.897767i	$0.645188\pi$
$12$	− 455.793i	− 0.913723i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−394.376	−0.537763
$15$	0	0
$16$	574.462	0.560998
$17$	− 1545.40i	− 1.29694i	−0.761242	−	0.648468i	$-0.775410\pi$
$17$	− 1545.40i	− 1.29694i	0.761242	−	0.648468i	$-0.224590\pi$
$18$	90.2967i	0.0656887i
$19$	132.374	0.0841236	0.0420618	−	0.999115i	$-0.486607\pi$
$19$	132.374	0.0841236	0.0420618	+	0.999115i	$0.486607\pi$
$20$	0	0
$21$	−2989.40	−1.47923
$22$	785.489i	0.346006i
$23$	3897.96i	1.53645i	0.640181	+	0.768224i	$0.278859\pi$
$23$	3897.96i	1.53645i	−0.640181	+	0.768224i	$0.721141\pi$
$24$	−2210.11	−0.783224
$25$	0	0
$26$	−375.489	−0.108934
$27$	− 3408.06i	− 0.899701i
$28$	− 4803.78i	− 1.15795i
$29$	−825.458	−0.182264	−0.0911318	−	0.995839i	$-0.529048\pi$
$29$	−825.458	−0.182264	−0.0911318	+	0.995839i	$0.529048\pi$
$30$	0	0
$31$	−5360.62	−1.00187	−0.500934	−	0.865485i	$-0.667010\pi$
$31$	−5360.62	−1.00187	−0.500934	+	0.865485i	$0.667010\pi$
$32$	− 5475.69i	− 0.945287i
$33$	5954.06i	0.951762i
$34$	−3433.62	−0.509395
$35$	0	0
$36$	−1099.88	−0.141445
$37$	5251.49i	0.630636i	0.948986	+	0.315318i	$0.102111\pi$
$37$	5251.49i	0.630636i	−0.948986	+	0.315318i	$0.897889\pi$
$38$	− 294.112i	− 0.0330411i
$39$	−2846.24	−0.299647
$40$	0	0
$41$	17324.0	1.60949	0.804745	−	0.593620i	$-0.202302\pi$
$41$	17324.0	1.60949	0.804745	+	0.593620i	$0.202302\pi$
$42$	6641.94i	0.580994i
$43$	− 11155.8i	− 0.920089i	−0.887896	−	0.460044i	$-0.847833\pi$
$43$	− 11155.8i	− 0.920089i	0.887896	−	0.460044i	$-0.152167\pi$
$44$	−9567.80	−0.745042
$45$	0	0
$46$	8660.61	0.603467
$47$	5169.48i	0.341352i	0.985327	+	0.170676i	$0.0545951\pi$
$47$	5169.48i	0.341352i	−0.985327	+	0.170676i	$0.945405\pi$
$48$	− 9674.88i	− 0.606098i
$49$	−14699.4	−0.874602
$50$	0	0
$51$	−26027.1	−1.40120
$52$	− 4573.73i	− 0.234564i
$53$	− 37052.1i	− 1.81186i	−0.423432	−	0.905928i	$-0.639175\pi$
$53$	− 37052.1i	− 1.81186i	0.423432	−	0.905928i	$-0.360825\pi$
$54$	−7572.14	−0.353374
$55$	0	0
$56$	−23293.2	−0.992566
$57$	− 2229.39i	− 0.0908864i
$58$	1834.03i	0.0715873i
$59$	22894.2	0.856240	0.428120	−	0.903722i	$-0.359176\pi$
$59$	22894.2	0.856240	0.428120	+	0.903722i	$0.359176\pi$
$60$	0	0
$61$	−56415.9	−1.94123	−0.970615	−	0.240639i	$-0.922643\pi$
$61$	−56415.9	−1.94123	−0.970615	+	0.240639i	$0.922643\pi$
$62$	11910.4i	0.393502i
$63$	7213.74i	0.228986i
$64$	6216.74	0.189720
$65$	0	0
$66$	13228.9	0.373822
$67$	33700.6i	0.917173i	0.888650	+	0.458586i	$0.151644\pi$
$67$	33700.6i	0.917173i	−0.888650	+	0.458586i	$0.848356\pi$
$68$	− 41823.9i	− 1.09686i
$69$	65648.0	1.65996
$70$	0	0
$71$	19320.2	0.454848	0.227424	−	0.973796i	$-0.426970\pi$
$71$	19320.2	0.454848	0.227424	+	0.973796i	$0.426970\pi$
$72$	5333.24i	0.121244i
$73$	47632.1i	1.04615i	0.852287	+	0.523074i	$0.175215\pi$
$73$	47632.1i	1.04615i	−0.852287	+	0.523074i	$0.824785\pi$
$74$	11667.9	0.247693
$75$	0	0
$76$	3582.49	0.0711461
$77$	62752.1i	1.20615i
$78$	6323.86i	0.117692i
$79$	−20458.1	−0.368805	−0.184403	−	0.982851i	$-0.559035\pi$
$79$	−20458.1	−0.368805	−0.184403	+	0.982851i	$0.559035\pi$
$80$	0	0
$81$	−67273.0	−1.13927
$82$	− 38491.0i	− 0.632156i
$83$	− 16270.2i	− 0.259237i	−0.991564	−	0.129619i	$-0.958625\pi$
$83$	− 16270.2i	− 0.259237i	0.991564	−	0.129619i	$-0.0413753\pi$
$84$	−80903.5	−1.25103
$85$	0	0
$86$	−24786.3	−0.361381
$87$	13902.1i	0.196916i
$88$	46393.7i	0.638634i
$89$	73643.6	0.985508	0.492754	−	0.870169i	$-0.335990\pi$
$89$	73643.6	0.985508	0.492754	+	0.870169i	$0.335990\pi$
$90$	0	0
$91$	−29997.6	−0.379737
$92$	105492.i	1.29942i
$93$	90281.5i	1.08241i
$94$	11485.7	0.134072
$95$	0	0
$96$	−92219.5	−1.02128
$97$	140209.i	1.51303i	0.653976	+	0.756516i	$0.273100\pi$
$97$	140209.i	1.51303i	−0.653976	+	0.756516i	$0.726900\pi$
$98$	32659.7i	0.343516i
$99$	14367.8	0.147334
$100$	0	0
$101$	−45049.4	−0.439426	−0.219713	−	0.975565i	$-0.570512\pi$
$101$	−45049.4	−0.439426	−0.219713	+	0.975565i	$0.570512\pi$
$102$	57827.7i	0.550346i
$103$	125731.i	1.16774i	0.811845	+	0.583872i	$0.198463\pi$
$103$	125731.i	1.16774i	−0.811845	+	0.583872i	$0.801537\pi$
$104$	−22177.7	−0.201064
$105$	0	0
$106$	−82323.6	−0.711639
$107$	9971.50i	0.0841979i	0.999113	+	0.0420989i	$0.0134045\pi$
$107$	9971.50i	0.0841979i	−0.999113	+	0.0420989i	$0.986596\pi$
$108$	− 92234.0i	− 0.760907i
$109$	−121034.	−0.975756	−0.487878	−	0.872912i	$-0.662229\pi$
$109$	−121034.	−0.975756	−0.487878	+	0.872912i	$0.662229\pi$
$110$	0	0
$111$	88443.7	0.681333
$112$	− 101967.i	− 0.768097i
$113$	130644.i	0.962480i	0.876589	+	0.481240i	$0.159813\pi$
$113$	130644.i	0.962480i	−0.876589	+	0.481240i	$0.840187\pi$
$114$	−4953.33	−0.0356973
$115$	0	0
$116$	−22339.8	−0.154146
$117$	6868.27i	0.0463856i
$118$	− 50867.1i	− 0.336304i
$119$	−274309.	−1.77571
$120$	0	0
$121$	−36066.0	−0.223942
$122$	125347.i	0.762453i
$123$	− 291764.i	− 1.73888i
$124$	−145077.	−0.847313
$125$	0	0
$126$	16027.7	0.0899384
$127$	− 157630.i	− 0.867220i	−0.901101	−	0.433610i	$-0.857240\pi$
$127$	− 157630.i	− 0.867220i	0.901101	−	0.433610i	$-0.142760\pi$
$128$	− 189035.i	− 1.01980i
$129$	−187882.	−0.994056
$130$	0	0
$131$	−176730.	−0.899774	−0.449887	−	0.893086i	$-0.648536\pi$
$131$	−176730.	−0.899774	−0.449887	+	0.893086i	$0.648536\pi$
$132$	161137.i	0.804937i
$133$	− 23496.4i	− 0.115179i
$134$	74877.2	0.360236
$135$	0	0
$136$	−202801.	−0.940207
$137$	− 85660.1i	− 0.389921i	−0.980811	−	0.194961i	$-0.937542\pi$
$137$	− 85660.1i	− 0.389921i	0.980811	−	0.194961i	$-0.0624579\pi$
$138$	− 145859.i	− 0.651981i
$139$	221477.	0.972281	0.486141	−	0.873881i	$-0.338404\pi$
$139$	221477.	0.972281	0.486141	+	0.873881i	$0.338404\pi$
$140$	0	0
$141$	87062.4	0.368793
$142$	− 42926.3i	− 0.178650i
$143$	59746.9i	0.244329i
$144$	−23346.5	−0.0938245
$145$	0	0
$146$	105831.	0.410893
$147$	247562.i	0.944912i
$148$	142124.i	0.533350i
$149$	121702.	0.449089	0.224545	−	0.974464i	$-0.427911\pi$
$149$	121702.	0.449089	0.224545	+	0.974464i	$0.427911\pi$
$150$	0	0
$151$	144774.	0.516710	0.258355	−	0.966050i	$-0.416820\pi$
$151$	144774.	0.516710	0.258355	+	0.966050i	$0.416820\pi$
$152$	− 17371.3i	− 0.0609850i
$153$	62806.1i	0.216907i
$154$	139425.	0.473737
$155$	0	0
$156$	−77029.0	−0.253421
$157$	− 548818.i	− 1.77697i	−0.458910	−	0.888483i	$-0.651760\pi$
$157$	− 548818.i	− 1.77697i	0.458910	−	0.888483i	$-0.348240\pi$
$158$	45454.4i	0.144855i
$159$	−624019.	−1.95751
$160$	0	0
$161$	691890.	2.10364
$162$	149469.i	0.447471i
$163$	− 1504.59i	− 0.00443557i	−0.999998	−	0.00221779i	$-0.999294\pi$
$163$	− 1504.59i	− 0.00443557i	0.999998	−	0.00221779i	$-0.000705944\pi$
$164$	468847.	1.36120
$165$	0	0
$166$	−36149.6	−0.101820
$167$	437343.i	1.21348i	0.794902	+	0.606738i	$0.207522\pi$
$167$	437343.i	1.21348i	−0.794902	+	0.606738i	$0.792478\pi$
$168$	392296.i	1.07236i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−5379.76	−0.0140693
$172$	− 301915.i	− 0.778150i
$173$	− 511893.i	− 1.30036i	−0.759779	−	0.650181i	$-0.774693\pi$
$173$	− 511893.i	− 1.30036i	0.759779	−	0.650181i	$-0.225307\pi$
$174$	30888.0	0.0773423
$175$	0	0
$176$	−203091.	−0.494207
$177$	− 385576.i	− 0.925074i
$178$	− 163624.i	− 0.387076i
$179$	424470.	0.990180	0.495090	−	0.868842i	$-0.335135\pi$
$179$	424470.	0.990180	0.495090	+	0.868842i	$0.335135\pi$
$180$	0	0
$181$	−374185.	−0.848965	−0.424482	−	0.905436i	$-0.639544\pi$
$181$	−374185.	−0.848965	−0.424482	+	0.905436i	$0.639544\pi$
$182$	66649.6i	0.149149i
$183$	950136.i	2.09729i
$184$	511526.	1.11384
$185$	0	0
$186$	200590.	0.425136
$187$	546348.i	1.14252i
$188$	139904.i	0.288692i
$189$	−604933.	−1.23184
$190$	0	0
$191$	−358814.	−0.711683	−0.355841	−	0.934546i	$-0.615806\pi$
$191$	−358814.	−0.711683	−0.355841	+	0.934546i	$0.615806\pi$
$192$	− 104700.i	− 0.204972i
$193$	− 841799.i	− 1.62673i	−0.581755	−	0.813364i	$-0.697634\pi$
$193$	− 841799.i	− 1.62673i	0.581755	−	0.813364i	$-0.302366\pi$
$194$	311522.	0.594270
$195$	0	0
$196$	−397818.	−0.739680
$197$	− 694841.i	− 1.27561i	−0.770196	−	0.637807i	$-0.779842\pi$
$197$	− 694841.i	− 1.27561i	0.770196	−	0.637807i	$-0.220158\pi$
$198$	− 31922.8i	− 0.0578679i
$199$	552511.	0.989027	0.494513	−	0.869170i	$-0.335346\pi$
$199$	552511.	0.989027	0.494513	+	0.869170i	$0.335346\pi$
$200$	0	0
$201$	567574.	0.990906
$202$	100092.i	0.172592i
$203$	146519.i	0.249548i
$204$	−704382.	−1.18504
$205$	0	0
$206$	279352.	0.458653
$207$	− 158416.i	− 0.256964i
$208$	− 97084.1i	− 0.155593i
$209$	−46798.4	−0.0741080
$210$	0	0
$211$	−117103.	−0.181076	−0.0905380	−	0.995893i	$-0.528859\pi$
$211$	−117103.	−0.181076	−0.0905380	+	0.995893i	$0.528859\pi$
$212$	− 1.00276e6i	− 1.53235i
$213$	− 325384.i	− 0.491414i
$214$	22155.0	0.0330702
$215$	0	0
$216$	−447237.	−0.652233
$217$	951512.i	1.37172i
$218$	268917.i	0.383246i
$219$	802203.	1.13025
$220$	0	0
$221$	−261173.	−0.359705
$222$	− 196507.i	− 0.267606i
$223$	885756.i	1.19276i	0.802703	+	0.596379i	$0.203394\pi$
$223$	885756.i	1.19276i	−0.802703	+	0.596379i	$0.796606\pi$
$224$	−971937.	−1.29425
$225$	0	0
$226$	290268.	0.378031
$227$	− 114582.i	− 0.147588i	−0.997273	−	0.0737941i	$-0.976489\pi$
$227$	− 114582.i	− 0.147588i	0.997273	−	0.0737941i	$-0.0235108\pi$
$228$	− 60335.0i	− 0.0768657i
$229$	1.37389e6	1.73126	0.865630	−	0.500683i	$-0.166918\pi$
$229$	1.37389e6	1.73126	0.865630	+	0.500683i	$0.166918\pi$
$230$	0	0
$231$	1.05685e6	1.30312
$232$	108324.i	0.132131i
$233$	− 1.28998e6i	− 1.55666i	−0.627857	−	0.778329i	$-0.716068\pi$
$233$	− 1.28998e6i	− 1.55666i	0.627857	−	0.778329i	$-0.283932\pi$
$234$	15260.1	0.0182188
$235$	0	0
$236$	619597.	0.724151
$237$	344547.i	0.398454i
$238$	609469.i	0.697444i
$239$	626246.	0.709170	0.354585	−	0.935024i	$-0.384622\pi$
$239$	626246.	0.709170	0.354585	+	0.935024i	$0.384622\pi$
$240$	0	0
$241$	876030.	0.971575	0.485788	−	0.874077i	$-0.338533\pi$
$241$	876030.	0.971575	0.485788	+	0.874077i	$0.338533\pi$
$242$	80132.6i	0.0879571i
$243$	304829.i	0.331162i
$244$	−1.52681e6	−1.64176
$245$	0	0
$246$	−648251.	−0.682976
$247$	− 22371.2i	− 0.0233317i
$248$	703469.i	0.726299i
$249$	−274016.	−0.280078
$250$	0	0
$251$	1.69324e6	1.69642	0.848209	−	0.529661i	$-0.177681\pi$
$251$	1.69324e6	1.69642	0.848209	+	0.529661i	$0.177681\pi$
$252$	195229.i	0.193661i
$253$	− 1.37805e6i	− 1.35352i
$254$	−350227.	−0.340616
$255$	0	0
$256$	−221067.	−0.210826
$257$	775274.i	0.732188i	0.930578	+	0.366094i	$0.119305\pi$
$257$	775274.i	0.732188i	−0.930578	+	0.366094i	$0.880695\pi$
$258$	417442.i	0.390433i
$259$	932143.	0.863442
$260$	0	0
$261$	33547.2	0.0304828
$262$	392665.i	0.353402i
$263$	− 1.64046e6i	− 1.46244i	−0.682143	−	0.731218i	$-0.738952\pi$
$263$	− 1.64046e6i	− 1.46244i	0.682143	−	0.731218i	$-0.261048\pi$
$264$	781346.	0.689975
$265$	0	0
$266$	−52205.0	−0.0452385
$267$	− 1.24028e6i	− 1.06473i
$268$	912056.i	0.775684i
$269$	−1.00150e6	−0.843860	−0.421930	−	0.906629i	$-0.638647\pi$
$269$	−1.00150e6	−0.843860	−0.421930	+	0.906629i	$0.638647\pi$
$270$	0	0
$271$	−1.29197e6	−1.06864	−0.534318	−	0.845284i	$-0.679431\pi$
$271$	−1.29197e6	−1.06864	−0.534318	+	0.845284i	$0.679431\pi$
$272$	− 887774.i	− 0.727579i
$273$	505208.i	0.410265i
$274$	−190322.	−0.153149
$275$	0	0
$276$	1.77666e6	1.40389
$277$	− 2.50509e6i	− 1.96166i	−0.194873	−	0.980829i	$-0.562429\pi$
$277$	− 2.50509e6i	− 1.96166i	0.194873	−	0.980829i	$-0.437571\pi$
$278$	− 492085.i	− 0.381881i
$279$	217859.	0.167558
$280$	0	0
$281$	965963.	0.729785	0.364892	−	0.931050i	$-0.381106\pi$
$281$	965963.	0.729785	0.364892	+	0.931050i	$0.381106\pi$
$282$	− 193438.i	− 0.144850i
$283$	1.70072e6i	1.26231i	0.775657	+	0.631155i	$0.217419\pi$
$283$	1.70072e6i	1.26231i	−0.775657	+	0.631155i	$0.782581\pi$
$284$	522873.	0.384680
$285$	0	0
$286$	132748.	0.0959647
$287$	− 3.07502e6i	− 2.20365i
$288$	222536.i	0.158095i
$289$	−968403.	−0.682042
$290$	0	0
$291$	2.36136e6	1.63467
$292$	1.28909e6i	0.884762i
$293$	− 507714.i	− 0.345502i	−0.984966	−	0.172751i	$-0.944734\pi$
$293$	− 507714.i	− 0.345502i	0.984966	−	0.172751i	$-0.0552656\pi$
$294$	550042.	0.371131
$295$	0	0
$296$	689148.	0.457176
$297$	1.20486e6i	0.792584i
$298$	− 270402.i	− 0.176388i
$299$	658755.	0.426134
$300$	0	0
$301$	−1.98016e6	−1.25975
$302$	− 321662.i	− 0.202947i
$303$	758706.i	0.474752i
$304$	76043.7	0.0471932
$305$	0	0
$306$	139545.	0.0851940
$307$	− 529122.i	− 0.320413i	−0.987084	−	0.160206i	$-0.948784\pi$
$307$	− 529122.i	− 0.320413i	0.987084	−	0.160206i	$-0.0512160\pi$
$308$	1.69829e6i	1.02008i
$309$	2.11751e6	1.26162
$310$	0	0
$311$	−524988.	−0.307785	−0.153893	−	0.988088i	$-0.549181\pi$
$311$	−524988.	−0.307785	−0.153893	+	0.988088i	$0.549181\pi$
$312$	373509.i	0.217227i
$313$	− 2.13481e6i	− 1.23168i	−0.787870	−	0.615842i	$-0.788816\pi$
$313$	− 2.13481e6i	− 1.23168i	0.787870	−	0.615842i	$-0.211184\pi$
$314$	−1.21938e6	−0.697935
$315$	0	0
$316$	−553667.	−0.311911
$317$	− 1.52498e6i	− 0.852348i	−0.904641	−	0.426174i	$-0.859861\pi$
$317$	− 1.52498e6i	− 0.852348i	0.904641	−	0.426174i	$-0.140139\pi$
$318$	1.38646e6i	0.768849i
$319$	291826.	0.160564
$320$	0	0
$321$	167936.	0.0909667
$322$	− 1.53726e6i	− 0.826244i
$323$	− 204570.i	− 0.109103i
$324$	−1.82064e6	−0.963522
$325$	0	0
$326$	−3342.95	−0.00174215
$327$	2.03841e6i	1.05420i
$328$	− 2.27341e6i	− 1.16679i
$329$	917585.	0.467365
$330$	0	0
$331$	−835573.	−0.419194	−0.209597	−	0.977788i	$-0.567215\pi$
$331$	−835573.	−0.419194	−0.209597	+	0.977788i	$0.567215\pi$
$332$	− 440327.i	− 0.219245i
$333$	− 213424.i	− 0.105471i
$334$	971703.	0.476615
$335$	0	0
$336$	−1.71730e6	−0.829845
$337$	− 1.15640e6i	− 0.554670i	−0.960773	−	0.277335i	$-0.910549\pi$
$337$	− 1.15640e6i	− 0.554670i	0.960773	−	0.277335i	$-0.0894512\pi$
$338$	63457.7i	0.0302129i
$339$	2.20025e6	1.03986
$340$	0	0
$341$	1.89515e6	0.882587
$342$	11952.9i	0.00552597i
$343$	− 374094.i	− 0.171690i
$344$	−1.46397e6	−0.667014
$345$	0	0
$346$	−1.13734e6	−0.510740
$347$	− 229963.i	− 0.102526i	−0.998685	−	0.0512630i	$-0.983675\pi$
$347$	− 229963.i	− 0.102526i	0.998685	−	0.0512630i	$-0.0163247\pi$
$348$	376238.i	0.166538i
$349$	−4.06529e6	−1.78660	−0.893301	−	0.449460i	$-0.851616\pi$
$349$	−4.06529e6	−1.78660	−0.893301	+	0.449460i	$0.851616\pi$
$350$	0	0
$351$	−575962.	−0.249532
$352$	1.93583e6i	0.832743i
$353$	− 2.23732e6i	− 0.955634i	−0.878459	−	0.477817i	$-0.841428\pi$
$353$	− 2.23732e6i	− 0.955634i	0.878459	−	0.477817i	$-0.158572\pi$
$354$	−856684.	−0.363340
$355$	0	0
$356$	1.99305e6	0.833477
$357$	4.61982e6i	1.91847i
$358$	− 943100.i	− 0.388911i
$359$	−1.05065e6	−0.430250	−0.215125	−	0.976587i	$-0.569016\pi$
$359$	−1.05065e6	−0.430250	−0.215125	+	0.976587i	$0.569016\pi$
$360$	0	0
$361$	−2.45858e6	−0.992923
$362$	831376.i	0.333446i
$363$	607411.i	0.241945i
$364$	−811839.	−0.321156
$365$	0	0
$366$	2.11104e6	0.823747
$367$	481650.i	0.186666i	0.995635	+	0.0933332i	$0.0297522\pi$
$367$	481650.i	0.186666i	−0.995635	+	0.0933332i	$0.970248\pi$
$368$	2.23923e6i	0.861944i
$369$	−704059.	−0.269180
$370$	0	0
$371$	−6.57677e6	−2.48072
$372$	2.44333e6i	0.915430i
$373$	519900.i	0.193485i	0.995309	+	0.0967426i	$0.0308424\pi$
$373$	519900.i	0.193485i	−0.995309	+	0.0967426i	$0.969158\pi$
$374$	1.21389e6	0.448747
$375$	0	0
$376$	678386.	0.247461
$377$	139502.i	0.0505508i
$378$	1.34406e6i	0.483825i
$379$	5.03926e6	1.80206	0.901029	−	0.433759i	$-0.142813\pi$
$379$	5.03926e6	1.80206	0.901029	+	0.433759i	$0.142813\pi$
$380$	0	0
$381$	−2.65475e6	−0.936937
$382$	797225.i	0.279526i
$383$	− 2.06181e6i	− 0.718212i	−0.933297	−	0.359106i	$-0.883082\pi$
$383$	− 2.06181e6i	− 0.718212i	0.933297	−	0.359106i	$-0.116918\pi$
$384$	−3.18365e6	−1.10179
$385$	0	0
$386$	−1.87034e6	−0.638927
$387$	453379.i	0.153881i
$388$	3.79455e6i	1.27962i
$389$	3.43440e6	1.15074	0.575370	−	0.817893i	$-0.304858\pi$
$389$	3.43440e6	1.15074	0.575370	+	0.817893i	$0.304858\pi$
$390$	0	0
$391$	6.02390e6	1.99267
$392$	1.92899e6i	0.634038i
$393$	2.97643e6i	0.972108i
$394$	−1.54382e6	−0.501021
$395$	0	0
$396$	388842.	0.124605
$397$	1.81167e6i	0.576903i	0.957494	+	0.288452i	$0.0931404\pi$
$397$	1.81167e6i	0.576903i	−0.957494	+	0.288452i	$0.906860\pi$
$398$	− 1.22759e6i	− 0.388458i
$399$	−395718.	−0.124438
$400$	0	0
$401$	546555.	0.169736	0.0848678	−	0.996392i	$-0.472953\pi$
$401$	546555.	0.169736	0.0848678	+	0.996392i	$0.472953\pi$
$402$	− 1.26105e6i	− 0.389196i
$403$	905944.i	0.277868i
$404$	−1.21919e6	−0.371637
$405$	0	0
$406$	325541.	0.0980145
$407$	− 1.85657e6i	− 0.555553i
$408$	3.41551e6i	1.01579i
$409$	−4.17249e6	−1.23335	−0.616676	−	0.787217i	$-0.711521\pi$
$409$	−4.17249e6	−1.23335	−0.616676	+	0.787217i	$0.711521\pi$
$410$	0	0
$411$	−1.44266e6	−0.421268
$412$	3.40271e6i	0.987601i
$413$	− 4.06373e6i	− 1.17233i
$414$	−351973.	−0.100927
$415$	0	0
$416$	−925391.	−0.262175
$417$	− 3.73004e6i	− 1.05044i
$418$	103978.i	0.0291072i
$419$	2.08208e6	0.579379	0.289689	−	0.957121i	$-0.406448\pi$
$419$	2.08208e6	0.579379	0.289689	+	0.957121i	$0.406448\pi$
$420$	0	0
$421$	5.16730e6	1.42088	0.710442	−	0.703756i	$-0.248495\pi$
$421$	5.16730e6	1.42088	0.710442	+	0.703756i	$0.248495\pi$
$422$	260183.i	0.0711209i
$423$	− 210091.i	− 0.0570895i
$424$	−4.86232e6	−1.31350
$425$	0	0
$426$	−722949.	−0.193012
$427$	1.00139e7i	2.65786i
$428$	269863.i	0.0712090i
$429$	1.00624e6	0.263971
$430$	0	0
$431$	−5.84842e6	−1.51651	−0.758256	−	0.651957i	$-0.773948\pi$
$431$	−5.84842e6	−1.51651	−0.758256	+	0.651957i	$0.773948\pi$
$432$	− 1.95780e6i	− 0.504730i
$433$	− 1.57061e6i	− 0.402577i	−0.979532	−	0.201289i	$-0.935487\pi$
$433$	− 1.57061e6i	− 0.402577i	0.979532	−	0.201289i	$-0.0645129\pi$
$434$	2.11410e6	0.538767
$435$	0	0
$436$	−3.27560e6	−0.825229
$437$	515987.i	0.129251i
$438$	− 1.78236e6i	− 0.443925i
$439$	4.17392e6	1.03367	0.516837	−	0.856084i	$-0.327109\pi$
$439$	4.17392e6	1.03367	0.516837	+	0.856084i	$0.327109\pi$
$440$	0	0
$441$	597395.	0.146273
$442$	580281.i	0.141281i
$443$	− 3.07291e6i	− 0.743944i	−0.928244	−	0.371972i	$-0.878682\pi$
$443$	− 3.07291e6i	− 0.743944i	0.928244	−	0.371972i	$-0.121318\pi$
$444$	2.39359e6	0.576226
$445$	0	0
$446$	1.96800e6	0.468477
$447$	− 2.04966e6i	− 0.485192i
$448$	− 1.10347e6i	− 0.259757i
$449$	−5.77522e6	−1.35192	−0.675962	−	0.736936i	$-0.736272\pi$
$449$	−5.77522e6	−1.35192	−0.675962	+	0.736936i	$0.736272\pi$
$450$	0	0
$451$	−6.12459e6	−1.41787
$452$	3.53567e6i	0.814001i
$453$	− 2.43822e6i	− 0.558249i
$454$	−254582.	−0.0579679
$455$	0	0
$456$	−292561.	−0.0658876
$457$	190877.i	0.0427526i	0.999772	+	0.0213763i	$0.00680481\pi$
$457$	190877.i	0.0427526i	−0.999772	+	0.0213763i	$0.993195\pi$
$458$	− 3.05255e6i	− 0.679984i
$459$	−5.26682e6	−1.16685
$460$	0	0
$461$	−7.30776e6	−1.60152	−0.800760	−	0.598986i	$-0.795571\pi$
$461$	−7.30776e6	−1.60152	−0.800760	+	0.598986i	$0.795571\pi$
$462$	− 2.34814e6i	− 0.511822i
$463$	4.81629e6i	1.04414i	0.852902	+	0.522072i	$0.174841\pi$
$463$	4.81629e6i	1.04414i	−0.852902	+	0.522072i	$0.825159\pi$
$464$	−474194.	−0.102250
$465$	0	0
$466$	−2.86612e6	−0.611405
$467$	6.73997e6i	1.43010i	0.699075	+	0.715049i	$0.253595\pi$
$467$	6.73997e6i	1.43010i	−0.699075	+	0.715049i	$0.746405\pi$
$468$	185879.i	0.0392298i
$469$	5.98188e6	1.25576
$470$	0	0
$471$	−9.24299e6	−1.91982
$472$	− 3.00438e6i	− 0.620727i
$473$	3.94394e6i	0.810545i
$474$	765526.	0.156500
$475$	0	0
$476$	−7.42376e6	−1.50178
$477$	1.50582e6i	0.303025i
$478$	− 1.39141e6i	− 0.278539i
$479$	7.26335e6	1.44643	0.723217	−	0.690621i	$-0.242663\pi$
$479$	7.26335e6	1.44643	0.723217	+	0.690621i	$0.242663\pi$
$480$	0	0
$481$	887502.	0.174907
$482$	− 1.94639e6i	− 0.381604i
$483$	− 1.16526e7i	− 2.27276i
$484$	−976072.	−0.189395
$485$	0	0
$486$	677278.	0.130070
$487$	453817.i	0.0867079i	0.999060	+	0.0433539i	$0.0138043\pi$
$487$	453817.i	0.0867079i	−0.999060	+	0.0433539i	$0.986196\pi$
$488$	7.40340e6i	1.40728i
$489$	−25339.8	−0.00479215
$490$	0	0
$491$	7.33509e6	1.37310	0.686550	−	0.727083i	$-0.259124\pi$
$491$	7.33509e6	1.37310	0.686550	+	0.727083i	$0.259124\pi$
$492$	− 7.89616e6i	− 1.47063i
$493$	1.27566e6i	0.236384i
$494$	−49704.9	−0.00916394
$495$	0	0
$496$	−3.07947e6	−0.562046
$497$	− 3.42935e6i	− 0.622761i
$498$	608818.i	0.110005i
$499$	2.82351e6	0.507620	0.253810	−	0.967254i	$-0.418316\pi$
$499$	2.82351e6	0.507620	0.253810	+	0.967254i	$0.418316\pi$
$500$	0	0
$501$	7.36558e6	1.31103
$502$	− 3.76208e6i	− 0.666299i
$503$	− 5.55958e6i	− 0.979765i	−0.871789	−	0.489882i	$-0.837040\pi$
$503$	− 5.55958e6i	− 0.979765i	0.871789	−	0.489882i	$-0.162960\pi$
$504$	946652.	0.166002
$505$	0	0
$506$	−3.06180e6	−0.531620
$507$	481014.i	0.0831070i
$508$	− 4.26601e6i	− 0.733437i
$509$	6.75257e6	1.15525	0.577623	−	0.816304i	$-0.303980\pi$
$509$	6.75257e6	1.15525	0.577623	+	0.816304i	$0.303980\pi$
$510$	0	0
$511$	8.45473e6	1.43234
$512$	− 5.55793e6i	− 0.936997i
$513$	− 451138.i	− 0.0756861i
$514$	1.72253e6	0.287580
$515$	0	0
$516$	−5.08474e6	−0.840706
$517$	− 1.82758e6i	− 0.300711i
$518$	− 2.07106e6i	− 0.339132i
$519$	−8.62112e6	−1.40490
$520$	0	0
$521$	6.06655e6	0.979146	0.489573	−	0.871962i	$-0.337153\pi$
$521$	6.06655e6	0.979146	0.489573	+	0.871962i	$0.337153\pi$
$522$	− 74536.1i	− 0.0119727i
$523$	− 2.28846e6i	− 0.365838i	−0.983128	−	0.182919i	$-0.941445\pi$
$523$	− 2.28846e6i	− 0.365838i	0.983128	−	0.182919i	$-0.0585547\pi$
$524$	−4.78294e6	−0.760968
$525$	0	0
$526$	−3.64483e6	−0.574398
$527$	8.28429e6i	1.29936i
$528$	3.42038e6i	0.533937i
$529$	−8.75775e6	−1.36067
$530$	0	0
$531$	−930436.	−0.143202
$532$	− 635894.i	− 0.0974105i
$533$	− 2.92776e6i	− 0.446392i
$534$	−2.75569e6	−0.418194
$535$	0	0
$536$	4.42250e6	0.664900
$537$	− 7.14877e6i	− 1.06978i
$538$	2.22516e6i	0.331441i
$539$	5.19672e6	0.770473
$540$	0	0
$541$	1.01976e7	1.49797	0.748986	−	0.662586i	$-0.230541\pi$
$541$	1.01976e7	1.49797	0.748986	+	0.662586i	$0.230541\pi$
$542$	2.87054e6i	0.419726i
$543$	6.30188e6i	0.917214i
$544$	−8.46212e6	−1.22598
$545$	0	0
$546$	1.12249e6	0.161139
$547$	9.57008e6i	1.36756i	0.729687	+	0.683781i	$0.239666\pi$
$547$	9.57008e6i	1.36756i	−0.729687	+	0.683781i	$0.760334\pi$
$548$	− 2.31826e6i	− 0.329770i
$549$	2.29278e6	0.324662
$550$	0	0
$551$	−109269.	−0.0153327
$552$	− 8.61493e6i	− 1.20338i
$553$	3.63132e6i	0.504954i
$554$	−5.56588e6	−0.770476
$555$	0	0
$556$	5.99394e6	0.822291
$557$	3.80767e6i	0.520021i	0.965606	+	0.260010i	$0.0837260\pi$
$557$	3.80767e6i	0.520021i	−0.965606	+	0.260010i	$0.916274\pi$
$558$	− 484046.i	− 0.0658114i
$559$	−1.88533e6	−0.255187
$560$	0	0
$561$	9.20140e6	1.23437
$562$	− 2.14621e6i	− 0.286636i
$563$	2.91611e6i	0.387733i	0.981028	+	0.193866i	$0.0621028\pi$
$563$	2.91611e6i	0.387733i	−0.981028	+	0.193866i	$0.937897\pi$
$564$	2.35621e6	0.311901
$565$	0	0
$566$	3.77871e6	0.495795
$567$	1.19410e7i	1.55985i
$568$	− 2.53538e6i	− 0.329740i
$569$	1.01859e6	0.131892	0.0659459	−	0.997823i	$-0.478994\pi$
$569$	1.01859e6	0.131892	0.0659459	+	0.997823i	$0.478994\pi$
$570$	0	0
$571$	5.49132e6	0.704833	0.352417	−	0.935843i	$-0.385360\pi$
$571$	5.49132e6	0.704833	0.352417	+	0.935843i	$0.385360\pi$
$572$	1.61696e6i	0.206637i
$573$	6.04302e6i	0.768896i
$574$	−6.83217e6	−0.865524
$575$	0	0
$576$	−252652.	−0.0317298
$577$	− 6.62872e6i	− 0.828877i	−0.910077	−	0.414439i	$-0.863978\pi$
$577$	− 6.62872e6i	− 0.828877i	0.910077	−	0.414439i	$-0.136022\pi$
$578$	2.15163e6i	0.267884i
$579$	−1.41773e7	−1.75750
$580$	0	0
$581$	−2.88797e6	−0.354937
$582$	− 5.24653e6i	− 0.642044i
$583$	1.30991e7i	1.59614i
$584$	6.25072e6	0.758399
$585$	0	0
$586$	−1.12806e6	−0.135702
$587$	− 9.21203e6i	− 1.10347i	−0.834020	−	0.551734i	$-0.813966\pi$
$587$	− 9.21203e6i	− 1.10347i	0.834020	−	0.551734i	$-0.186034\pi$
$588$	6.69990e6i	0.799144i
$589$	−709605.	−0.0842807
$590$	0	0
$591$	−1.17023e7	−1.37816
$592$	3.01678e6i	0.353785i
$593$	− 1.40550e6i	− 0.164133i	−0.996627	−	0.0820664i	$-0.973848\pi$
$593$	− 1.40550e6i	− 0.164133i	0.996627	−	0.0820664i	$-0.0261520\pi$
$594$	2.67699e6	0.311302
$595$	0	0
$596$	3.29368e6	0.379810
$597$	− 9.30518e6i	− 1.06854i
$598$	− 1.46364e6i	− 0.167372i
$599$	9.54275e6	1.08669	0.543346	−	0.839509i	$-0.317157\pi$
$599$	9.54275e6	1.08669	0.543346	+	0.839509i	$0.317157\pi$
$600$	0	0
$601$	−1.61935e6	−0.182875	−0.0914377	−	0.995811i	$-0.529146\pi$
$601$	−1.61935e6	−0.182875	−0.0914377	+	0.995811i	$0.529146\pi$
$602$	4.39958e6i	0.494789i
$603$	− 1.36962e6i	− 0.153393i
$604$	3.91807e6	0.436999
$605$	0	0
$606$	1.68572e6	0.186467
$607$	− 1.02926e7i	− 1.13385i	−0.823770	−	0.566924i	$-0.808133\pi$
$607$	− 1.02926e7i	− 1.13385i	0.823770	−	0.566924i	$-0.191867\pi$
$608$	− 724837.i	− 0.0795209i
$609$	2.46762e6	0.269610
$610$	0	0
$611$	873642.	0.0946739
$612$	1.69975e6i	0.183445i
$613$	− 2.64065e6i	− 0.283831i	−0.989879	−	0.141916i	$-0.954674\pi$
$613$	− 2.64065e6i	− 0.283831i	0.989879	−	0.141916i	$-0.0453262\pi$
$614$	−1.17562e6	−0.125848
$615$	0	0
$616$	8.23490e6	0.874393
$617$	1.18557e7i	1.25376i	0.779114	+	0.626882i	$0.215669\pi$
$617$	1.18557e7i	1.25376i	−0.779114	+	0.626882i	$0.784331\pi$
$618$	− 4.70475e6i	− 0.495524i
$619$	1.38298e7	1.45074	0.725370	−	0.688359i	$-0.241669\pi$
$619$	1.38298e7	1.45074	0.725370	+	0.688359i	$0.241669\pi$
$620$	0	0
$621$	1.32845e7	1.38234
$622$	1.16643e6i	0.120888i
$623$	− 1.30718e7i	− 1.34932i
$624$	−1.63506e6	−0.168101
$625$	0	0
$626$	−4.74320e6	−0.483766
$627$	788161.i	0.0800656i
$628$	− 1.48529e7i	− 1.50284i
$629$	8.11565e6	0.817894
$630$	0	0
$631$	−621762.	−0.0621657	−0.0310829	−	0.999517i	$-0.509896\pi$
$631$	−621762.	−0.0621657	−0.0310829	+	0.999517i	$0.509896\pi$
$632$	2.68469e6i	0.267363i
$633$	1.97220e6i	0.195633i
$634$	−3.38825e6	−0.334775
$635$	0	0
$636$	−1.68881e7	−1.65553
$637$	2.48420e6i	0.242571i
$638$	− 648388.i	− 0.0630642i
$639$	−785188.	−0.0760714
$640$	0	0
$641$	−4.56137e6	−0.438481	−0.219240	−	0.975671i	$-0.570358\pi$
$641$	−4.56137e6	−0.438481	−0.219240	+	0.975671i	$0.570358\pi$
$642$	− 373126.i	− 0.0357288i
$643$	1.30425e7i	1.24404i	0.783003	+	0.622018i	$0.213687\pi$
$643$	1.30425e7i	1.24404i	−0.783003	+	0.622018i	$0.786313\pi$
$644$	1.87249e7	1.77912
$645$	0	0
$646$	−454521.	−0.0428521
$647$	− 2.42397e6i	− 0.227649i	−0.993501	−	0.113825i	$-0.963690\pi$
$647$	− 2.42397e6i	− 0.227649i	0.993501	−	0.113825i	$-0.0363102\pi$
$648$	8.82817e6i	0.825911i
$649$	−8.09384e6	−0.754298
$650$	0	0
$651$	1.60250e7	1.48199
$652$	− 40719.5i	− 0.00375131i
$653$	1.21724e7i	1.11711i	0.829469	+	0.558553i	$0.188643\pi$
$653$	1.21724e7i	1.11711i	−0.829469	+	0.558553i	$0.811357\pi$
$654$	4.52900e6	0.414055
$655$	0	0
$656$	9.95198e6	0.902921
$657$	− 1.93580e6i	− 0.174964i
$658$	− 2.03872e6i	− 0.183566i
$659$	2.00313e6	0.179678	0.0898392	−	0.995956i	$-0.471365\pi$
$659$	2.00313e6	0.179678	0.0898392	+	0.995956i	$0.471365\pi$
$660$	0	0
$661$	−1.32025e7	−1.17531	−0.587657	−	0.809110i	$-0.699950\pi$
$661$	−1.32025e7	−1.17531	−0.587657	+	0.809110i	$0.699950\pi$
$662$	1.85650e6i	0.164646i
$663$	4.39857e6i	0.388622i
$664$	−2.13512e6	−0.187933
$665$	0	0
$666$	−474193.	−0.0414256
$667$	− 3.21760e6i	− 0.280038i
$668$	1.18360e7i	1.02628i
$669$	1.49176e7	1.28865
$670$	0	0
$671$	1.99448e7	1.71011
$672$	1.63690e7i	1.39830i
$673$	1.25393e7i	1.06718i	0.845744	+	0.533590i	$0.179157\pi$
$673$	1.25393e7i	1.06718i	−0.845744	+	0.533590i	$0.820843\pi$
$674$	−2.56933e6	−0.217857
$675$	0	0
$676$	−772960.	−0.0650564
$677$	2.58077e6i	0.216410i	0.994129	+	0.108205i	$0.0345103\pi$
$677$	2.58077e6i	0.216410i	−0.994129	+	0.108205i	$0.965490\pi$
$678$	− 4.88858e6i	− 0.408422i
$679$	2.48872e7	2.07158
$680$	0	0
$681$	−1.92975e6	−0.159453
$682$	− 4.21070e6i	− 0.346652i
$683$	2.33466e6i	0.191502i	0.995405	+	0.0957509i	$0.0305252\pi$
$683$	2.33466e6i	0.191502i	−0.995405	+	0.0957509i	$0.969475\pi$
$684$	−145595.	−0.0118989
$685$	0	0
$686$	−831175.	−0.0674345
$687$	− 2.31385e7i	− 1.87044i
$688$	− 6.40859e6i	− 0.516168i
$689$	−6.26181e6	−0.502518
$690$	0	0
$691$	−9.56099e6	−0.761742	−0.380871	−	0.924628i	$-0.624376\pi$
$691$	−9.56099e6	−0.761742	−0.380871	+	0.924628i	$0.624376\pi$
$692$	− 1.38536e7i	− 1.09976i
$693$	− 2.55029e6i	− 0.201723i
$694$	−510938.	−0.0402689
$695$	0	0
$696$	1.82435e6	0.142753
$697$	− 2.67725e7i	− 2.08741i
$698$	9.03238e6i	0.701720i
$699$	−2.17254e7	−1.68180
$700$	0	0
$701$	1.75263e7	1.34708	0.673542	−	0.739149i	$-0.264772\pi$
$701$	1.75263e7	1.34708	0.673542	+	0.739149i	$0.264772\pi$
$702$	1.27969e6i	0.0980082i
$703$	695160.i	0.0530513i
$704$	−2.19782e6	−0.167132
$705$	0	0
$706$	−4.97095e6	−0.375342
$707$	7.99629e6i	0.601645i
$708$	− 1.04350e7i	− 0.782366i
$709$	1.04230e7	0.778714	0.389357	−	0.921087i	$-0.372697\pi$
$709$	1.04230e7	0.778714	0.389357	+	0.921087i	$0.372697\pi$
$710$	0	0
$711$	831430.	0.0616811
$712$	− 9.66418e6i	− 0.714439i
$713$	− 2.08955e7i	− 1.53932i
$714$	1.02645e7	0.753512
$715$	0	0
$716$	1.14876e7	0.837428
$717$	− 1.05470e7i	− 0.766181i
$718$	2.33436e6i	0.168988i
$719$	1.40272e7	1.01193	0.505964	−	0.862554i	$-0.331137\pi$
$719$	1.40272e7	1.01193	0.505964	+	0.862554i	$0.331137\pi$
$720$	0	0
$721$	2.23172e7	1.59883
$722$	5.46254e6i	0.389988i
$723$	− 1.47538e7i	− 1.04968i
$724$	−1.01267e7	−0.717998
$725$	0	0
$726$	1.34956e6	0.0950281
$727$	1.43818e7i	1.00920i	0.863354	+	0.504599i	$0.168360\pi$
$727$	1.43818e7i	1.00920i	−0.863354	+	0.504599i	$0.831640\pi$
$728$	3.93655e6i	0.275288i
$729$	−1.12135e7	−0.781490
$730$	0	0
$731$	−1.72402e7	−1.19330
$732$	2.57140e7i	1.77375i
$733$	1.55999e7i	1.07241i	0.844087	+	0.536207i	$0.180143\pi$
$733$	1.55999e7i	1.07241i	−0.844087	+	0.536207i	$0.819857\pi$
$734$	1.07014e6	0.0733166
$735$	0	0
$736$	2.13440e7	1.45238
$737$	− 1.19143e7i	− 0.807976i
$738$	1.56430e6i	0.105725i
$739$	−1.09689e7	−0.738840	−0.369420	−	0.929263i	$-0.620444\pi$
$739$	−1.09689e7	−0.738840	−0.369420	+	0.929263i	$0.620444\pi$
$740$	0	0
$741$	−376767.	−0.0252074
$742$	1.46125e7i	0.974348i
$743$	− 2.14327e7i	− 1.42431i	−0.702021	−	0.712156i	$-0.747719\pi$
$743$	− 2.14327e7i	− 1.42431i	0.702021	−	0.712156i	$-0.252281\pi$
$744$	1.18476e7	0.784687
$745$	0	0
$746$	1.15513e6	0.0759948
$747$	661231.i	0.0433563i
$748$	1.47861e7i	0.966271i
$749$	1.76995e6	0.115280
$750$	0	0
$751$	−2.79453e7	−1.80804	−0.904022	−	0.427486i	$-0.859399\pi$
$751$	−2.79453e7	−1.80804	−0.904022	+	0.427486i	$0.859399\pi$
$752$	2.96967e6i	0.191498i
$753$	− 2.85169e7i	− 1.83280i
$754$	309951.	0.0198547
$755$	0	0
$756$	−1.63716e7	−1.04180
$757$	9.90774e6i	0.628398i	0.949357	+	0.314199i	$0.101736\pi$
$757$	9.90774e6i	0.628398i	−0.949357	+	0.314199i	$0.898264\pi$
$758$	− 1.11964e7i	− 0.707790i
$759$	−2.32087e7	−1.46233
$760$	0	0
$761$	−7.18134e6	−0.449515	−0.224757	−	0.974415i	$-0.572159\pi$
$761$	−7.18134e6	−0.449515	−0.224757	+	0.974415i	$0.572159\pi$
$762$	5.89840e6i	0.367999i
$763$	2.14836e7i	1.33597i
$764$	−9.71076e6	−0.601894
$765$	0	0
$766$	−4.58100e6	−0.282091
$767$	− 3.86912e6i	− 0.237478i
$768$	3.72313e6i	0.227775i
$769$	−4.84119e6	−0.295213	−0.147607	−	0.989046i	$-0.547157\pi$
$769$	−4.84119e6	−0.295213	−0.147607	+	0.989046i	$0.547157\pi$
$770$	0	0
$771$	1.30569e7	0.791050
$772$	− 2.27820e7i	− 1.37578i
$773$	− 1.57169e7i	− 0.946059i	−0.881046	−	0.473030i	$-0.843160\pi$
$773$	− 1.57169e7i	− 0.946059i	0.881046	−	0.473030i	$-0.156840\pi$
$774$	1.00733e6	0.0604394
$775$	0	0
$776$	1.83995e7	1.09686
$777$	− 1.56988e7i	− 0.932855i
$778$	− 7.63066e6i	− 0.451974i
$779$	2.29324e6	0.135396
$780$	0	0
$781$	−6.83033e6	−0.400695
$782$	− 1.33841e7i	− 0.782658i
$783$	2.81321e6i	0.163983i
$784$	−8.44427e6	−0.490650
$785$	0	0
$786$	6.61313e6	0.381813
$787$	− 1.32624e6i	− 0.0763281i	−0.999271	−	0.0381640i	$-0.987849\pi$
$787$	− 1.32624e6i	− 0.0763281i	0.999271	−	0.0381640i	$-0.0121509\pi$
$788$	− 1.88048e7i	− 1.07883i
$789$	−2.76281e7	−1.58000
$790$	0	0
$791$	2.31893e7	1.31779
$792$	− 1.88547e6i	− 0.106809i
$793$	9.53429e6i	0.538400i
$794$	4.02523e6	0.226589
$795$	0	0
$796$	1.49529e7	0.836453
$797$	− 2.03514e7i	− 1.13488i	−0.823416	−	0.567438i	$-0.807935\pi$
$797$	− 2.03514e7i	− 1.13488i	0.823416	−	0.567438i	$-0.192065\pi$
$798$	879218.i	0.0488753i
$799$	7.98891e6	0.442711
$800$	0	0
$801$	−2.99293e6	−0.164822
$802$	− 1.21435e6i	− 0.0666667i
$803$	− 1.68395e7i	− 0.921595i
$804$	1.53605e7	0.838042
$805$	0	0
$806$	2.01285e6	0.109138
$807$	1.68669e7i	0.911699i
$808$	5.91179e6i	0.318560i
$809$	−2.72733e7	−1.46510	−0.732548	−	0.680716i	$-0.761669\pi$
$809$	−2.72733e7	−1.46510	−0.732548	+	0.680716i	$0.761669\pi$
$810$	0	0
$811$	−2.97325e7	−1.58738	−0.793688	−	0.608325i	$-0.791842\pi$
$811$	−2.97325e7	−1.58738	−0.793688	+	0.608325i	$0.791842\pi$
$812$	3.96532e6i	0.211051i
$813$	2.17589e7i	1.15454i
$814$	−4.12499e6	−0.218203
$815$	0	0
$816$	−1.49516e7	−0.786070
$817$	− 1.47674e6i	− 0.0774012i
$818$	9.27058e6i	0.484422i
$819$	1.21912e6	0.0635093
$820$	0	0
$821$	4.79065e6	0.248048	0.124024	−	0.992279i	$-0.460420\pi$
$821$	4.79065e6	0.248048	0.124024	+	0.992279i	$0.460420\pi$
$822$	3.20534e6i	0.165460i
$823$	− 9.06060e6i	− 0.466291i	−0.972442	−	0.233146i	$-0.925098\pi$
$823$	− 9.06060e6i	− 0.466291i	0.972442	−	0.233146i	$-0.0749019\pi$
$824$	1.64995e7	0.846551
$825$	0	0
$826$	−9.02893e6	−0.460454
$827$	3.26848e7i	1.66181i	0.556414	+	0.830906i	$0.312177\pi$
$827$	3.26848e7i	1.66181i	−0.556414	+	0.830906i	$0.687823\pi$
$828$	− 4.28728e6i	− 0.217323i
$829$	−1.08535e6	−0.0548506	−0.0274253	−	0.999624i	$-0.508731\pi$
$829$	−1.08535e6	−0.0548506	−0.0274253	+	0.999624i	$0.508731\pi$
$830$	0	0
$831$	−4.21897e7	−2.11936
$832$	− 1.05063e6i	− 0.0526188i
$833$	2.27165e7i	1.13430i
$834$	−8.28751e6	−0.412581
$835$	0	0
$836$	−1.26653e6	−0.0626756
$837$	1.82693e7i	0.901381i
$838$	− 4.62603e6i	− 0.227561i
$839$	−1.30918e7	−0.642089	−0.321045	−	0.947064i	$-0.604034\pi$
$839$	−1.30918e7	−0.642089	−0.321045	+	0.947064i	$0.604034\pi$
$840$	0	0
$841$	−1.98298e7	−0.966780
$842$	− 1.14809e7i	− 0.558078i
$843$	− 1.62684e7i	− 0.788453i
$844$	−3.16921e6	−0.153142
$845$	0	0
$846$	−466787.	−0.0224229
$847$	6.40174e6i	0.306612i
$848$	− 2.12851e7i	− 1.01645i
$849$	2.86429e7	1.36379
$850$	0	0
$851$	−2.04701e7	−0.968938
$852$	− 8.80603e6i	− 0.415605i
$853$	2.08437e7i	0.980848i	0.871484	+	0.490424i	$0.163158\pi$
$853$	2.08437e7i	0.980848i	−0.871484	+	0.490424i	$0.836842\pi$
$854$	2.22491e7	1.04392
$855$	0	0
$856$	1.30855e6	0.0610388
$857$	− 2.39931e7i	− 1.11592i	−0.829866	−	0.557962i	$-0.811583\pi$
$857$	− 2.39931e7i	− 1.11592i	0.829866	−	0.557962i	$-0.188417\pi$
$858$	− 2.23569e6i	− 0.103679i
$859$	−3.57929e7	−1.65506	−0.827531	−	0.561419i	$-0.810255\pi$
$859$	−3.57929e7	−1.65506	−0.827531	+	0.561419i	$0.810255\pi$
$860$	0	0
$861$	−5.17883e7	−2.38081
$862$	1.29942e7i	0.595637i
$863$	− 3.10281e6i	− 0.141817i	−0.997483	−	0.0709086i	$-0.977410\pi$
$863$	− 3.10281e6i	− 0.141817i	0.997483	−	0.0709086i	$-0.0225899\pi$
$864$	−1.86615e7	−0.850475
$865$	0	0
$866$	−3.48963e6	−0.158119
$867$	1.63095e7i	0.736873i
$868$	2.57512e7i	1.16011i
$869$	7.23259e6	0.324896
$870$	0	0
$871$	5.69541e6	0.254378
$872$	1.58832e7i	0.707369i
$873$	− 5.69821e6i	− 0.253048i
$874$	1.14644e6	0.0507658
$875$	0	0
$876$	2.17104e7	0.955889
$877$	− 3.29223e7i	− 1.44541i	−0.691155	−	0.722706i	$-0.742898\pi$
$877$	− 3.29223e7i	− 1.44541i	0.691155	−	0.722706i	$-0.257102\pi$
$878$	− 9.27376e6i	− 0.405994i
$879$	−8.55074e6	−0.373277
$880$	0	0
$881$	7.18368e6	0.311822	0.155911	−	0.987771i	$-0.450169\pi$
$881$	7.18368e6	0.311822	0.155911	+	0.987771i	$0.450169\pi$
$882$	− 1.32731e6i	− 0.0574515i
$883$	− 1.55599e7i	− 0.671593i	−0.941935	−	0.335797i	$-0.890995\pi$
$883$	− 1.55599e7i	− 0.671593i	0.941935	−	0.335797i	$-0.109005\pi$
$884$	−7.06823e6	−0.304215
$885$	0	0
$886$	−6.82749e6	−0.292198
$887$	1.70664e7i	0.728336i	0.931333	+	0.364168i	$0.118647\pi$
$887$	1.70664e7i	0.728336i	−0.931333	+	0.364168i	$0.881353\pi$
$888$	− 1.16064e7i	− 0.493929i
$889$	−2.79794e7	−1.18736
$890$	0	0
$891$	2.37832e7	1.00363
$892$	2.39716e7i	1.00875i
$893$	684303.i	0.0287157i
$894$	−4.55401e6	−0.190568
$895$	0	0
$896$	−3.35537e7	−1.39627
$897$	− 1.10945e7i	− 0.460391i
$898$	1.28316e7i	0.530993i
$899$	4.42496e6	0.182604
$900$	0	0
$901$	−5.72604e7	−2.34986
$902$	1.36078e7i	0.556893i
$903$	3.33491e7i	1.36102i
$904$	1.71442e7	0.697745
$905$	0	0
$906$	−5.41732e6	−0.219262
$907$	2.26122e7i	0.912691i	0.889803	+	0.456345i	$0.150842\pi$
$907$	2.26122e7i	0.912691i	−0.889803	+	0.456345i	$0.849158\pi$
$908$	− 3.10099e6i	− 0.124820i
$909$	1.83084e6	0.0734920
$910$	0	0
$911$	2.42960e7	0.969927	0.484963	−	0.874534i	$-0.338833\pi$
$911$	2.42960e7	0.969927	0.484963	+	0.874534i	$0.338833\pi$
$912$	− 1.28070e6i	− 0.0509871i
$913$	5.75203e6i	0.228373i
$914$	424096.	0.0167919
$915$	0	0
$916$	3.71822e7	1.46419
$917$	3.13698e7i	1.23193i
$918$	1.17020e7i	0.458303i
$919$	2.72862e7	1.06575	0.532873	−	0.846195i	$-0.321112\pi$
$919$	2.72862e7	1.06575	0.532873	+	0.846195i	$0.321112\pi$
$920$	0	0
$921$	−8.91128e6	−0.346171
$922$	1.62366e7i	0.629026i
$923$	− 3.26512e6i	− 0.126152i
$924$	2.86020e7	1.10209
$925$	0	0
$926$	1.07010e7	0.410106
$927$	− 5.10978e6i	− 0.195300i
$928$	4.51995e6i	0.172291i
$929$	2.89100e7	1.09903	0.549515	−	0.835484i	$-0.314813\pi$
$929$	2.89100e7	1.09903	0.549515	+	0.835484i	$0.314813\pi$
$930$	0	0
$931$	−1.94582e6	−0.0735746
$932$	− 3.49113e7i	− 1.31652i
$933$	8.84165e6i	0.332529i
$934$	1.49751e7	0.561696
$935$	0	0
$936$	901317.	0.0336270
$937$	1.09147e7i	0.406127i	0.979166	+	0.203063i	$0.0650897\pi$
$937$	1.09147e7i	0.406127i	−0.979166	+	0.203063i	$0.934910\pi$
$938$	− 1.32907e7i	− 0.493221i
$939$	−3.59538e7	−1.33070
$940$	0	0
$941$	4.12025e7	1.51687	0.758437	−	0.651746i	$-0.225963\pi$
$941$	4.12025e7	1.51687	0.758437	+	0.651746i	$0.225963\pi$
$942$	2.05364e7i	0.754043i
$943$	6.75282e7i	2.47290i
$944$	1.31519e7	0.480349
$945$	0	0
$946$	8.76276e6	0.318356
$947$	1.50182e6i	0.0544180i	0.999630	+	0.0272090i	$0.00866197\pi$
$947$	1.50182e6i	0.0544180i	−0.999630	+	0.0272090i	$0.991338\pi$
$948$	9.32465e6i	0.336986i
$949$	8.04983e6	0.290149
$950$	0	0
$951$	−2.56832e7	−0.920869
$952$	3.59973e7i	1.28729i
$953$	− 2.96005e6i	− 0.105576i	−0.998606	−	0.0527882i	$-0.983189\pi$
$953$	− 2.96005e6i	− 0.105576i	0.998606	−	0.0527882i	$-0.0168108\pi$
$954$	3.34569e6	0.119018
$955$	0	0
$956$	1.69484e7	0.599768
$957$	− 4.91483e6i	− 0.173472i
$958$	− 1.61379e7i	− 0.568113i
$959$	−1.52047e7	−0.533865
$960$	0	0
$961$	107057.	0.00373946
$962$	− 1.97188e6i	− 0.0686978i
$963$	− 405249.i	− 0.0140817i
$964$	2.37084e7	0.821693
$965$	0	0
$966$	−2.58900e7	−0.892667
$967$	4.68047e7i	1.60962i	0.593532	+	0.804810i	$0.297733\pi$
$967$	4.68047e7i	1.60962i	−0.593532	+	0.804810i	$0.702267\pi$
$968$	4.73291e6i	0.162345i
$969$	−3.44530e6	−0.117874
$970$	0	0
$971$	3.91779e7	1.33350	0.666750	−	0.745281i	$-0.267685\pi$
$971$	3.91779e7	1.33350	0.666750	+	0.745281i	$0.267685\pi$
$972$	8.24972e6i	0.280074i
$973$	− 3.93123e7i	− 1.33121i
$974$	1.00831e6	0.0340561
$975$	0	0
$976$	−3.24088e7	−1.08903
$977$	3.27316e6i	0.109706i	0.998494	+	0.0548530i	$0.0174690\pi$
$977$	3.27316e6i	0.109706i	−0.998494	+	0.0548530i	$0.982531\pi$
$978$	56300.7i	0.00188220i
$979$	−2.60354e7	−0.868175
$980$	0	0
$981$	4.91890e6	0.163191
$982$	− 1.62973e7i	− 0.539309i
$983$	2.58761e7i	0.854111i	0.904225	+	0.427055i	$0.140449\pi$
$983$	2.58761e7i	0.854111i	−0.904225	+	0.427055i	$0.859551\pi$
$984$	−3.82880e7	−1.26059
$985$	0	0
$986$	2.83431e6	0.0928441
$987$	− 1.54536e7i	− 0.504937i
$988$	− 605441.i	− 0.0197324i
$989$	4.34849e7	1.41367
$990$	0	0
$991$	−4.94262e6	−0.159872	−0.0799362	−	0.996800i	$-0.525472\pi$
$991$	−4.94262e6	−0.159872	−0.0799362	+	0.996800i	$0.525472\pi$
$992$	2.93531e7i	0.947053i
$993$	1.40724e7i	0.452893i
$994$	−7.61944e6	−0.244600
$995$	0	0
$996$	−7.41583e6	−0.236871
$997$	− 2.01815e6i	− 0.0643007i	−0.999483	−	0.0321503i	$-0.989764\pi$
$997$	− 2.01815e6i	− 0.0643007i	0.999483	−	0.0321503i	$-0.0102355\pi$
$998$	− 6.27337e6i	− 0.199377i
$999$	1.78974e7	0.567383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.d.274.3		6
5.2	odd	4		325.6.a.d.1.2		3
5.3	odd	4		65.6.a.b.1.2	✓	3
5.4	even	2	inner	325.6.b.d.274.4		6
15.8	even	4		585.6.a.c.1.2		3
20.3	even	4		1040.6.a.k.1.1		3
65.38	odd	4		845.6.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.b.1.2	✓	3	5.3	odd	4
325.6.a.d.1.2		3	5.2	odd	4
325.6.b.d.274.3		6	1.1	even	1	trivial
325.6.b.d.274.4		6	5.4	even	2	inner
585.6.a.c.1.2		3	15.8	even	4
845.6.a.c.1.2		3	65.38	odd	4
1040.6.a.k.1.1		3	20.3	even	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.22183i q^{2} -16.8416i q^{3} +27.0635 q^{4} -37.4193 q^{6} -177.501i q^{7} -131.229i q^{8} -40.6407 q^{9} -353.532 q^{11} -455.793i q^{12} -169.000i q^{13} -394.376 q^{14} +574.462 q^{16} -1545.40i q^{17} +90.2967i q^{18} +132.374 q^{19} -2989.40 q^{21} +785.489i q^{22} +3897.96i q^{23} -2210.11 q^{24} -375.489 q^{26} -3408.06i q^{27} -4803.78i q^{28} -825.458 q^{29} -5360.62 q^{31} -5475.69i q^{32} +5954.06i q^{33} -3433.62 q^{34} -1099.88 q^{36} +5251.49i q^{37} -294.112i q^{38} -2846.24 q^{39} +17324.0 q^{41} +6641.94i q^{42} -11155.8i q^{43} -9567.80 q^{44} +8660.61 q^{46} +5169.48i q^{47} -9674.88i q^{48} -14699.4 q^{49} -26027.1 q^{51} -4573.73i q^{52} -37052.1i q^{53} -7572.14 q^{54} -23293.2 q^{56} -2229.39i q^{57} +1834.03i q^{58} +22894.2 q^{59} -56415.9 q^{61} +11910.4i q^{62} +7213.74i q^{63} +6216.74 q^{64} +13228.9 q^{66} +33700.6i q^{67} -41823.9i q^{68} +65648.0 q^{69} +19320.2 q^{71} +5333.24i q^{72} +47632.1i q^{73} +11667.9 q^{74} +3582.49 q^{76} +62752.1i q^{77} +6323.86i q^{78} -20458.1 q^{79} -67273.0 q^{81} -38491.0i q^{82} -16270.2i q^{83} -80903.5 q^{84} -24786.3 q^{86} +13902.1i q^{87} +46393.7i q^{88} +73643.6 q^{89} -29997.6 q^{91} +105492. i q^{92} +90281.5i q^{93} +11485.7 q^{94} -92219.5 q^{96} +140209. i q^{97} +32659.7i q^{98} +14367.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 20 q^{4} + 44 q^{6} - 286 q^{9} - 1232 q^{11} + 2568 q^{14} - 3740 q^{16} + 6720 q^{19} - 7584 q^{21} - 1044 q^{24} - 676 q^{26} - 4612 q^{29} - 12760 q^{31} + 5720 q^{34} - 1372 q^{36} + 5408 q^{39}+ \cdots + 138896 q^{99}+O(q^{100}) \) 6 * q + 20 * q^4 + 44 * q^6 - 286 * q^9 - 1232 * q^11 + 2568 * q^14 - 3740 * q^16 + 6720 * q^19 - 7584 * q^21 - 1044 * q^24 - 676 * q^26 - 4612 * q^29 - 12760 * q^31 + 5720 * q^34 - 1372 * q^36 + 5408 * q^39 + 16572 * q^41 - 15380 * q^44 + 28372 * q^46 - 32342 * q^49 - 86656 * q^51 - 60008 * q^54 + 31560 * q^56 - 9504 * q^59 - 173788 * q^61 - 67236 * q^64 - 42376 * q^66 + 17776 * q^69 - 246504 * q^71 - 187776 * q^74 - 51820 * q^76 + 221392 * q^79 - 304154 * q^81 - 159888 * q^84 - 237932 * q^86 + 224420 * q^89 - 70304 * q^91 - 255032 * q^94 - 212876 * q^96 + 138896 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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