LMFDB - Embedded newform 325.6.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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([0, 0]))

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

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

chi := DirichletCharacter("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 $ x^11 - 5x^10 - 257x^9 + 1165x^8 + 22234x^7 - 90282x^6 - 751180x^5 + 2564400x^4 + 9571584x^3 - 23191072x^2 - 33154496x + 44115200
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.96479$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.96479 q^{2} -29.7675 q^{3} -28.1396 q^{4} +58.4868 q^{6} +176.972 q^{7} +118.162 q^{8} +643.103 q^{9} -62.7908 q^{11} +837.645 q^{12} +169.000 q^{13} -347.712 q^{14} +668.304 q^{16} -1825.19 q^{17} -1263.56 q^{18} +2509.78 q^{19} -5268.00 q^{21} +123.371 q^{22} +137.706 q^{23} -3517.37 q^{24} -332.049 q^{26} -11910.0 q^{27} -4979.91 q^{28} +5978.98 q^{29} -5203.47 q^{31} -5094.25 q^{32} +1869.12 q^{33} +3586.12 q^{34} -18096.7 q^{36} -4275.33 q^{37} -4931.19 q^{38} -5030.70 q^{39} +16780.1 q^{41} +10350.5 q^{42} -21508.6 q^{43} +1766.91 q^{44} -270.563 q^{46} +16741.5 q^{47} -19893.7 q^{48} +14511.9 q^{49} +54331.4 q^{51} -4755.59 q^{52} +2749.61 q^{53} +23400.7 q^{54} +20911.2 q^{56} -74709.8 q^{57} -11747.4 q^{58} +16631.5 q^{59} -25005.7 q^{61} +10223.7 q^{62} +113811. q^{63} -11376.6 q^{64} -3672.44 q^{66} +29425.1 q^{67} +51360.2 q^{68} -4099.16 q^{69} -26909.1 q^{71} +75990.1 q^{72} +33664.0 q^{73} +8400.13 q^{74} -70624.2 q^{76} -11112.2 q^{77} +9884.27 q^{78} -13528.9 q^{79} +198258. q^{81} -32969.4 q^{82} +790.985 q^{83} +148239. q^{84} +42259.9 q^{86} -177979. q^{87} -7419.47 q^{88} -72223.8 q^{89} +29908.2 q^{91} -3874.99 q^{92} +154894. q^{93} -32893.6 q^{94} +151643. q^{96} -5458.83 q^{97} -28512.9 q^{98} -40380.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 5 q^{2} + 11 q^{3} + 187 q^{4} + 351 q^{6} + 208 q^{7} + 165 q^{8} + 1372 q^{9} + 1276 q^{11} + 1533 q^{12} + 1859 q^{13} + 578 q^{14} + 5707 q^{16} + 2218 q^{17} - 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) $ 11 * q + 5 * q^2 + 11 * q^3 + 187 * q^4 + 351 * q^6 + 208 * q^7 + 165 * q^8 + 1372 * q^9 + 1276 * q^11 + 1533 * q^12 + 1859 * q^13 + 578 * q^14 + 5707 * q^16 + 2218 * q^17 - 6776 * q^18 + 3520 * q^19 + 1706 * q^21 - 6785 * q^22 - 1215 * q^23 + 7723 * q^24 + 845 * q^26 + 161 * q^27 + 12746 * q^28 + 18425 * q^29 + 9068 * q^31 + 21165 * q^32 + 19748 * q^33 - 14455 * q^34 + 37684 * q^36 - 2280 * q^37 - 23783 * q^38 + 1859 * q^39 - 10010 * q^41 - 44184 * q^42 + 6763 * q^43 + 60651 * q^44 - 9858 * q^46 + 4110 * q^47 + 63289 * q^48 + 73313 * q^49 + 36606 * q^51 + 31603 * q^52 + 15633 * q^53 + 48013 * q^54 - 1262 * q^56 - 205974 * q^57 + 77652 * q^58 + 131642 * q^59 - 43365 * q^61 - 115940 * q^62 + 218466 * q^63 + 180071 * q^64 - 58607 * q^66 - 100904 * q^67 + 371277 * q^68 + 59345 * q^69 + 68420 * q^71 - 210708 * q^72 + 124436 * q^73 + 371396 * q^74 + 22517 * q^76 - 150978 * q^77 + 59319 * q^78 + 52239 * q^79 + 107507 * q^81 - 351137 * q^82 + 211538 * q^83 + 852716 * q^84 + 202246 * q^86 - 28273 * q^87 - 28107 * q^88 + 327712 * q^89 + 35152 * q^91 - 413252 * q^92 + 243338 * q^93 + 649974 * q^94 + 899663 * q^96 - 110796 * q^97 + 27143 * q^98 + 426698 * q^99

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 $

$2$	−1.96479	−0.347329	−0.173664	−	0.984805i	$-0.555561\pi$
$2$	−1.96479	−0.347329	−0.173664	+	0.984805i	$0.555561\pi$
$3$	−29.7675	−1.90958	−0.954792	−	0.297274i	$-0.903923\pi$
$3$	−29.7675	−1.90958	−0.954792	+	0.297274i	$0.903923\pi$
$4$	−28.1396	−0.879363
$5$	0	0
$5$	0	0
$6$	58.4868	0.663254
$7$	176.972	1.36508	0.682540	−	0.730848i	$-0.260875\pi$
$7$	176.972	1.36508	0.682540	+	0.730848i	$0.260875\pi$
$8$	118.162	0.652757
$9$	643.103	2.64651
$10$	0	0
$11$	−62.7908	−0.156464	−0.0782320	−	0.996935i	$-0.524927\pi$
$11$	−62.7908	−0.156464	−0.0782320	+	0.996935i	$0.524927\pi$
$12$	837.645	1.67922
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−347.712	−0.474132
$15$	0	0
$16$	668.304	0.652641
$17$	−1825.19	−1.53175	−0.765873	−	0.642992i	$-0.777693\pi$
$17$	−1825.19	−1.53175	−0.765873	+	0.642992i	$0.777693\pi$
$18$	−1263.56	−0.919211
$19$	2509.78	1.59497	0.797483	−	0.603342i	$-0.206165\pi$
$19$	2509.78	1.59497	0.797483	+	0.603342i	$0.206165\pi$
$20$	0	0
$21$	−5268.00	−2.60674
$22$	123.371	0.0543445
$23$	137.706	0.0542791	0.0271396	−	0.999632i	$-0.491360\pi$
$23$	137.706	0.0542791	0.0271396	+	0.999632i	$0.491360\pi$
$24$	−3517.37	−1.24649
$25$	0	0
$26$	−332.049	−0.0963317
$27$	−11910.0	−3.14416
$28$	−4979.91	−1.20040
$29$	5978.98	1.32018	0.660088	−	0.751188i	$-0.270519\pi$
$29$	5978.98	1.32018	0.660088	+	0.751188i	$0.270519\pi$
$30$	0	0
$31$	−5203.47	−0.972499	−0.486249	−	0.873820i	$-0.661635\pi$
$31$	−5203.47	−0.972499	−0.486249	+	0.873820i	$0.661635\pi$
$32$	−5094.25	−0.879438
$33$	1869.12	0.298781
$34$	3586.12	0.532020
$35$	0	0
$36$	−18096.7	−2.32724
$37$	−4275.33	−0.513412	−0.256706	−	0.966490i	$-0.582637\pi$
$37$	−4275.33	−0.513412	−0.256706	+	0.966490i	$0.582637\pi$
$38$	−4931.19	−0.553978
$39$	−5030.70	−0.529623
$40$	0	0
$41$	16780.1	1.55896	0.779481	−	0.626426i	$-0.215483\pi$
$41$	16780.1	1.55896	0.779481	+	0.626426i	$0.215483\pi$
$42$	10350.5	0.905395
$43$	−21508.6	−1.77395	−0.886974	−	0.461819i	$-0.847197\pi$
$43$	−21508.6	−1.77395	−0.886974	+	0.461819i	$0.847197\pi$
$44$	1766.91	0.137589
$45$	0	0
$46$	−270.563	−0.0188527
$47$	16741.5	1.10548	0.552739	−	0.833354i	$-0.313583\pi$
$47$	16741.5	1.10548	0.552739	+	0.833354i	$0.313583\pi$
$48$	−19893.7	−1.24627
$49$	14511.9	0.863445
$50$	0	0
$51$	54331.4	2.92500
$52$	−4755.59	−0.243891
$53$	2749.61	0.134457	0.0672283	−	0.997738i	$-0.478584\pi$
$53$	2749.61	0.134457	0.0672283	+	0.997738i	$0.478584\pi$
$54$	23400.7	1.09206
$55$	0	0
$56$	20911.2	0.891066
$57$	−74709.8	−3.04572
$58$	−11747.4	−0.458536
$59$	16631.5	0.622015	0.311007	−	0.950407i	$-0.399334\pi$
$59$	16631.5	0.622015	0.311007	+	0.950407i	$0.399334\pi$
$60$	0	0
$61$	−25005.7	−0.860429	−0.430214	−	0.902727i	$-0.641562\pi$
$61$	−25005.7	−0.860429	−0.430214	+	0.902727i	$0.641562\pi$
$62$	10223.7	0.337777
$63$	113811.	3.61270
$64$	−11376.6	−0.347187
$65$	0	0
$66$	−3672.44	−0.103775
$67$	29425.1	0.800812	0.400406	−	0.916338i	$-0.368869\pi$
$67$	29425.1	0.800812	0.400406	+	0.916338i	$0.368869\pi$
$68$	51360.2	1.34696
$69$	−4099.16	−0.103651
$70$	0	0
$71$	−26909.1	−0.633509	−0.316754	−	0.948508i	$-0.602593\pi$
$71$	−26909.1	−0.633509	−0.316754	+	0.948508i	$0.602593\pi$
$72$	75990.1	1.72753
$73$	33664.0	0.739365	0.369683	−	0.929158i	$-0.379466\pi$
$73$	33664.0	0.739365	0.369683	+	0.929158i	$0.379466\pi$
$74$	8400.13	0.178323
$75$	0	0
$76$	−70624.2	−1.40255
$77$	−11112.2	−0.213586
$78$	9884.27	0.183954
$79$	−13528.9	−0.243891	−0.121945	−	0.992537i	$-0.538913\pi$
$79$	−13528.9	−0.243891	−0.121945	+	0.992537i	$0.538913\pi$
$80$	0	0
$81$	198258.	3.35752
$82$	−32969.4	−0.541473
$83$	790.985	0.0126030	0.00630148	−	0.999980i	$-0.497994\pi$
$83$	790.985	0.0126030	0.00630148	+	0.999980i	$0.497994\pi$
$84$	148239.	2.29227
$85$	0	0
$86$	42259.9	0.616144
$87$	−177979.	−2.52099
$88$	−7419.47	−0.102133
$89$	−72223.8	−0.966508	−0.483254	−	0.875480i	$-0.660545\pi$
$89$	−72223.8	−0.966508	−0.483254	+	0.875480i	$0.660545\pi$
$90$	0	0
$91$	29908.2	0.378605
$92$	−3874.99	−0.0477310
$93$	154894.	1.85707
$94$	−32893.6	−0.383965
$95$	0	0
$96$	151643.	1.67936
$97$	−5458.83	−0.0589075	−0.0294538	−	0.999566i	$-0.509377\pi$
$97$	−5458.83	−0.0589075	−0.0294538	+	0.999566i	$0.509377\pi$
$98$	−28512.9	−0.299899
$99$	−40380.9	−0.414084
$100$	0	0
$101$	−87461.4	−0.853125	−0.426563	−	0.904458i	$-0.640276\pi$
$101$	−87461.4	−0.853125	−0.426563	+	0.904458i	$0.640276\pi$
$102$	−106750.	−1.01594
$103$	−75287.7	−0.699248	−0.349624	−	0.936890i	$-0.613691\pi$
$103$	−75287.7	−0.699248	−0.349624	+	0.936890i	$0.613691\pi$
$104$	19969.3	0.181042
$105$	0	0
$106$	−5402.41	−0.0467007
$107$	77781.0	0.656771	0.328386	−	0.944544i	$-0.393495\pi$
$107$	77781.0	0.656771	0.328386	+	0.944544i	$0.393495\pi$
$108$	335144.	2.76485
$109$	175666.	1.41619	0.708096	−	0.706116i	$-0.249554\pi$
$109$	175666.	1.41619	0.708096	+	0.706116i	$0.249554\pi$
$110$	0	0
$111$	127266.	0.980403
$112$	118271.	0.890908
$113$	−2350.11	−0.0173138	−0.00865691	−	0.999963i	$-0.502756\pi$
$113$	−2350.11	−0.0173138	−0.00865691	+	0.999963i	$0.502756\pi$
$114$	146789.	1.05787
$115$	0	0
$116$	−168246.	−1.16091
$117$	108684.	0.734011
$118$	−32677.4	−0.216044
$119$	−323007.	−2.09096
$120$	0	0
$121$	−157108.	−0.975519
$122$	49131.0	0.298852
$123$	−499502.	−2.97697
$124$	146424.	0.855179
$125$	0	0
$126$	−223614.	−1.25480
$127$	−266534.	−1.46637	−0.733183	−	0.680031i	$-0.761966\pi$
$127$	−266534.	−1.46637	−0.733183	+	0.680031i	$0.761966\pi$
$128$	185369.	1.00003
$129$	640257.	3.38750
$130$	0	0
$131$	118703.	0.604341	0.302170	−	0.953254i	$-0.402289\pi$
$131$	118703.	0.604341	0.302170	+	0.953254i	$0.402289\pi$
$132$	−52596.4	−0.262737
$133$	444159.	2.17726
$134$	−57814.1	−0.278145
$135$	0	0
$136$	−215668.	−0.999858
$137$	−163850.	−0.745838	−0.372919	−	0.927864i	$-0.621643\pi$
$137$	−163850.	−0.745838	−0.372919	+	0.927864i	$0.621643\pi$
$138$	8053.98	0.0360008
$139$	218277.	0.958231	0.479115	−	0.877752i	$-0.340958\pi$
$139$	218277.	0.958231	0.479115	+	0.877752i	$0.340958\pi$
$140$	0	0
$141$	−498353.	−2.11100
$142$	52870.6	0.220036
$143$	−10611.6	−0.0433953
$144$	429788.	1.72722
$145$	0	0
$146$	−66142.7	−0.256803
$147$	−431983.	−1.64882
$148$	120306.	0.451475
$149$	−79310.9	−0.292663	−0.146331	−	0.989236i	$-0.546747\pi$
$149$	−79310.9	−0.292663	−0.146331	+	0.989236i	$0.546747\pi$
$150$	0	0
$151$	368505.	1.31523	0.657615	−	0.753355i	$-0.271566\pi$
$151$	368505.	1.31523	0.657615	+	0.753355i	$0.271566\pi$
$152$	296560.	1.04113
$153$	−1.17379e6	−4.05378
$154$	21833.1	0.0741846
$155$	0	0
$156$	141562.	0.465731
$157$	30957.2	0.100233	0.0501166	−	0.998743i	$-0.484041\pi$
$157$	30957.2	0.100233	0.0501166	+	0.998743i	$0.484041\pi$
$158$	26581.5	0.0847103
$159$	−81849.1	−0.256756
$160$	0	0
$161$	24370.0	0.0740954
$162$	−389535.	−1.16616
$163$	−204546.	−0.603006	−0.301503	−	0.953465i	$-0.597488\pi$
$163$	−204546.	−0.603006	−0.301503	+	0.953465i	$0.597488\pi$
$164$	−472186.	−1.37089
$165$	0	0
$166$	−1554.12	−0.00437738
$167$	158449.	0.439641	0.219821	−	0.975540i	$-0.429453\pi$
$167$	158449.	0.439641	0.219821	+	0.975540i	$0.429453\pi$
$168$	−622475.	−1.70157
$169$	28561.0	0.0769231
$170$	0	0
$171$	1.61404e6	4.22110
$172$	605243.	1.55994
$173$	230663.	0.585952	0.292976	−	0.956120i	$-0.405354\pi$
$173$	230663.	0.585952	0.292976	+	0.956120i	$0.405354\pi$
$174$	349692.	0.875613
$175$	0	0
$176$	−41963.4	−0.102115
$177$	−495077.	−1.18779
$178$	141905.	0.335696
$179$	791891.	1.84728	0.923640	−	0.383260i	$-0.125199\pi$
$179$	791891.	1.84728	0.923640	+	0.383260i	$0.125199\pi$
$180$	0	0
$181$	−272694.	−0.618699	−0.309350	−	0.950948i	$-0.600111\pi$
$181$	−272694.	−0.618699	−0.309350	+	0.950948i	$0.600111\pi$
$182$	−58763.3	−0.131501
$183$	744358.	1.64306
$184$	16271.5	0.0354311
$185$	0	0
$186$	−304335.	−0.645014
$187$	114605.	0.239663
$188$	−471100.	−0.972116
$189$	−2.10774e6	−4.29203
$190$	0	0
$191$	603069.	1.19614	0.598072	−	0.801442i	$-0.295933\pi$
$191$	603069.	1.19614	0.598072	+	0.801442i	$0.295933\pi$
$192$	338653.	0.662982
$193$	−59043.0	−0.114097	−0.0570486	−	0.998371i	$-0.518169\pi$
$193$	−59043.0	−0.114097	−0.0570486	+	0.998371i	$0.518169\pi$
$194$	10725.5	0.0204603
$195$	0	0
$196$	−408360.	−0.759281
$197$	815964.	1.49798	0.748989	−	0.662582i	$-0.230539\pi$
$197$	815964.	1.49798	0.748989	+	0.662582i	$0.230539\pi$
$198$	79340.1	0.143823
$199$	490593.	0.878190	0.439095	−	0.898441i	$-0.355299\pi$
$199$	490593.	0.878190	0.439095	+	0.898441i	$0.355299\pi$
$200$	0	0
$201$	−875911.	−1.52922
$202$	171843.	0.296315
$203$	1.05811e6	1.80215
$204$	−1.52886e6	−2.57213
$205$	0	0
$206$	147925.	0.242869
$207$	88559.0	0.143650
$208$	112943.	0.181010
$209$	−157591.	−0.249555
$210$	0	0
$211$	593914.	0.918370	0.459185	−	0.888341i	$-0.348142\pi$
$211$	593914.	0.918370	0.459185	+	0.888341i	$0.348142\pi$
$212$	−77373.0	−0.118236
$213$	801015.	1.20974
$214$	−152823.	−0.228116
$215$	0	0
$216$	−1.40731e6	−2.05237
$217$	−920866.	−1.32754
$218$	−345147.	−0.491884
$219$	−1.00209e6	−1.41188
$220$	0	0
$221$	−308458.	−0.424830
$222$	−250051.	−0.340522
$223$	−893543.	−1.20324	−0.601621	−	0.798781i	$-0.705478\pi$
$223$	−893543.	−1.20324	−0.601621	+	0.798781i	$0.705478\pi$
$224$	−901537.	−1.20050
$225$	0	0
$226$	4617.48	0.00601359
$227$	1.10400e6	1.42201	0.711006	−	0.703186i	$-0.248240\pi$
$227$	1.10400e6	1.42201	0.711006	+	0.703186i	$0.248240\pi$
$228$	2.10230e6	2.67829
$229$	−662093.	−0.834315	−0.417158	−	0.908834i	$-0.636974\pi$
$229$	−662093.	−0.834315	−0.417158	+	0.908834i	$0.636974\pi$
$230$	0	0
$231$	330782.	0.407860
$232$	706486.	0.861755
$233$	828333.	0.999574	0.499787	−	0.866148i	$-0.333412\pi$
$233$	828333.	0.999574	0.499787	+	0.866148i	$0.333412\pi$
$234$	−213542.	−0.254943
$235$	0	0
$236$	−468003.	−0.546977
$237$	402722.	0.465730
$238$	634641.	0.726250
$239$	359216.	0.406781	0.203391	−	0.979098i	$-0.434804\pi$
$239$	359216.	0.406781	0.203391	+	0.979098i	$0.434804\pi$
$240$	0	0
$241$	−928812.	−1.03011	−0.515057	−	0.857156i	$-0.672229\pi$
$241$	−928812.	−1.03011	−0.515057	+	0.857156i	$0.672229\pi$
$242$	308685.	0.338826
$243$	−3.00750e6	−3.26731
$244$	703651.	0.756629
$245$	0	0
$246$	981416.	1.03399
$247$	424152.	0.442364
$248$	−614851.	−0.634805
$249$	−23545.6	−0.0240664
$250$	0	0
$251$	−102334.	−0.102526	−0.0512631	−	0.998685i	$-0.516325\pi$
$251$	−102334.	−0.102526	−0.0512631	+	0.998685i	$0.516325\pi$
$252$	−3.20259e6	−3.17688
$253$	−8646.66	−0.00849273
$254$	523682.	0.509312
$255$	0	0
$256$	−158.767	−0.000151412 0
$257$	1.17461e6	1.10933	0.554664	−	0.832075i	$-0.312847\pi$
$257$	1.17461e6	1.10933	0.554664	+	0.832075i	$0.312847\pi$
$258$	−1.25797e6	−1.17658
$259$	−756612.	−0.700848
$260$	0	0
$261$	3.84510e6	3.49387
$262$	−233226.	−0.209905
$263$	99506.0	0.0887074	0.0443537	−	0.999016i	$-0.485877\pi$
$263$	99506.0	0.0887074	0.0443537	+	0.999016i	$0.485877\pi$
$264$	220859.	0.195032
$265$	0	0
$266$	−872679.	−0.756224
$267$	2.14992e6	1.84563
$268$	−828010.	−0.704204
$269$	−558985.	−0.470999	−0.235499	−	0.971874i	$-0.575673\pi$
$269$	−558985.	−0.470999	−0.235499	+	0.971874i	$0.575673\pi$
$270$	0	0
$271$	−503763.	−0.416680	−0.208340	−	0.978056i	$-0.566806\pi$
$271$	−503763.	−0.416680	−0.208340	+	0.978056i	$0.566806\pi$
$272$	−1.21978e6	−0.999680
$273$	−890291.	−0.722979
$274$	321930.	0.259051
$275$	0	0
$276$	115349.	0.0911464
$277$	−1.08351e6	−0.848466	−0.424233	−	0.905553i	$-0.639456\pi$
$277$	−1.08351e6	−0.848466	−0.424233	+	0.905553i	$0.639456\pi$
$278$	−428867.	−0.332821
$279$	−3.34637e6	−2.57373
$280$	0	0
$281$	−1.46593e6	−1.10751	−0.553756	−	0.832679i	$-0.686806\pi$
$281$	−1.46593e6	−1.10751	−0.553756	+	0.832679i	$0.686806\pi$
$282$	979158.	0.733213
$283$	2.11337e6	1.56859	0.784294	−	0.620390i	$-0.213026\pi$
$283$	2.11337e6	1.56859	0.784294	+	0.620390i	$0.213026\pi$
$284$	757210.	0.557084
$285$	0	0
$286$	20849.7	0.0150724
$287$	2.96960e6	2.12811
$288$	−3.27613e6	−2.32744
$289$	1.91147e6	1.34624
$290$	0	0
$291$	162496.	0.112489
$292$	−947293.	−0.650170
$293$	−2.71115e6	−1.84495	−0.922475	−	0.386057i	$-0.873837\pi$
$293$	−2.71115e6	−1.84495	−0.922475	+	0.386057i	$0.873837\pi$
$294$	848756.	0.572683
$295$	0	0
$296$	−505180.	−0.335133
$297$	747842.	0.491947
$298$	155829.	0.101650
$299$	23272.3	0.0150543
$300$	0	0
$301$	−3.80641e6	−2.42158
$302$	−724035.	−0.456817
$303$	2.60350e6	1.62911
$304$	1.67730e6	1.04094
$305$	0	0
$306$	2.30624e6	1.40800
$307$	1.55774e6	0.943297	0.471648	−	0.881787i	$-0.343659\pi$
$307$	1.55774e6	0.943297	0.471648	+	0.881787i	$0.343659\pi$
$308$	312693.	0.187820
$309$	2.24113e6	1.33527
$310$	0	0
$311$	−2.57327e6	−1.50863	−0.754316	−	0.656511i	$-0.772032\pi$
$311$	−2.57327e6	−1.50863	−0.754316	+	0.656511i	$0.772032\pi$
$312$	−594436.	−0.345715
$313$	−563667.	−0.325208	−0.162604	−	0.986691i	$-0.551989\pi$
$313$	−563667.	−0.325208	−0.162604	+	0.986691i	$0.551989\pi$
$314$	−60824.3	−0.0348139
$315$	0	0
$316$	380698.	0.214468
$317$	−1.49727e6	−0.836858	−0.418429	−	0.908250i	$-0.637419\pi$
$317$	−1.49727e6	−0.836858	−0.418429	+	0.908250i	$0.637419\pi$
$318$	160816.	0.0891789
$319$	−375425.	−0.206560
$320$	0	0
$321$	−2.31534e6	−1.25416
$322$	−47881.9	−0.0257355
$323$	−4.58083e6	−2.44308
$324$	−5.57890e6	−2.95248
$325$	0	0
$326$	401889.	0.209441
$327$	−5.22914e6	−2.70434
$328$	1.98277e6	1.01762
$329$	2.96277e6	1.50907
$330$	0	0
$331$	2.48241e6	1.24538	0.622692	−	0.782467i	$-0.286039\pi$
$331$	2.48241e6	1.24538	0.622692	+	0.782467i	$0.286039\pi$
$332$	−22258.0	−0.0110826
$333$	−2.74948e6	−1.35875
$334$	−311319.	−0.152700
$335$	0	0
$336$	−3.52063e6	−1.70126
$337$	2.17057e6	1.04112	0.520558	−	0.853827i	$-0.325724\pi$
$337$	2.17057e6	1.04112	0.520558	+	0.853827i	$0.325724\pi$
$338$	−56116.4	−0.0267176
$339$	69957.0	0.0330622
$340$	0	0
$341$	326730.	0.152161
$342$	−3.17126e6	−1.46611
$343$	−406164.	−0.186409
$344$	−2.54149e6	−1.15796
$345$	0	0
$346$	−453204.	−0.203518
$347$	3.93089e6	1.75254	0.876269	−	0.481821i	$-0.160025\pi$
$347$	3.93089e6	1.75254	0.876269	+	0.481821i	$0.160025\pi$
$348$	5.00826e6	2.21686
$349$	378915.	0.166524	0.0832622	−	0.996528i	$-0.473466\pi$
$349$	378915.	0.166524	0.0832622	+	0.996528i	$0.473466\pi$
$350$	0	0
$351$	−2.01280e6	−0.872032
$352$	319872.	0.137600
$353$	−1.13031e6	−0.482792	−0.241396	−	0.970427i	$-0.577605\pi$
$353$	−1.13031e6	−0.482792	−0.241396	+	0.970427i	$0.577605\pi$
$354$	972722.	0.412554
$355$	0	0
$356$	2.03235e6	0.849911
$357$	9.61511e6	3.99286
$358$	−1.55590e6	−0.641614
$359$	3.35579e6	1.37423	0.687114	−	0.726550i	$-0.258877\pi$
$359$	3.35579e6	1.37423	0.687114	+	0.726550i	$0.258877\pi$
$360$	0	0
$361$	3.82289e6	1.54391
$362$	535787.	0.214892
$363$	4.67672e6	1.86284
$364$	−841605.	−0.332931
$365$	0	0
$366$	−1.46251e6	−0.570683
$367$	1.99093e6	0.771597	0.385799	−	0.922583i	$-0.373926\pi$
$367$	1.99093e6	0.771597	0.385799	+	0.922583i	$0.373926\pi$
$368$	92029.4	0.0354248
$369$	1.07913e7	4.12581
$370$	0	0
$371$	486603.	0.183544
$372$	−4.35866e6	−1.63304
$373$	1.39554e6	0.519362	0.259681	−	0.965694i	$-0.416383\pi$
$373$	1.39554e6	0.519362	0.259681	+	0.965694i	$0.416383\pi$
$374$	−225175.	−0.0832419
$375$	0	0
$376$	1.97821e6	0.721609
$377$	1.01045e6	0.366151
$378$	4.14126e6	1.49075
$379$	3.81965e6	1.36592	0.682960	−	0.730455i	$-0.260692\pi$
$379$	3.81965e6	1.36592	0.682960	+	0.730455i	$0.260692\pi$
$380$	0	0
$381$	7.93403e6	2.80015
$382$	−1.18490e6	−0.415456
$383$	4.30052e6	1.49804	0.749020	−	0.662547i	$-0.230525\pi$
$383$	4.30052e6	1.49804	0.749020	+	0.662547i	$0.230525\pi$
$384$	−5.51796e6	−1.90963
$385$	0	0
$386$	116007.	0.0396293
$387$	−1.38322e7	−4.69478
$388$	153609.	0.0518011
$389$	5.63218e6	1.88713	0.943567	−	0.331182i	$-0.107447\pi$
$389$	5.63218e6	1.88713	0.943567	+	0.331182i	$0.107447\pi$
$390$	0	0
$391$	−251340.	−0.0831418
$392$	1.71475e6	0.563620
$393$	−3.53348e6	−1.15404
$394$	−1.60320e6	−0.520291
$395$	0	0
$396$	1.13630e6	0.364130
$397$	1.02692e6	0.327011	0.163505	−	0.986542i	$-0.447720\pi$
$397$	1.02692e6	0.327011	0.163505	+	0.986542i	$0.447720\pi$
$398$	−963911.	−0.305021
$399$	−1.32215e7	−4.15766
$400$	0	0
$401$	−1.07636e6	−0.334269	−0.167134	−	0.985934i	$-0.553451\pi$
$401$	−1.07636e6	−0.334269	−0.167134	+	0.985934i	$0.553451\pi$
$402$	1.72098e6	0.531142
$403$	−879387.	−0.269723
$404$	2.46113e6	0.750206
$405$	0	0
$406$	−2.07896e6	−0.625938
$407$	268452.	0.0803304
$408$	6.41989e6	1.90931
$409$	−4.48956e6	−1.32708	−0.663538	−	0.748143i	$-0.730946\pi$
$409$	−4.48956e6	−1.32708	−0.663538	+	0.748143i	$0.730946\pi$
$410$	0	0
$411$	4.87739e6	1.42424
$412$	2.11857e6	0.614892
$413$	2.94330e6	0.849100
$414$	−174000.	−0.0498939
$415$	0	0
$416$	−860928.	−0.243912
$417$	−6.49754e6	−1.82982
$418$	309633.	0.0866776
$419$	−2.90095e6	−0.807245	−0.403622	−	0.914926i	$-0.632249\pi$
$419$	−2.90095e6	−0.807245	−0.403622	+	0.914926i	$0.632249\pi$
$420$	0	0
$421$	−5.69840e6	−1.56692	−0.783461	−	0.621441i	$-0.786548\pi$
$421$	−5.69840e6	−1.56692	−0.783461	+	0.621441i	$0.786548\pi$
$422$	−1.16692e6	−0.318977
$423$	1.07665e7	2.92566
$424$	324899.	0.0877675
$425$	0	0
$426$	−1.57383e6	−0.420177
$427$	−4.42530e6	−1.17455
$428$	−2.18873e6	−0.577540
$429$	315882.	0.0828670
$430$	0	0
$431$	7.25477e6	1.88118	0.940590	−	0.339545i	$-0.110273\pi$
$431$	7.25477e6	1.88118	0.940590	+	0.339545i	$0.110273\pi$
$432$	−7.95954e6	−2.05201
$433$	2.92650e6	0.750115	0.375058	−	0.927001i	$-0.377623\pi$
$433$	2.92650e6	0.750115	0.375058	+	0.927001i	$0.377623\pi$
$434$	1.80931e6	0.461093
$435$	0	0
$436$	−4.94318e6	−1.24535
$437$	345611.	0.0865733
$438$	1.96890e6	0.490387
$439$	2.15422e6	0.533492	0.266746	−	0.963767i	$-0.414051\pi$
$439$	2.15422e6	0.533492	0.266746	+	0.963767i	$0.414051\pi$
$440$	0	0
$441$	9.33266e6	2.28512
$442$	606054.	0.147556
$443$	7.40975e6	1.79388	0.896942	−	0.442148i	$-0.145783\pi$
$443$	7.40975e6	1.79388	0.896942	+	0.442148i	$0.145783\pi$
$444$	−3.58121e6	−0.862130
$445$	0	0
$446$	1.75562e6	0.417921
$447$	2.36088e6	0.558864
$448$	−2.01334e6	−0.473938
$449$	−1.90284e6	−0.445438	−0.222719	−	0.974883i	$-0.571493\pi$
$449$	−1.90284e6	−0.445438	−0.222719	+	0.974883i	$0.571493\pi$
$450$	0	0
$451$	−1.05364e6	−0.243921
$452$	66131.3	0.0152251
$453$	−1.09695e7	−2.51154
$454$	−2.16912e6	−0.493906
$455$	0	0
$456$	−8.82783e6	−1.98812
$457$	2.79693e6	0.626457	0.313229	−	0.949678i	$-0.398589\pi$
$457$	2.79693e6	0.626457	0.313229	+	0.949678i	$0.398589\pi$
$458$	1.30087e6	0.289782
$459$	2.17381e7	4.81605
$460$	0	0
$461$	4.18307e6	0.916733	0.458366	−	0.888763i	$-0.348435\pi$
$461$	4.18307e6	0.916733	0.458366	+	0.888763i	$0.348435\pi$
$462$	−649917.	−0.141662
$463$	−2.51126e6	−0.544427	−0.272213	−	0.962237i	$-0.587756\pi$
$463$	−2.51126e6	−0.544427	−0.272213	+	0.962237i	$0.587756\pi$
$464$	3.99578e6	0.861602
$465$	0	0
$466$	−1.62750e6	−0.347181
$467$	2.17175e6	0.460806	0.230403	−	0.973095i	$-0.425996\pi$
$467$	2.17175e6	0.460806	0.230403	+	0.973095i	$0.425996\pi$
$468$	−3.05833e6	−0.645462
$469$	5.20740e6	1.09317
$470$	0	0
$471$	−921517.	−0.191404
$472$	1.96520e6	0.406025
$473$	1.35054e6	0.277559
$474$	−791263.	−0.161761
$475$	0	0
$476$	9.08930e6	1.83871
$477$	1.76828e6	0.355841
$478$	−705784.	−0.141287
$479$	−9.46791e6	−1.88545	−0.942726	−	0.333568i	$-0.891747\pi$
$479$	−9.46791e6	−1.88545	−0.942726	+	0.333568i	$0.891747\pi$
$480$	0	0
$481$	−722531.	−0.142395
$482$	1.82492e6	0.357788
$483$	−725434.	−0.141491
$484$	4.42097e6	0.857835
$485$	0	0
$486$	5.90911e6	1.13483
$487$	4.87785e6	0.931979	0.465989	−	0.884790i	$-0.345698\pi$
$487$	4.87785e6	0.931979	0.465989	+	0.884790i	$0.345698\pi$
$488$	−2.95472e6	−0.561651
$489$	6.08881e6	1.15149
$490$	0	0
$491$	−3.86456e6	−0.723429	−0.361715	−	0.932289i	$-0.617808\pi$
$491$	−3.86456e6	−0.723429	−0.361715	+	0.932289i	$0.617808\pi$
$492$	1.40558e7	2.61784
$493$	−1.09128e7	−2.02217
$494$	−833370.	−0.153646
$495$	0	0
$496$	−3.47750e6	−0.634693
$497$	−4.76214e6	−0.864791
$498$	46262.2	0.00835897
$499$	5.75229e6	1.03416	0.517082	−	0.855936i	$-0.327018\pi$
$499$	5.75229e6	1.03416	0.517082	+	0.855936i	$0.327018\pi$
$500$	0	0
$501$	−4.71663e6	−0.839532
$502$	201064.	0.0356103
$503$	−5.93696e6	−1.04627	−0.523136	−	0.852249i	$-0.675238\pi$
$503$	−5.93696e6	−1.04627	−0.523136	+	0.852249i	$0.675238\pi$
$504$	1.34481e7	2.35822
$505$	0	0
$506$	16988.9	0.00294977
$507$	−850189.	−0.146891
$508$	7.50015e6	1.28947
$509$	−563930.	−0.0964785	−0.0482393	−	0.998836i	$-0.515361\pi$
$509$	−563930.	−0.0964785	−0.0482393	+	0.998836i	$0.515361\pi$
$510$	0	0
$511$	5.95758e6	1.00929
$512$	−5.93149e6	−0.999974
$513$	−2.98916e7	−5.01482
$514$	−2.30786e6	−0.385302
$515$	0	0
$516$	−1.80166e7	−2.97884
$517$	−1.05121e6	−0.172968
$518$	1.48658e6	0.243425
$519$	−6.86625e6	−1.11892
$520$	0	0
$521$	1.09272e6	0.176366	0.0881828	−	0.996104i	$-0.471894\pi$
$521$	1.09272e6	0.176366	0.0881828	+	0.996104i	$0.471894\pi$
$522$	−7.55481e6	−1.21352
$523$	2.47626e6	0.395861	0.197930	−	0.980216i	$-0.436578\pi$
$523$	2.47626e6	0.395861	0.197930	+	0.980216i	$0.436578\pi$
$524$	−3.34024e6	−0.531435
$525$	0	0
$526$	−195508.	−0.0308107
$527$	9.49734e6	1.48962
$528$	1.24914e6	0.194997
$529$	−6.41738e6	−0.997054
$530$	0	0
$531$	1.06957e7	1.64617
$532$	−1.24985e7	−1.91460
$533$	2.83584e6	0.432378
$534$	−4.22414e6	−0.641041
$535$	0	0
$536$	3.47692e6	0.522736
$537$	−2.35726e7	−3.52754
$538$	1.09829e6	0.163592
$539$	−911215.	−0.135098
$540$	0	0
$541$	−5.78515e6	−0.849810	−0.424905	−	0.905238i	$-0.639692\pi$
$541$	−5.78515e6	−0.849810	−0.424905	+	0.905238i	$0.639692\pi$
$542$	989788.	0.144725
$543$	8.11742e6	1.18146
$544$	9.29799e6	1.34708
$545$	0	0
$546$	1.74923e6	0.251111
$547$	−1.26151e7	−1.80269	−0.901345	−	0.433102i	$-0.857419\pi$
$547$	−1.26151e7	−1.80269	−0.901345	+	0.433102i	$0.857419\pi$
$548$	4.61067e6	0.655862
$549$	−1.60813e7	−2.27714
$550$	0	0
$551$	1.50059e7	2.10564
$552$	−484363.	−0.0676586
$553$	−2.39423e6	−0.332930
$554$	2.12887e6	0.294697
$555$	0	0
$556$	−6.14222e6	−0.842632
$557$	−9.93482e6	−1.35682	−0.678409	−	0.734684i	$-0.737330\pi$
$557$	−9.93482e6	−1.35682	−0.678409	+	0.734684i	$0.737330\pi$
$558$	6.57491e6	0.893931
$559$	−3.63495e6	−0.492005
$560$	0	0
$561$	−3.41151e6	−0.457657
$562$	2.88025e6	0.384671
$563$	8.72418e6	1.15999	0.579994	−	0.814621i	$-0.303055\pi$
$563$	8.72418e6	1.15999	0.579994	+	0.814621i	$0.303055\pi$
$564$	1.40234e7	1.85634
$565$	0	0
$566$	−4.15232e6	−0.544816
$567$	3.50860e7	4.58328
$568$	−3.17962e6	−0.413527
$569$	5.33272e6	0.690508	0.345254	−	0.938509i	$-0.387793\pi$
$569$	5.33272e6	0.690508	0.345254	+	0.938509i	$0.387793\pi$
$570$	0	0
$571$	−1.46290e7	−1.87769	−0.938847	−	0.344334i	$-0.888105\pi$
$571$	−1.46290e7	−1.87769	−0.938847	+	0.344334i	$0.888105\pi$
$572$	298608.	0.0381602
$573$	−1.79519e7	−2.28414
$574$	−5.83465e6	−0.739154
$575$	0	0
$576$	−7.31633e6	−0.918834
$577$	7.62278e6	0.953177	0.476589	−	0.879126i	$-0.341873\pi$
$577$	7.62278e6	0.953177	0.476589	+	0.879126i	$0.341873\pi$
$578$	−3.75564e6	−0.467590
$579$	1.75756e6	0.217878
$580$	0	0
$581$	139982.	0.0172041
$582$	−319270.	−0.0390706
$583$	−172650.	−0.0210376
$584$	3.97780e6	0.482626
$585$	0	0
$586$	5.32684e6	0.640805
$587$	7.92637e6	0.949465	0.474732	−	0.880130i	$-0.342545\pi$
$587$	7.92637e6	0.949465	0.474732	+	0.880130i	$0.342545\pi$
$588$	1.21558e7	1.44991
$589$	−1.30596e7	−1.55110
$590$	0	0
$591$	−2.42892e7	−2.86052
$592$	−2.85722e6	−0.335074
$593$	1.13806e7	1.32901	0.664507	−	0.747282i	$-0.268642\pi$
$593$	1.13806e7	1.32901	0.664507	+	0.747282i	$0.268642\pi$
$594$	−1.46935e6	−0.170868
$595$	0	0
$596$	2.23178e6	0.257356
$597$	−1.46037e7	−1.67698
$598$	−45725.1	−0.00522880
$599$	7.12203e6	0.811030	0.405515	−	0.914088i	$-0.367092\pi$
$599$	7.12203e6	0.811030	0.405515	+	0.914088i	$0.367092\pi$
$600$	0	0
$601$	1.50084e6	0.169492	0.0847458	−	0.996403i	$-0.472992\pi$
$601$	1.50084e6	0.169492	0.0847458	+	0.996403i	$0.472992\pi$
$602$	7.47879e6	0.841086
$603$	1.89234e7	2.11936
$604$	−1.03696e7	−1.15656
$605$	0	0
$606$	−5.11534e6	−0.565839
$607$	−1.39958e7	−1.54179	−0.770897	−	0.636960i	$-0.780192\pi$
$607$	−1.39958e7	−1.54179	−0.770897	+	0.636960i	$0.780192\pi$
$608$	−1.27854e7	−1.40267
$609$	−3.14972e7	−3.44135
$610$	0	0
$611$	2.82932e6	0.306604
$612$	3.30299e7	3.56475
$613$	4.60575e6	0.495050	0.247525	−	0.968882i	$-0.420383\pi$
$613$	4.60575e6	0.495050	0.247525	+	0.968882i	$0.420383\pi$
$614$	−3.06063e6	−0.327634
$615$	0	0
$616$	−1.31303e6	−0.139420
$617$	5.55212e6	0.587146	0.293573	−	0.955937i	$-0.405156\pi$
$617$	5.55212e6	0.587146	0.293573	+	0.955937i	$0.405156\pi$
$618$	−4.40334e6	−0.463779
$619$	−2.71815e6	−0.285133	−0.142566	−	0.989785i	$-0.545535\pi$
$619$	−2.71815e6	−0.285133	−0.142566	+	0.989785i	$0.545535\pi$
$620$	0	0
$621$	−1.64008e6	−0.170662
$622$	5.05592e6	0.523992
$623$	−1.27816e7	−1.31936
$624$	−3.36204e6	−0.345654
$625$	0	0
$626$	1.10749e6	0.112954
$627$	4.69109e6	0.476546
$628$	−871122.	−0.0881414
$629$	7.80331e6	0.786416
$630$	0	0
$631$	−1.93426e6	−0.193393	−0.0966964	−	0.995314i	$-0.530828\pi$
$631$	−1.93426e6	−0.193393	−0.0966964	+	0.995314i	$0.530828\pi$
$632$	−1.59860e6	−0.159201
$633$	−1.76793e7	−1.75371
$634$	2.94182e6	0.290665
$635$	0	0
$636$	2.30320e6	0.225782
$637$	2.45251e6	0.239477
$638$	737631.	0.0717443
$639$	−1.73053e7	−1.67659
$640$	0	0
$641$	9.58248e6	0.921155	0.460578	−	0.887619i	$-0.347642\pi$
$641$	9.58248e6	0.921155	0.460578	+	0.887619i	$0.347642\pi$
$642$	4.54916e6	0.435606
$643$	2.46659e6	0.235272	0.117636	−	0.993057i	$-0.462468\pi$
$643$	2.46659e6	0.235272	0.117636	+	0.993057i	$0.462468\pi$
$644$	−685762.	−0.0651567
$645$	0	0
$646$	9.00037e6	0.848553
$647$	−2.72804e6	−0.256207	−0.128103	−	0.991761i	$-0.540889\pi$
$647$	−2.72804e6	−0.256207	−0.128103	+	0.991761i	$0.540889\pi$
$648$	2.34265e7	2.19164
$649$	−1.04430e6	−0.0973229
$650$	0	0
$651$	2.74119e7	2.53505
$652$	5.75584e6	0.530261
$653$	9.94071e6	0.912293	0.456147	−	0.889905i	$-0.349229\pi$
$653$	9.94071e6	0.912293	0.456147	+	0.889905i	$0.349229\pi$
$654$	1.02742e7	0.939295
$655$	0	0
$656$	1.12142e7	1.01744
$657$	2.16494e7	1.95674
$658$	−5.82122e6	−0.524143
$659$	440385.	0.0395019	0.0197510	−	0.999805i	$-0.493713\pi$
$659$	440385.	0.0395019	0.0197510	+	0.999805i	$0.493713\pi$
$660$	0	0
$661$	4.85089e6	0.431835	0.215918	−	0.976412i	$-0.430726\pi$
$661$	4.85089e6	0.431835	0.215918	+	0.976412i	$0.430726\pi$
$662$	−4.87741e6	−0.432558
$663$	9.18201e6	0.811248
$664$	93464.0	0.00822668
$665$	0	0
$666$	5.40215e6	0.471933
$667$	823340.	0.0716580
$668$	−4.45869e6	−0.386604
$669$	2.65985e7	2.29769
$670$	0	0
$671$	1.57013e6	0.134626
$672$	2.68365e7	2.29246
$673$	2.15372e6	0.183296	0.0916479	−	0.995791i	$-0.470787\pi$
$673$	2.15372e6	0.183296	0.0916479	+	0.995791i	$0.470787\pi$
$674$	−4.26471e6	−0.361609
$675$	0	0
$676$	−803695.	−0.0676433
$677$	4.93215e6	0.413585	0.206792	−	0.978385i	$-0.433698\pi$
$677$	4.93215e6	0.413585	0.206792	+	0.978385i	$0.433698\pi$
$678$	−137451.	−0.0114835
$679$	−966058.	−0.0804135
$680$	0	0
$681$	−3.28632e7	−2.71545
$682$	−641956.	−0.0528499
$683$	−1.39352e7	−1.14304	−0.571521	−	0.820588i	$-0.693646\pi$
$683$	−1.39352e7	−1.14304	−0.571521	+	0.820588i	$0.693646\pi$
$684$	−4.54186e7	−3.71187
$685$	0	0
$686$	798026.	0.0647451
$687$	1.97088e7	1.59320
$688$	−1.43743e7	−1.15775
$689$	464685.	0.0372915
$690$	0	0
$691$	1.71835e7	1.36904	0.684520	−	0.728994i	$-0.260012\pi$
$691$	1.71835e7	1.36904	0.684520	+	0.728994i	$0.260012\pi$
$692$	−6.49076e6	−0.515264
$693$	−7.14628e6	−0.565258
$694$	−7.72338e6	−0.608708
$695$	0	0
$696$	−2.10303e7	−1.64559
$697$	−3.06270e7	−2.38793
$698$	−744488.	−0.0578387
$699$	−2.46574e7	−1.90877
$700$	0	0
$701$	4.18640e6	0.321770	0.160885	−	0.986973i	$-0.448565\pi$
$701$	4.18640e6	0.321770	0.160885	+	0.986973i	$0.448565\pi$
$702$	3.95472e6	0.302882
$703$	−1.07301e7	−0.818874
$704$	714347.	0.0543222
$705$	0	0
$706$	2.22082e6	0.167688
$707$	−1.54782e7	−1.16458
$708$	1.39313e7	1.04450
$709$	−7.30623e6	−0.545855	−0.272928	−	0.962035i	$-0.587992\pi$
$709$	−7.30623e6	−0.545855	−0.272928	+	0.962035i	$0.587992\pi$
$710$	0	0
$711$	−8.70048e6	−0.645460
$712$	−8.53409e6	−0.630895
$713$	−716548.	−0.0527864
$714$	−1.88917e7	−1.38684
$715$	0	0
$716$	−2.22835e7	−1.62443
$717$	−1.06930e7	−0.776783
$718$	−6.59342e6	−0.477309
$719$	−1.22435e7	−0.883249	−0.441625	−	0.897200i	$-0.645598\pi$
$719$	−1.22435e7	−0.883249	−0.441625	+	0.897200i	$0.645598\pi$
$720$	0	0
$721$	−1.33238e7	−0.954530
$722$	−7.51117e6	−0.536246
$723$	2.76484e7	1.96709
$724$	7.67351e6	0.544061
$725$	0	0
$726$	−9.18877e6	−0.647017
$727$	612630.	0.0429895	0.0214947	−	0.999769i	$-0.493157\pi$
$727$	612630.	0.0429895	0.0214947	+	0.999769i	$0.493157\pi$
$728$	3.53400e6	0.247137
$729$	4.13490e7	2.88169
$730$	0	0
$731$	3.92574e7	2.71724
$732$	−2.09459e7	−1.44485
$733$	1.54190e7	1.05998	0.529989	−	0.848004i	$-0.322196\pi$
$733$	1.54190e7	1.05998	0.529989	+	0.848004i	$0.322196\pi$
$734$	−3.91176e6	−0.267998
$735$	0	0
$736$	−701508.	−0.0477351
$737$	−1.84763e6	−0.125298
$738$	−2.12027e7	−1.43301
$739$	1.10414e7	0.743724	0.371862	−	0.928288i	$-0.378720\pi$
$739$	1.10414e7	0.743724	0.371862	+	0.928288i	$0.378720\pi$
$740$	0	0
$741$	−1.26259e7	−0.844731
$742$	−956073.	−0.0637502
$743$	−1.66010e7	−1.10322	−0.551611	−	0.834102i	$-0.685987\pi$
$743$	−1.66010e7	−1.10322	−0.551611	+	0.834102i	$0.685987\pi$
$744$	1.83026e7	1.21221
$745$	0	0
$746$	−2.74194e6	−0.180390
$747$	508684.	0.0333539
$748$	−3.22495e6	−0.210751
$749$	1.37650e7	0.896546
$750$	0	0
$751$	1.04497e6	0.0676089	0.0338045	−	0.999428i	$-0.489238\pi$
$751$	1.04497e6	0.0676089	0.0338045	+	0.999428i	$0.489238\pi$
$752$	1.11884e7	0.721481
$753$	3.04622e6	0.195782
$754$	−1.98532e6	−0.127175
$755$	0	0
$756$	5.93109e7	3.77425
$757$	5.93460e6	0.376402	0.188201	−	0.982131i	$-0.439734\pi$
$757$	5.93460e6	0.376402	0.188201	+	0.982131i	$0.439734\pi$
$758$	−7.50481e6	−0.474424
$759$	257389.	0.0162176
$760$	0	0
$761$	5.89549e6	0.369027	0.184514	−	0.982830i	$-0.440929\pi$
$761$	5.89549e6	0.369027	0.184514	+	0.982830i	$0.440929\pi$
$762$	−1.55887e7	−0.972574
$763$	3.10879e7	1.93322
$764$	−1.69701e7	−1.05185
$765$	0	0
$766$	−8.44961e6	−0.520313
$767$	2.81072e6	0.172516
$768$	4726.10	0.000289134 0
$769$	−1.30498e7	−0.795769	−0.397884	−	0.917436i	$-0.630256\pi$
$769$	−1.30498e7	−0.795769	−0.397884	+	0.917436i	$0.630256\pi$
$770$	0	0
$771$	−3.49651e7	−2.11835
$772$	1.66145e6	0.100333
$773$	−2.29371e7	−1.38067	−0.690334	−	0.723491i	$-0.742536\pi$
$773$	−2.29371e7	−1.38067	−0.690334	+	0.723491i	$0.742536\pi$
$774$	2.71774e7	1.63063
$775$	0	0
$776$	−645025.	−0.0384523
$777$	2.25224e7	1.33833
$778$	−1.10661e7	−0.655456
$779$	4.21144e7	2.48649
$780$	0	0
$781$	1.68964e6	0.0991213
$782$	493830.	0.0288775
$783$	−7.12099e7	−4.15084
$784$	9.69838e6	0.563520
$785$	0	0
$786$	6.94254e6	0.400831
$787$	−1.68766e7	−0.971291	−0.485645	−	0.874156i	$-0.661415\pi$
$787$	−1.68766e7	−0.971291	−0.485645	+	0.874156i	$0.661415\pi$
$788$	−2.29609e7	−1.31727
$789$	−2.96204e6	−0.169394
$790$	0	0
$791$	−415903.	−0.0236348
$792$	−4.77148e6	−0.270296
$793$	−4.22597e6	−0.238640
$794$	−2.01769e6	−0.113580
$795$	0	0
$796$	−1.38051e7	−0.772247
$797$	−4.13009e6	−0.230311	−0.115155	−	0.993348i	$-0.536737\pi$
$797$	−4.13009e6	−0.230311	−0.115155	+	0.993348i	$0.536737\pi$
$798$	2.59775e7	1.44407
$799$	−3.05565e7	−1.69331
$800$	0	0
$801$	−4.64474e7	−2.55788
$802$	2.11482e6	0.116101
$803$	−2.11379e6	−0.115684
$804$	2.46478e7	1.34474
$805$	0	0
$806$	1.72781e6	0.0936825
$807$	1.66396e7	0.899412
$808$	−1.03346e7	−0.556884
$809$	1.01709e7	0.546369	0.273185	−	0.961962i	$-0.411923\pi$
$809$	1.01709e7	0.546369	0.273185	+	0.961962i	$0.411923\pi$
$810$	0	0
$811$	−2.22723e7	−1.18909	−0.594543	−	0.804064i	$-0.702667\pi$
$811$	−2.22723e7	−1.18909	−0.594543	+	0.804064i	$0.702667\pi$
$812$	−2.97748e7	−1.58474
$813$	1.49957e7	0.795686
$814$	−527451.	−0.0279011
$815$	0	0
$816$	3.63099e7	1.90897
$817$	−5.39818e7	−2.82939
$818$	8.82104e6	0.460932
$819$	1.92340e7	1.00198
$820$	0	0
$821$	3.65506e6	0.189250	0.0946251	−	0.995513i	$-0.469835\pi$
$821$	3.65506e6	0.189250	0.0946251	+	0.995513i	$0.469835\pi$
$822$	−9.58305e6	−0.494680
$823$	−7.29867e6	−0.375616	−0.187808	−	0.982206i	$-0.560138\pi$
$823$	−7.29867e6	−0.375616	−0.187808	+	0.982206i	$0.560138\pi$
$824$	−8.89612e6	−0.456439
$825$	0	0
$826$	−5.78296e6	−0.294917
$827$	3.19608e7	1.62500	0.812502	−	0.582958i	$-0.198105\pi$
$827$	3.19608e7	1.62500	0.812502	+	0.582958i	$0.198105\pi$
$828$	−2.49201e6	−0.126321
$829$	9.74730e6	0.492604	0.246302	−	0.969193i	$-0.420785\pi$
$829$	9.74730e6	0.492604	0.246302	+	0.969193i	$0.420785\pi$
$830$	0	0
$831$	3.22534e7	1.62022
$832$	−1.92265e6	−0.0962923
$833$	−2.64871e7	−1.32258
$834$	1.27663e7	0.635550
$835$	0	0
$836$	4.43455e6	0.219449
$837$	6.19736e7	3.05769
$838$	5.69976e6	0.280380
$839$	−6.75711e6	−0.331403	−0.165701	−	0.986176i	$-0.552989\pi$
$839$	−6.75711e6	−0.331403	−0.165701	+	0.986176i	$0.552989\pi$
$840$	0	0
$841$	1.52371e7	0.742867
$842$	1.11962e7	0.544238
$843$	4.36371e7	2.11489
$844$	−1.67125e7	−0.807580
$845$	0	0
$846$	−2.11539e7	−1.01617
$847$	−2.78037e7	−1.33166
$848$	1.83758e6	0.0877519
$849$	−6.29096e7	−2.99535
$850$	0	0
$851$	−588738.	−0.0278675
$852$	−2.25402e7	−1.06380
$853$	2.85698e7	1.34442	0.672209	−	0.740362i	$-0.265346\pi$
$853$	2.85698e7	1.34442	0.672209	+	0.740362i	$0.265346\pi$
$854$	8.69479e6	0.407957
$855$	0	0
$856$	9.19073e6	0.428712
$857$	2.35895e6	0.109715	0.0548577	−	0.998494i	$-0.482529\pi$
$857$	2.35895e6	0.109715	0.0548577	+	0.998494i	$0.482529\pi$
$858$	−620642.	−0.0287821
$859$	1.07995e7	0.499368	0.249684	−	0.968327i	$-0.419673\pi$
$859$	1.07995e7	0.499368	0.249684	+	0.968327i	$0.419673\pi$
$860$	0	0
$861$	−8.83976e7	−4.06380
$862$	−1.42541e7	−0.653388
$863$	−2.27989e7	−1.04204	−0.521022	−	0.853543i	$-0.674449\pi$
$863$	−2.27989e7	−1.04204	−0.521022	+	0.853543i	$0.674449\pi$
$864$	6.06728e7	2.76509
$865$	0	0
$866$	−5.74995e6	−0.260537
$867$	−5.68998e7	−2.57077
$868$	2.59128e7	1.16739
$869$	849491.	0.0381601
$870$	0	0
$871$	4.97284e6	0.222105
$872$	2.07570e7	0.924429
$873$	−3.51059e6	−0.155899
$874$	−679053.	−0.0300694
$875$	0	0
$876$	2.81985e7	1.24155
$877$	3.23124e7	1.41863	0.709317	−	0.704890i	$-0.249003\pi$
$877$	3.23124e7	1.41863	0.709317	+	0.704890i	$0.249003\pi$
$878$	−4.23258e6	−0.185297
$879$	8.07041e7	3.52309
$880$	0	0
$881$	1.40863e7	0.611447	0.305723	−	0.952120i	$-0.401102\pi$
$881$	1.40863e7	0.611447	0.305723	+	0.952120i	$0.401102\pi$
$882$	−1.83367e7	−0.793688
$883$	4.14803e7	1.79036	0.895180	−	0.445705i	$-0.147047\pi$
$883$	4.14803e7	1.79036	0.895180	+	0.445705i	$0.147047\pi$
$884$	8.67988e6	0.373579
$885$	0	0
$886$	−1.45586e7	−0.623068
$887$	−6.64591e6	−0.283626	−0.141813	−	0.989893i	$-0.545293\pi$
$887$	−6.64591e6	−0.283626	−0.141813	+	0.989893i	$0.545293\pi$
$888$	1.50379e7	0.639965
$889$	−4.71689e7	−2.00171
$890$	0	0
$891$	−1.24488e7	−0.525331
$892$	2.51439e7	1.05809
$893$	4.20175e7	1.76320
$894$	−4.63864e6	−0.194110
$895$	0	0
$896$	3.28050e7	1.36512
$897$	−692757.	−0.0287475
$898$	3.73869e6	0.154713
$899$	−3.11115e7	−1.28387
$900$	0	0
$901$	−5.01858e6	−0.205953
$902$	2.07018e6	0.0847210
$903$	1.13307e8	4.62422
$904$	−277693.	−0.0113017
$905$	0	0
$906$	2.15527e7	0.872331
$907$	1.12437e7	0.453827	0.226914	−	0.973915i	$-0.427137\pi$
$907$	1.12437e7	0.453827	0.226914	+	0.973915i	$0.427137\pi$
$908$	−3.10660e7	−1.25046
$909$	−5.62466e7	−2.25781
$910$	0	0
$911$	2.28294e7	0.911377	0.455689	−	0.890139i	$-0.349393\pi$
$911$	2.28294e7	0.911377	0.455689	+	0.890139i	$0.349393\pi$
$912$	−4.99289e7	−1.98776
$913$	−49666.6	−0.00197191
$914$	−5.49538e6	−0.217587
$915$	0	0
$916$	1.86310e7	0.733666
$917$	2.10070e7	0.824974
$918$	−4.27109e7	−1.67275
$919$	3.58183e7	1.39899	0.699497	−	0.714635i	$-0.253407\pi$
$919$	3.58183e7	1.39899	0.699497	+	0.714635i	$0.253407\pi$
$920$	0	0
$921$	−4.63699e7	−1.80130
$922$	−8.21885e6	−0.318408
$923$	−4.54763e6	−0.175704
$924$	−9.30807e6	−0.358657
$925$	0	0
$926$	4.93410e6	0.189095
$927$	−4.84177e7	−1.85057
$928$	−3.04584e7	−1.16101
$929$	−1.37009e7	−0.520847	−0.260424	−	0.965494i	$-0.583862\pi$
$929$	−1.37009e7	−0.520847	−0.260424	+	0.965494i	$0.583862\pi$
$930$	0	0
$931$	3.64217e7	1.37716
$932$	−2.33090e7	−0.878988
$933$	7.65996e7	2.88086
$934$	−4.26704e6	−0.160051
$935$	0	0
$936$	1.28423e7	0.479131
$937$	2.45732e7	0.914352	0.457176	−	0.889376i	$-0.348861\pi$
$937$	2.45732e7	0.914352	0.457176	+	0.889376i	$0.348861\pi$
$938$	−1.02315e7	−0.379691
$939$	1.67789e7	0.621013
$940$	0	0
$941$	3.53611e7	1.30182	0.650912	−	0.759153i	$-0.274387\pi$
$941$	3.53611e7	1.30182	0.650912	+	0.759153i	$0.274387\pi$
$942$	1.81059e6	0.0664801
$943$	2.31072e6	0.0846191
$944$	1.11149e7	0.405952
$945$	0	0
$946$	−2.65353e6	−0.0964043
$947$	4.90338e7	1.77673	0.888364	−	0.459140i	$-0.151842\pi$
$947$	4.90338e7	1.77673	0.888364	+	0.459140i	$0.151842\pi$
$948$	−1.13324e7	−0.409545
$949$	5.68922e6	0.205063
$950$	0	0
$951$	4.45699e7	1.59805
$952$	−3.81671e7	−1.36489
$953$	−1.81532e7	−0.647472	−0.323736	−	0.946147i	$-0.604939\pi$
$953$	−1.81532e7	−0.647472	−0.323736	+	0.946147i	$0.604939\pi$
$954$	−3.47431e6	−0.123594
$955$	0	0
$956$	−1.01082e7	−0.357708
$957$	1.11755e7	0.394444
$958$	1.86025e7	0.654872
$959$	−2.89967e7	−1.01813
$960$	0	0
$961$	−1.55304e6	−0.0542466
$962$	1.41962e6	0.0494578
$963$	5.00212e7	1.73815
$964$	2.61364e7	0.905844
$965$	0	0
$966$	1.42532e6	0.0491441
$967$	5.23618e7	1.80073	0.900365	−	0.435135i	$-0.143299\pi$
$967$	5.23618e7	1.80073	0.900365	+	0.435135i	$0.143299\pi$
$968$	−1.85642e7	−0.636777
$969$	1.36360e8	4.66527
$970$	0	0
$971$	2.95200e7	1.00477	0.502387	−	0.864643i	$-0.332455\pi$
$971$	2.95200e7	1.00477	0.502387	+	0.864643i	$0.332455\pi$
$972$	8.46299e7	2.87315
$973$	3.86287e7	1.30806
$974$	−9.58395e6	−0.323703
$975$	0	0
$976$	−1.67114e7	−0.561551
$977$	−1.21759e7	−0.408098	−0.204049	−	0.978961i	$-0.565410\pi$
$977$	−1.21759e7	−0.408098	−0.204049	+	0.978961i	$0.565410\pi$
$978$	−1.19632e7	−0.399946
$979$	4.53499e6	0.151224
$980$	0	0
$981$	1.12971e8	3.74797
$982$	7.59304e6	0.251268
$983$	−1.46087e6	−0.0482200	−0.0241100	−	0.999709i	$-0.507675\pi$
$983$	−1.46087e6	−0.0482200	−0.0241100	+	0.999709i	$0.507675\pi$
$984$	−5.90220e7	−1.94324
$985$	0	0
$986$	2.14413e7	0.702360
$987$	−8.81942e7	−2.88169
$988$	−1.19355e7	−0.388998
$989$	−2.96186e6	−0.0962883
$990$	0	0
$991$	3.16160e7	1.02264	0.511320	−	0.859390i	$-0.329157\pi$
$991$	3.16160e7	1.02264	0.511320	+	0.859390i	$0.329157\pi$
$992$	2.65078e7	0.855252
$993$	−7.38951e7	−2.37817
$994$	9.35660e6	0.300367
$995$	0	0
$996$	662564.	0.0211631
$997$	−3.93744e7	−1.25452	−0.627258	−	0.778812i	$-0.715823\pi$
$997$	−3.93744e7	−1.25452	−0.627258	+	0.778812i	$0.715823\pi$
$998$	−1.13020e7	−0.359195
$999$	5.09194e7	1.61425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.k.1.5	yes	11
5.2	odd	4		325.6.b.i.274.10		22
5.3	odd	4		325.6.b.i.274.13		22
5.4	even	2		325.6.a.j.1.7	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.7	✓	11	5.4	even	2
325.6.a.k.1.5	yes	11	1.1	even	1	trivial
325.6.b.i.274.10		22	5.2	odd	4
325.6.b.i.274.13		22	5.3	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q-1.96479 q^{2} -29.7675 q^{3} -28.1396 q^{4} +58.4868 q^{6} +176.972 q^{7} +118.162 q^{8} +643.103 q^{9} -62.7908 q^{11} +837.645 q^{12} +169.000 q^{13} -347.712 q^{14} +668.304 q^{16} -1825.19 q^{17} -1263.56 q^{18} +2509.78 q^{19} -5268.00 q^{21} +123.371 q^{22} +137.706 q^{23} -3517.37 q^{24} -332.049 q^{26} -11910.0 q^{27} -4979.91 q^{28} +5978.98 q^{29} -5203.47 q^{31} -5094.25 q^{32} +1869.12 q^{33} +3586.12 q^{34} -18096.7 q^{36} -4275.33 q^{37} -4931.19 q^{38} -5030.70 q^{39} +16780.1 q^{41} +10350.5 q^{42} -21508.6 q^{43} +1766.91 q^{44} -270.563 q^{46} +16741.5 q^{47} -19893.7 q^{48} +14511.9 q^{49} +54331.4 q^{51} -4755.59 q^{52} +2749.61 q^{53} +23400.7 q^{54} +20911.2 q^{56} -74709.8 q^{57} -11747.4 q^{58} +16631.5 q^{59} -25005.7 q^{61} +10223.7 q^{62} +113811. q^{63} -11376.6 q^{64} -3672.44 q^{66} +29425.1 q^{67} +51360.2 q^{68} -4099.16 q^{69} -26909.1 q^{71} +75990.1 q^{72} +33664.0 q^{73} +8400.13 q^{74} -70624.2 q^{76} -11112.2 q^{77} +9884.27 q^{78} -13528.9 q^{79} +198258. q^{81} -32969.4 q^{82} +790.985 q^{83} +148239. q^{84} +42259.9 q^{86} -177979. q^{87} -7419.47 q^{88} -72223.8 q^{89} +29908.2 q^{91} -3874.99 q^{92} +154894. q^{93} -32893.6 q^{94} +151643. q^{96} -5458.83 q^{97} -28512.9 q^{98} -40380.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 5 q^{2} + 11 q^{3} + 187 q^{4} + 351 q^{6} + 208 q^{7} + 165 q^{8} + 1372 q^{9} + 1276 q^{11} + 1533 q^{12} + 1859 q^{13} + 578 q^{14} + 5707 q^{16} + 2218 q^{17} - 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 11 * q + 5 * q^2 + 11 * q^3 + 187 * q^4 + 351 * q^6 + 208 * q^7 + 165 * q^8 + 1372 * q^9 + 1276 * q^11 + 1533 * q^12 + 1859 * q^13 + 578 * q^14 + 5707 * q^16 + 2218 * q^17 - 6776 * q^18 + 3520 * q^19 + 1706 * q^21 - 6785 * q^22 - 1215 * q^23 + 7723 * q^24 + 845 * q^26 + 161 * q^27 + 12746 * q^28 + 18425 * q^29 + 9068 * q^31 + 21165 * q^32 + 19748 * q^33 - 14455 * q^34 + 37684 * q^36 - 2280 * q^37 - 23783 * q^38 + 1859 * q^39 - 10010 * q^41 - 44184 * q^42 + 6763 * q^43 + 60651 * q^44 - 9858 * q^46 + 4110 * q^47 + 63289 * q^48 + 73313 * q^49 + 36606 * q^51 + 31603 * q^52 + 15633 * q^53 + 48013 * q^54 - 1262 * q^56 - 205974 * q^57 + 77652 * q^58 + 131642 * q^59 - 43365 * q^61 - 115940 * q^62 + 218466 * q^63 + 180071 * q^64 - 58607 * q^66 - 100904 * q^67 + 371277 * q^68 + 59345 * q^69 + 68420 * q^71 - 210708 * q^72 + 124436 * q^73 + 371396 * q^74 + 22517 * q^76 - 150978 * q^77 + 59319 * q^78 + 52239 * q^79 + 107507 * q^81 - 351137 * q^82 + 211538 * q^83 + 852716 * q^84 + 202246 * q^86 - 28273 * q^87 - 28107 * q^88 + 327712 * q^89 + 35152 * q^91 - 413252 * q^92 + 243338 * q^93 + 649974 * q^94 + 899663 * q^96 - 110796 * q^97 + 27143 * q^98 + 426698 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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