LMFDB - Embedded newform 325.6.a.h.1.2

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 $ x^9 - 4x^8 - 181x^7 + 688x^6 + 10455x^5 - 37904x^4 - 197375x^3 + 702868x^2 + 262812x - 366960
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.11691$ 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)

$f(q)$	$=$	\(q-8.11691 q^{2} -3.22994 q^{3} +33.8842 q^{4} +26.2172 q^{6} -20.7248 q^{7} -15.2938 q^{8} -232.567 q^{9} +134.772 q^{11} -109.444 q^{12} +169.000 q^{13} +168.222 q^{14} -960.156 q^{16} +2192.81 q^{17} +1887.73 q^{18} -1981.97 q^{19} +66.9401 q^{21} -1093.93 q^{22} -4227.53 q^{23} +49.3980 q^{24} -1371.76 q^{26} +1536.06 q^{27} -702.244 q^{28} +2547.33 q^{29} -1368.81 q^{31} +8282.90 q^{32} -435.305 q^{33} -17798.9 q^{34} -7880.36 q^{36} +14161.1 q^{37} +16087.5 q^{38} -545.861 q^{39} +11750.1 q^{41} -543.347 q^{42} +6627.63 q^{43} +4566.63 q^{44} +34314.5 q^{46} -14822.2 q^{47} +3101.25 q^{48} -16377.5 q^{49} -7082.67 q^{51} +5726.43 q^{52} +24261.4 q^{53} -12468.0 q^{54} +316.961 q^{56} +6401.66 q^{57} -20676.4 q^{58} +1438.11 q^{59} +16295.7 q^{61} +11110.5 q^{62} +4819.92 q^{63} -36506.5 q^{64} +3533.33 q^{66} +16700.2 q^{67} +74301.7 q^{68} +13654.7 q^{69} +27153.9 q^{71} +3556.83 q^{72} +63222.7 q^{73} -114944. q^{74} -67157.5 q^{76} -2793.12 q^{77} +4430.70 q^{78} -58165.8 q^{79} +51552.5 q^{81} -95374.1 q^{82} -121286. q^{83} +2268.21 q^{84} -53795.9 q^{86} -8227.72 q^{87} -2061.17 q^{88} -49897.1 q^{89} -3502.50 q^{91} -143246. q^{92} +4421.18 q^{93} +120311. q^{94} -26753.3 q^{96} -21395.4 q^{97} +132934. q^{98} -31343.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) $ 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9 - 1422 * q^11 + 1567 * q^12 + 1521 * q^13 - 342 * q^14 - 1061 * q^16 + 648 * q^17 + 418 * q^18 - 408 * q^19 - 3912 * q^21 + 4345 * q^22 + 1839 * q^23 - 8469 * q^24 - 845 * q^26 - 7649 * q^27 - 2836 * q^28 - 8737 * q^29 + 748 * q^31 - 423 * q^32 - 356 * q^33 - 17789 * q^34 + 512 * q^36 - 15486 * q^37 - 3425 * q^38 - 1859 * q^39 - 28676 * q^41 + 6876 * q^42 + 28665 * q^43 - 30599 * q^44 - 12056 * q^46 - 29452 * q^47 + 64759 * q^48 - 40907 * q^49 - 31006 * q^51 + 15379 * q^52 + 75977 * q^53 - 102761 * q^54 - 23002 * q^56 - 38038 * q^57 + 142384 * q^58 - 88142 * q^59 + 28165 * q^61 - 137308 * q^62 + 41492 * q^63 - 100845 * q^64 + 42577 * q^66 - 94754 * q^67 + 89267 * q^68 - 181747 * q^69 - 70562 * q^71 - 263778 * q^72 + 60602 * q^73 - 135676 * q^74 + 46373 * q^76 - 140292 * q^77 - 14027 * q^78 - 164073 * q^79 - 69935 * q^81 - 72887 * q^82 - 22458 * q^83 - 345656 * q^84 - 294920 * q^86 - 87031 * q^87 + 430607 * q^88 - 252698 * q^89 + 2028 * q^91 - 237824 * q^92 + 56556 * q^93 - 501606 * q^94 - 319181 * q^96 + 137986 * q^97 + 378699 * q^98 - 757776 * 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 $

$2$	−8.11691	−1.43488	−0.717440	−	0.696620i	$-0.754686\pi$
$2$	−8.11691	−1.43488	−0.717440	+	0.696620i	$0.754686\pi$
$3$	−3.22994	−0.207201	−0.103601	−	0.994619i	$-0.533036\pi$
$3$	−3.22994	−0.207201	−0.103601	+	0.994619i	$0.533036\pi$
$4$	33.8842	1.05888
$5$	0	0
$5$	0	0
$6$	26.2172	0.297309
$7$	−20.7248	−0.159862	−0.0799312	−	0.996800i	$-0.525470\pi$
$7$	−20.7248	−0.159862	−0.0799312	+	0.996800i	$0.525470\pi$
$8$	−15.2938	−0.0844869
$9$	−232.567	−0.957068
$10$	0	0
$11$	134.772	0.335828	0.167914	−	0.985802i	$-0.446297\pi$
$11$	134.772	0.335828	0.167914	+	0.985802i	$0.446297\pi$
$12$	−109.444	−0.219401
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	168.222	0.229383
$15$	0	0
$16$	−960.156	−0.937652
$17$	2192.81	1.84026	0.920131	−	0.391612i	$-0.128082\pi$
$17$	2192.81	1.84026	0.920131	+	0.391612i	$0.128082\pi$
$18$	1887.73	1.37328
$19$	−1981.97	−1.25954	−0.629772	−	0.776780i	$-0.716852\pi$
$19$	−1981.97	−1.25954	−0.629772	+	0.776780i	$0.716852\pi$
$20$	0	0
$21$	66.9401	0.0331236
$22$	−1093.93	−0.481873
$23$	−4227.53	−1.66635	−0.833177	−	0.553006i	$-0.813481\pi$
$23$	−4227.53	−1.66635	−0.833177	+	0.553006i	$0.813481\pi$
$24$	49.3980	0.0175058
$25$	0	0
$26$	−1371.76	−0.397964
$27$	1536.06	0.405507
$28$	−702.244	−0.169275
$29$	2547.33	0.562457	0.281229	−	0.959641i	$-0.409258\pi$
$29$	2547.33	0.562457	0.281229	+	0.959641i	$0.409258\pi$
$30$	0	0
$31$	−1368.81	−0.255822	−0.127911	−	0.991786i	$-0.540827\pi$
$31$	−1368.81	−0.255822	−0.127911	+	0.991786i	$0.540827\pi$
$32$	8282.90	1.42991
$33$	−435.305	−0.0695840
$34$	−17798.9	−2.64055
$35$	0	0
$36$	−7880.36	−1.01342
$37$	14161.1	1.70056	0.850282	−	0.526327i	$-0.176431\pi$
$37$	14161.1	1.70056	0.850282	+	0.526327i	$0.176431\pi$
$38$	16087.5	1.80729
$39$	−545.861	−0.0574672
$40$	0	0
$41$	11750.1	1.09164	0.545821	−	0.837902i	$-0.316218\pi$
$41$	11750.1	1.09164	0.545821	+	0.837902i	$0.316218\pi$
$42$	−543.347	−0.0475285
$43$	6627.63	0.546622	0.273311	−	0.961926i	$-0.411881\pi$
$43$	6627.63	0.546622	0.273311	+	0.961926i	$0.411881\pi$
$44$	4566.63	0.355602
$45$	0	0
$46$	34314.5	2.39102
$47$	−14822.2	−0.978744	−0.489372	−	0.872075i	$-0.662774\pi$
$47$	−14822.2	−0.978744	−0.489372	+	0.872075i	$0.662774\pi$
$48$	3101.25	0.194283
$49$	−16377.5	−0.974444
$50$	0	0
$51$	−7082.67	−0.381304
$52$	5726.43	0.293681
$53$	24261.4	1.18639	0.593193	−	0.805060i	$-0.297867\pi$
$53$	24261.4	1.18639	0.593193	+	0.805060i	$0.297867\pi$
$54$	−12468.0	−0.581853
$55$	0	0
$56$	316.961	0.0135063
$57$	6401.66	0.260979
$58$	−20676.4	−0.807059
$59$	1438.11	0.0537850	0.0268925	−	0.999638i	$-0.491439\pi$
$59$	1438.11	0.0537850	0.0268925	+	0.999638i	$0.491439\pi$
$60$	0	0
$61$	16295.7	0.560724	0.280362	−	0.959894i	$-0.409546\pi$
$61$	16295.7	0.560724	0.280362	+	0.959894i	$0.409546\pi$
$62$	11110.5	0.367074
$63$	4819.92	0.152999
$64$	−36506.5	−1.11409
$65$	0	0
$66$	3533.33	0.0998446
$67$	16700.2	0.454502	0.227251	−	0.973836i	$-0.427026\pi$
$67$	16700.2	0.454502	0.227251	+	0.973836i	$0.427026\pi$
$68$	74301.7	1.94862
$69$	13654.7	0.345270
$70$	0	0
$71$	27153.9	0.639273	0.319636	−	0.947540i	$-0.396439\pi$
$71$	27153.9	0.639273	0.319636	+	0.947540i	$0.396439\pi$
$72$	3556.83	0.0808597
$73$	63222.7	1.38856	0.694282	−	0.719703i	$-0.255722\pi$
$73$	63222.7	1.38856	0.694282	+	0.719703i	$0.255722\pi$
$74$	−114944.	−2.44011
$75$	0	0
$76$	−67157.5	−1.33371
$77$	−2793.12	−0.0536863
$78$	4430.70	0.0824586
$79$	−58165.8	−1.04858	−0.524288	−	0.851541i	$-0.675669\pi$
$79$	−58165.8	−1.04858	−0.524288	+	0.851541i	$0.675669\pi$
$80$	0	0
$81$	51552.5	0.873046
$82$	−95374.1	−1.56638
$83$	−121286.	−1.93249	−0.966243	−	0.257632i	$-0.917058\pi$
$83$	−121286.	−1.93249	−0.966243	+	0.257632i	$0.917058\pi$
$84$	2268.21	0.0350740
$85$	0	0
$86$	−53795.9	−0.784337
$87$	−8227.72	−0.116542
$88$	−2061.17	−0.0283731
$89$	−49897.1	−0.667729	−0.333865	−	0.942621i	$-0.608353\pi$
$89$	−49897.1	−0.667729	−0.333865	+	0.942621i	$0.608353\pi$
$90$	0	0
$91$	−3502.50	−0.0443378
$92$	−143246.	−1.76447
$93$	4421.18	0.0530067
$94$	120311.	1.40438
$95$	0	0
$96$	−26753.3	−0.296278
$97$	−21395.4	−0.230883	−0.115441	−	0.993314i	$-0.536828\pi$
$97$	−21395.4	−0.230883	−0.115441	+	0.993314i	$0.536828\pi$
$98$	132934.	1.39821
$99$	−31343.5	−0.321410
$100$	0	0
$101$	−107983.	−1.05330	−0.526648	−	0.850084i	$-0.676551\pi$
$101$	−107983.	−1.05330	−0.526648	+	0.850084i	$0.676551\pi$
$102$	57489.4	0.547126
$103$	−40865.0	−0.379541	−0.189771	−	0.981828i	$-0.560774\pi$
$103$	−40865.0	−0.379541	−0.189771	+	0.981828i	$0.560774\pi$
$104$	−2584.65	−0.0234325
$105$	0	0
$106$	−196928.	−1.70232
$107$	−74698.6	−0.630744	−0.315372	−	0.948968i	$-0.602129\pi$
$107$	−74698.6	−0.630744	−0.315372	+	0.948968i	$0.602129\pi$
$108$	52048.0	0.429383
$109$	−84724.3	−0.683033	−0.341517	−	0.939876i	$-0.610941\pi$
$109$	−84724.3	−0.683033	−0.341517	+	0.939876i	$0.610941\pi$
$110$	0	0
$111$	−45739.6	−0.352359
$112$	19899.1	0.149895
$113$	−92231.9	−0.679493	−0.339747	−	0.940517i	$-0.610341\pi$
$113$	−92231.9	−0.679493	−0.339747	+	0.940517i	$0.610341\pi$
$114$	−51961.7	−0.374473
$115$	0	0
$116$	86314.1	0.595575
$117$	−39303.9	−0.265443
$118$	−11673.0	−0.0771751
$119$	−45445.7	−0.294188
$120$	0	0
$121$	−142888.	−0.887219
$122$	−132271.	−0.804572
$123$	−37952.0	−0.226189
$124$	−46381.0	−0.270885
$125$	0	0
$126$	−39122.9	−0.219535
$127$	−330445.	−1.81799	−0.908993	−	0.416812i	$-0.863147\pi$
$127$	−330445.	−1.81799	−0.908993	+	0.416812i	$0.863147\pi$
$128$	31267.3	0.168681
$129$	−21406.9	−0.113261
$130$	0	0
$131$	−73109.1	−0.372214	−0.186107	−	0.982529i	$-0.559587\pi$
$131$	−73109.1	−0.372214	−0.186107	+	0.982529i	$0.559587\pi$
$132$	−14750.0	−0.0736811
$133$	41076.0	0.201354
$134$	−135554.	−0.652156
$135$	0	0
$136$	−33536.4	−0.155478
$137$	−137616.	−0.626421	−0.313211	−	0.949684i	$-0.601405\pi$
$137$	−137616.	−0.626421	−0.313211	+	0.949684i	$0.601405\pi$
$138$	−110834.	−0.495422
$139$	366876.	1.61058	0.805291	−	0.592880i	$-0.202009\pi$
$139$	366876.	1.61058	0.805291	+	0.592880i	$0.202009\pi$
$140$	0	0
$141$	47875.0	0.202797
$142$	−220406.	−0.917280
$143$	22776.4	0.0931420
$144$	223301.	0.897397
$145$	0	0
$146$	−513173.	−1.99242
$147$	52898.4	0.201906
$148$	479838.	1.80070
$149$	−426714.	−1.57460	−0.787302	−	0.616567i	$-0.788523\pi$
$149$	−426714.	−1.57460	−0.787302	+	0.616567i	$0.788523\pi$
$150$	0	0
$151$	22573.4	0.0805663	0.0402832	−	0.999188i	$-0.487174\pi$
$151$	22573.4	0.0805663	0.0402832	+	0.999188i	$0.487174\pi$
$152$	30311.8	0.106415
$153$	−509977.	−1.76125
$154$	22671.5	0.0770334
$155$	0	0
$156$	−18496.0	−0.0608510
$157$	496911.	1.60890	0.804451	−	0.594018i	$-0.202459\pi$
$157$	496911.	1.60890	0.804451	+	0.594018i	$0.202459\pi$
$158$	472127.	1.50458
$159$	−78363.0	−0.245821
$160$	0	0
$161$	87614.9	0.266387
$162$	−418447.	−1.25272
$163$	129489.	0.381737	0.190868	−	0.981616i	$-0.438870\pi$
$163$	129489.	0.381737	0.190868	+	0.981616i	$0.438870\pi$
$164$	398141.	1.15592
$165$	0	0
$166$	984469.	2.77289
$167$	−413362.	−1.14694	−0.573469	−	0.819227i	$-0.694403\pi$
$167$	−413362.	−1.14694	−0.573469	+	0.819227i	$0.694403\pi$
$168$	−1023.77	−0.00279852
$169$	28561.0	0.0769231
$170$	0	0
$171$	460942.	1.20547
$172$	224572.	0.578808
$173$	−2949.45	−0.00749249	−0.00374625	−	0.999993i	$-0.501192\pi$
$173$	−2949.45	−0.00749249	−0.00374625	+	0.999993i	$0.501192\pi$
$174$	66783.7	0.167223
$175$	0	0
$176$	−129402.	−0.314890
$177$	−4645.01	−0.0111443
$178$	405010.	0.958112
$179$	−431301.	−1.00612	−0.503058	−	0.864253i	$-0.667792\pi$
$179$	−431301.	−1.00612	−0.503058	+	0.864253i	$0.667792\pi$
$180$	0	0
$181$	65154.9	0.147826	0.0739129	−	0.997265i	$-0.476451\pi$
$181$	65154.9	0.147826	0.0739129	+	0.997265i	$0.476451\pi$
$182$	28429.5	0.0636195
$183$	−52634.3	−0.116183
$184$	64654.9	0.140785
$185$	0	0
$186$	−35886.3	−0.0760582
$187$	295529.	0.618012
$188$	−502239.	−1.03637
$189$	−31834.5	−0.0648252
$190$	0	0
$191$	405278.	0.803840	0.401920	−	0.915675i	$-0.368343\pi$
$191$	405278.	0.803840	0.401920	+	0.915675i	$0.368343\pi$
$192$	117914.	0.230841
$193$	−596684.	−1.15306	−0.576529	−	0.817077i	$-0.695593\pi$
$193$	−596684.	−1.15306	−0.576529	+	0.817077i	$0.695593\pi$
$194$	173665.	0.331289
$195$	0	0
$196$	−554938.	−1.03182
$197$	240155.	0.440886	0.220443	−	0.975400i	$-0.429250\pi$
$197$	240155.	0.440886	0.220443	+	0.975400i	$0.429250\pi$
$198$	254413.	0.461185
$199$	438913.	0.785680	0.392840	−	0.919607i	$-0.371493\pi$
$199$	438913.	0.785680	0.392840	+	0.919607i	$0.371493\pi$
$200$	0	0
$201$	−53940.9	−0.0941733
$202$	876484.	1.51135
$203$	−52792.9	−0.0899157
$204$	−239990.	−0.403756
$205$	0	0
$206$	331698.	0.544596
$207$	983186.	1.59481
$208$	−162266.	−0.260058
$209$	−267114.	−0.422990
$210$	0	0
$211$	−107094.	−0.165600	−0.0828000	−	0.996566i	$-0.526386\pi$
$211$	−107094.	−0.165600	−0.0828000	+	0.996566i	$0.526386\pi$
$212$	822078.	1.25624
$213$	−87705.6	−0.132458
$214$	606321.	0.905041
$215$	0	0
$216$	−23492.1	−0.0342600
$217$	28368.3	0.0408964
$218$	687699.	0.980071
$219$	−204206.	−0.287712
$220$	0	0
$221$	370586.	0.510397
$222$	371264.	0.505593
$223$	−865096.	−1.16494	−0.582468	−	0.812853i	$-0.697913\pi$
$223$	−865096.	−1.16494	−0.582468	+	0.812853i	$0.697913\pi$
$224$	−171662.	−0.228588
$225$	0	0
$226$	748638.	0.974991
$227$	772248.	0.994700	0.497350	−	0.867550i	$-0.334307\pi$
$227$	772248.	0.994700	0.497350	+	0.867550i	$0.334307\pi$
$228$	216915.	0.276346
$229$	−615081.	−0.775075	−0.387538	−	0.921854i	$-0.626674\pi$
$229$	−615081.	−0.775075	−0.387538	+	0.921854i	$0.626674\pi$
$230$	0	0
$231$	9021.64	0.0111239
$232$	−38958.2	−0.0475203
$233$	−1.38212e6	−1.66784	−0.833922	−	0.551882i	$-0.813910\pi$
$233$	−1.38212e6	−1.66784	−0.833922	+	0.551882i	$0.813910\pi$
$234$	319026.	0.380879
$235$	0	0
$236$	48729.1	0.0569519
$237$	187872.	0.217266
$238$	368879.	0.422125
$239$	−1.15404e6	−1.30685	−0.653424	−	0.756993i	$-0.726668\pi$
$239$	−1.15404e6	−1.30685	−0.653424	+	0.756993i	$0.726668\pi$
$240$	0	0
$241$	−1.36874e6	−1.51802	−0.759011	−	0.651078i	$-0.774317\pi$
$241$	−1.36874e6	−1.51802	−0.759011	+	0.651078i	$0.774317\pi$
$242$	1.15981e6	1.27305
$243$	−539774.	−0.586403
$244$	552167.	0.593740
$245$	0	0
$246$	308053.	0.324555
$247$	−334953.	−0.349335
$248$	20934.2	0.0216136
$249$	391748.	0.400413
$250$	0	0
$251$	1.37571e6	1.37830	0.689149	−	0.724620i	$-0.257985\pi$
$251$	1.37571e6	1.37830	0.689149	+	0.724620i	$0.257985\pi$
$252$	163319.	0.162008
$253$	−569752.	−0.559609
$254$	2.68219e6	2.60859
$255$	0	0
$256$	914414.	0.872054
$257$	−594799.	−0.561742	−0.280871	−	0.959745i	$-0.590623\pi$
$257$	−594799.	−0.561742	−0.280871	+	0.959745i	$0.590623\pi$
$258$	173758.	0.162516
$259$	−293487.	−0.271856
$260$	0	0
$261$	−592425.	−0.538310
$262$	593420.	0.534083
$263$	1.60634e6	1.43202	0.716008	−	0.698092i	$-0.245967\pi$
$263$	1.60634e6	1.43202	0.716008	+	0.698092i	$0.245967\pi$
$264$	6657.46	0.00587894
$265$	0	0
$266$	−333410.	−0.288918
$267$	161165.	0.138354
$268$	565874.	0.481263
$269$	−764991.	−0.644578	−0.322289	−	0.946641i	$-0.604452\pi$
$269$	−764991.	−0.644578	−0.322289	+	0.946641i	$0.604452\pi$
$270$	0	0
$271$	479475.	0.396591	0.198296	−	0.980142i	$-0.436459\pi$
$271$	479475.	0.396591	0.198296	+	0.980142i	$0.436459\pi$
$272$	−2.10544e6	−1.72552
$273$	11312.9	0.00918685
$274$	1.11701e6	0.898839
$275$	0	0
$276$	462678.	0.365600
$277$	−218179.	−0.170849	−0.0854246	−	0.996345i	$-0.527225\pi$
$277$	−218179.	−0.170849	−0.0854246	+	0.996345i	$0.527225\pi$
$278$	−2.97790e6	−2.31099
$279$	318340.	0.244839
$280$	0	0
$281$	−824857.	−0.623179	−0.311590	−	0.950217i	$-0.600861\pi$
$281$	−824857.	−0.623179	−0.311590	+	0.950217i	$0.600861\pi$
$282$	−388597.	−0.290989
$283$	−1.37912e6	−1.02361	−0.511806	−	0.859101i	$-0.671023\pi$
$283$	−1.37912e6	−1.02361	−0.511806	+	0.859101i	$0.671023\pi$
$284$	920087.	0.676914
$285$	0	0
$286$	−184874.	−0.133648
$287$	−243518.	−0.174512
$288$	−1.92633e6	−1.36852
$289$	3.38858e6	2.38656
$290$	0	0
$291$	69106.1	0.0478392
$292$	2.14225e6	1.47032
$293$	2.22384e6	1.51334	0.756668	−	0.653799i	$-0.226826\pi$
$293$	2.22384e6	1.51334	0.756668	+	0.653799i	$0.226826\pi$
$294$	−429371.	−0.289711
$295$	0	0
$296$	−216577.	−0.143676
$297$	207017.	0.136181
$298$	3.46360e6	2.25937
$299$	−714453.	−0.462163
$300$	0	0
$301$	−137357.	−0.0873843
$302$	−183226.	−0.115603
$303$	348778.	0.218244
$304$	1.90300e6	1.18101
$305$	0	0
$306$	4.13944e6	2.52719
$307$	−2.71603e6	−1.64471	−0.822355	−	0.568975i	$-0.807340\pi$
$307$	−2.71603e6	−1.64471	−0.822355	+	0.568975i	$0.807340\pi$
$308$	−94642.7	−0.0568474
$309$	131992.	0.0786413
$310$	0	0
$311$	−2.07670e6	−1.21751	−0.608754	−	0.793359i	$-0.708330\pi$
$311$	−2.07670e6	−1.21751	−0.608754	+	0.793359i	$0.708330\pi$
$312$	8348.27	0.00485523
$313$	1.45231e6	0.837910	0.418955	−	0.908007i	$-0.362396\pi$
$313$	1.45231e6	0.837910	0.418955	+	0.908007i	$0.362396\pi$
$314$	−4.03338e6	−2.30858
$315$	0	0
$316$	−1.97090e6	−1.11032
$317$	−1.58790e6	−0.887511	−0.443755	−	0.896148i	$-0.646354\pi$
$317$	−1.58790e6	−0.887511	−0.443755	+	0.896148i	$0.646354\pi$
$318$	636065.	0.352723
$319$	343308.	0.188889
$320$	0	0
$321$	241272.	0.130691
$322$	−711162.	−0.382234
$323$	−4.34609e6	−2.31789
$324$	1.74681e6	0.924452
$325$	0	0
$326$	−1.05105e6	−0.547746
$327$	273655.	0.141525
$328$	−179703.	−0.0922295
$329$	307188.	0.156464
$330$	0	0
$331$	−2.91368e6	−1.46174	−0.730872	−	0.682515i	$-0.760886\pi$
$331$	−2.91368e6	−1.46174	−0.730872	+	0.682515i	$0.760886\pi$
$332$	−4.10969e6	−2.04627
$333$	−3.29341e6	−1.62756
$334$	3.35522e6	1.64572
$335$	0	0
$336$	−64272.9	−0.0310585
$337$	2.39665e6	1.14956	0.574778	−	0.818309i	$-0.305088\pi$
$337$	2.39665e6	1.14956	0.574778	+	0.818309i	$0.305088\pi$
$338$	−231827.	−0.110375
$339$	297904.	0.140792
$340$	0	0
$341$	−184477.	−0.0859124
$342$	−3.74142e6	−1.72970
$343$	687743.	0.315639
$344$	−101362.	−0.0461824
$345$	0	0
$346$	23940.4	0.0107508
$347$	529222.	0.235947	0.117973	−	0.993017i	$-0.462360\pi$
$347$	529222.	0.235947	0.117973	+	0.993017i	$0.462360\pi$
$348$	−278790.	−0.123404
$349$	−847145.	−0.372301	−0.186151	−	0.982521i	$-0.559601\pi$
$349$	−847145.	−0.372301	−0.186151	+	0.982521i	$0.559601\pi$
$350$	0	0
$351$	259594.	0.112467
$352$	1.11630e6	0.480203
$353$	−3.44913e6	−1.47324	−0.736619	−	0.676308i	$-0.763579\pi$
$353$	−3.44913e6	−1.47324	−0.736619	+	0.676308i	$0.763579\pi$
$354$	37703.1	0.0159908
$355$	0	0
$356$	−1.69072e6	−0.707046
$357$	146787.	0.0609562
$358$	3.50083e6	1.44366
$359$	−192156.	−0.0786895	−0.0393448	−	0.999226i	$-0.512527\pi$
$359$	−192156.	−0.0786895	−0.0393448	+	0.999226i	$0.512527\pi$
$360$	0	0
$361$	1.45211e6	0.586451
$362$	−528856.	−0.212112
$363$	461519.	0.183833
$364$	−118679.	−0.0469485
$365$	0	0
$366$	427228.	0.166708
$367$	1.34308e6	0.520518	0.260259	−	0.965539i	$-0.416192\pi$
$367$	1.34308e6	0.520518	0.260259	+	0.965539i	$0.416192\pi$
$368$	4.05909e6	1.56246
$369$	−2.73268e6	−1.04478
$370$	0	0
$371$	−502814.	−0.189658
$372$	149808.	0.0561277
$373$	3.94149e6	1.46686	0.733430	−	0.679765i	$-0.237918\pi$
$373$	3.94149e6	1.46686	0.733430	+	0.679765i	$0.237918\pi$
$374$	−2.39878e6	−0.886773
$375$	0	0
$376$	226688.	0.0826911
$377$	430498.	0.155998
$378$	258398.	0.0930164
$379$	−769080.	−0.275026	−0.137513	−	0.990500i	$-0.543911\pi$
$379$	−769080.	−0.275026	−0.137513	+	0.990500i	$0.543911\pi$
$380$	0	0
$381$	1.06732e6	0.376689
$382$	−3.28960e6	−1.15341
$383$	−1.33597e6	−0.465371	−0.232685	−	0.972552i	$-0.574751\pi$
$383$	−1.33597e6	−0.465371	−0.232685	+	0.972552i	$0.574751\pi$
$384$	−100992.	−0.0349509
$385$	0	0
$386$	4.84323e6	1.65450
$387$	−1.54137e6	−0.523155
$388$	−724967.	−0.244477
$389$	158354.	0.0530584	0.0265292	−	0.999648i	$-0.491554\pi$
$389$	158354.	0.0530584	0.0265292	+	0.999648i	$0.491554\pi$
$390$	0	0
$391$	−9.27019e6	−3.06653
$392$	250473.	0.0823278
$393$	236138.	0.0771232
$394$	−1.94932e6	−0.632619
$395$	0	0
$396$	−1.06205e6	−0.340335
$397$	2.11342e6	0.672992	0.336496	−	0.941685i	$-0.390758\pi$
$397$	2.11342e6	0.672992	0.336496	+	0.941685i	$0.390758\pi$
$398$	−3.56261e6	−1.12736
$399$	−132673.	−0.0417207
$400$	0	0
$401$	−2.39852e6	−0.744874	−0.372437	−	0.928057i	$-0.621478\pi$
$401$	−2.39852e6	−0.744874	−0.372437	+	0.928057i	$0.621478\pi$
$402$	437833.	0.135127
$403$	−231329.	−0.0709524
$404$	−3.65890e6	−1.11531
$405$	0	0
$406$	428515.	0.129018
$407$	1.90852e6	0.571098
$408$	108321.	0.0322152
$409$	−2.12794e6	−0.629000	−0.314500	−	0.949257i	$-0.601837\pi$
$409$	−2.12794e6	−0.629000	−0.314500	+	0.949257i	$0.601837\pi$
$410$	0	0
$411$	444491.	0.129795
$412$	−1.38468e6	−0.401889
$413$	−29804.6	−0.00859820
$414$	−7.98043e6	−2.28837
$415$	0	0
$416$	1.39981e6	0.396584
$417$	−1.18499e6	−0.333714
$418$	2.16814e6	0.606941
$419$	−3.35254e6	−0.932908	−0.466454	−	0.884545i	$-0.654469\pi$
$419$	−3.35254e6	−0.932908	−0.466454	+	0.884545i	$0.654469\pi$
$420$	0	0
$421$	205253.	0.0564397	0.0282198	−	0.999602i	$-0.491016\pi$
$421$	205253.	0.0564397	0.0282198	+	0.999602i	$0.491016\pi$
$422$	869275.	0.237616
$423$	3.44717e6	0.936724
$424$	−371048.	−0.100234
$425$	0	0
$426$	711898.	0.190061
$427$	−337726.	−0.0896386
$428$	−2.53110e6	−0.667882
$429$	−73566.6	−0.0192991
$430$	0	0
$431$	−6.22797e6	−1.61493	−0.807464	−	0.589917i	$-0.799161\pi$
$431$	−6.22797e6	−1.61493	−0.807464	+	0.589917i	$0.799161\pi$
$432$	−1.47485e6	−0.380224
$433$	4.07150e6	1.04360	0.521801	−	0.853067i	$-0.325260\pi$
$433$	4.07150e6	1.04360	0.521801	+	0.853067i	$0.325260\pi$
$434$	−230263.	−0.0586814
$435$	0	0
$436$	−2.87081e6	−0.723251
$437$	8.37885e6	2.09885
$438$	1.65752e6	0.412832
$439$	1.75484e6	0.434588	0.217294	−	0.976106i	$-0.430277\pi$
$439$	1.75484e6	0.434588	0.217294	+	0.976106i	$0.430277\pi$
$440$	0	0
$441$	3.80887e6	0.932609
$442$	−3.00801e6	−0.732358
$443$	−4.08504e6	−0.988979	−0.494489	−	0.869184i	$-0.664645\pi$
$443$	−4.08504e6	−0.988979	−0.494489	+	0.869184i	$0.664645\pi$
$444$	−1.54985e6	−0.373106
$445$	0	0
$446$	7.02191e6	1.67154
$447$	1.37826e6	0.326260
$448$	756592.	0.178101
$449$	4.32979e6	1.01356	0.506782	−	0.862074i	$-0.330835\pi$
$449$	4.32979e6	1.01356	0.506782	+	0.862074i	$0.330835\pi$
$450$	0	0
$451$	1.58358e6	0.366604
$452$	−3.12520e6	−0.719502
$453$	−72910.7	−0.0166934
$454$	−6.26826e6	−1.42728
$455$	0	0
$456$	−97905.5	−0.0220493
$457$	4.37096e6	0.979009	0.489504	−	0.872001i	$-0.337178\pi$
$457$	4.37096e6	0.979009	0.489504	+	0.872001i	$0.337178\pi$
$458$	4.99256e6	1.11214
$459$	3.36829e6	0.746238
$460$	0	0
$461$	−3.13672e6	−0.687421	−0.343711	−	0.939076i	$-0.611684\pi$
$461$	−3.13672e6	−0.687421	−0.343711	+	0.939076i	$0.611684\pi$
$462$	−73227.8	−0.0159614
$463$	6.47218e6	1.40313	0.701566	−	0.712605i	$-0.252485\pi$
$463$	6.47218e6	1.40313	0.701566	+	0.712605i	$0.252485\pi$
$464$	−2.44583e6	−0.527389
$465$	0	0
$466$	1.12185e7	2.39316
$467$	2.32552e6	0.493433	0.246717	−	0.969088i	$-0.420648\pi$
$467$	2.32552e6	0.493433	0.246717	+	0.969088i	$0.420648\pi$
$468$	−1.33178e6	−0.281072
$469$	−346110.	−0.0726577
$470$	0	0
$471$	−1.60500e6	−0.333366
$472$	−21994.1	−0.00454413
$473$	893218.	0.183571
$474$	−1.52494e6	−0.311751
$475$	0	0
$476$	−1.53989e6	−0.311510
$477$	−5.64241e6	−1.13545
$478$	9.36721e6	1.87517
$479$	−1.16074e6	−0.231150	−0.115575	−	0.993299i	$-0.536871\pi$
$479$	−1.16074e6	−0.231150	−0.115575	+	0.993299i	$0.536871\pi$
$480$	0	0
$481$	2.39323e6	0.471652
$482$	1.11099e7	2.17818
$483$	−282991.	−0.0551957
$484$	−4.84163e6	−0.939460
$485$	0	0
$486$	4.38129e6	0.841417
$487$	3.63836e6	0.695158	0.347579	−	0.937651i	$-0.387004\pi$
$487$	3.63836e6	0.695158	0.347579	+	0.937651i	$0.387004\pi$
$488$	−249223.	−0.0473739
$489$	−418242.	−0.0790962
$490$	0	0
$491$	8.45015e6	1.58183	0.790917	−	0.611924i	$-0.209604\pi$
$491$	8.45015e6	1.58183	0.790917	+	0.611924i	$0.209604\pi$
$492$	−1.28597e6	−0.239508
$493$	5.58581e6	1.03507
$494$	2.71878e6	0.501253
$495$	0	0
$496$	1.31427e6	0.239872
$497$	−562760.	−0.102196
$498$	−3.17978e6	−0.574545
$499$	−6.28285e6	−1.12955	−0.564775	−	0.825245i	$-0.691037\pi$
$499$	−6.28285e6	−1.12955	−0.564775	+	0.825245i	$0.691037\pi$
$500$	0	0
$501$	1.33514e6	0.237647
$502$	−1.11665e7	−1.97769
$503$	2.30761e6	0.406670	0.203335	−	0.979109i	$-0.434822\pi$
$503$	2.30761e6	0.406670	0.203335	+	0.979109i	$0.434822\pi$
$504$	−73714.8	−0.0129264
$505$	0	0
$506$	4.62462e6	0.802971
$507$	−92250.5	−0.0159385
$508$	−1.11969e7	−1.92503
$509$	−2.24701e6	−0.384423	−0.192212	−	0.981353i	$-0.561566\pi$
$509$	−2.24701e6	−0.384423	−0.192212	+	0.981353i	$0.561566\pi$
$510$	0	0
$511$	−1.31028e6	−0.221979
$512$	−8.42277e6	−1.41997
$513$	−3.04442e6	−0.510753
$514$	4.82793e6	0.806033
$515$	0	0
$516$	−725355.	−0.119930
$517$	−1.99762e6	−0.328690
$518$	2.38221e6	0.390081
$519$	9526.57	0.00155245
$520$	0	0
$521$	−1.20692e7	−1.94797	−0.973987	−	0.226602i	$-0.927238\pi$
$521$	−1.20692e7	−1.94797	−0.973987	+	0.226602i	$0.927238\pi$
$522$	4.80866e6	0.772410
$523$	1.19375e6	0.190835	0.0954174	−	0.995437i	$-0.469581\pi$
$523$	1.19375e6	0.190835	0.0954174	+	0.995437i	$0.469581\pi$
$524$	−2.47724e6	−0.394131
$525$	0	0
$526$	−1.30385e7	−2.05477
$527$	−3.00154e6	−0.470780
$528$	417961.	0.0652456
$529$	1.14357e7	1.77674
$530$	0	0
$531$	−334457.	−0.0514759
$532$	1.39183e6	0.213209
$533$	1.98576e6	0.302767
$534$	−1.30816e6	−0.198522
$535$	0	0
$536$	−255410.	−0.0383995
$537$	1.39308e6	0.208468
$538$	6.20936e6	0.924892
$539$	−2.20722e6	−0.327246
$540$	0	0
$541$	−9.83092e6	−1.44411	−0.722056	−	0.691834i	$-0.756803\pi$
$541$	−9.83092e6	−1.44411	−0.722056	+	0.691834i	$0.756803\pi$
$542$	−3.89186e6	−0.569061
$543$	−210447.	−0.0306297
$544$	1.81629e7	2.63140
$545$	0	0
$546$	−91825.6	−0.0131820
$547$	8.63291e6	1.23364	0.616821	−	0.787104i	$-0.288420\pi$
$547$	8.63291e6	1.23364	0.616821	+	0.787104i	$0.288420\pi$
$548$	−4.66299e6	−0.663305
$549$	−3.78986e6	−0.536651
$550$	0	0
$551$	−5.04873e6	−0.708440
$552$	−208832.	−0.0291708
$553$	1.20548e6	0.167628
$554$	1.77094e6	0.245148
$555$	0	0
$556$	1.24313e7	1.70541
$557$	−3.22690e6	−0.440704	−0.220352	−	0.975420i	$-0.570721\pi$
$557$	−3.22690e6	−0.440704	−0.220352	+	0.975420i	$0.570721\pi$
$558$	−2.58394e6	−0.351315
$559$	1.12007e6	0.151606
$560$	0	0
$561$	−954544.	−0.128053
$562$	6.69529e6	0.894187
$563$	−683340.	−0.0908586	−0.0454293	−	0.998968i	$-0.514466\pi$
$563$	−683340.	−0.0908586	−0.0454293	+	0.998968i	$0.514466\pi$
$564$	1.62221e6	0.214738
$565$	0	0
$566$	1.11942e7	1.46876
$567$	−1.06842e6	−0.139567
$568$	−415285.	−0.0540102
$569$	−5.95105e6	−0.770572	−0.385286	−	0.922797i	$-0.625897\pi$
$569$	−5.95105e6	−0.770572	−0.385286	+	0.922797i	$0.625897\pi$
$570$	0	0
$571$	−2.57116e6	−0.330019	−0.165009	−	0.986292i	$-0.552765\pi$
$571$	−2.57116e6	−0.330019	−0.165009	+	0.986292i	$0.552765\pi$
$572$	771761.	0.0986263
$573$	−1.30903e6	−0.166556
$574$	1.97661e6	0.250404
$575$	0	0
$576$	8.49023e6	1.06626
$577$	1.09233e6	0.136588	0.0682940	−	0.997665i	$-0.478244\pi$
$577$	1.09233e6	0.136588	0.0682940	+	0.997665i	$0.478244\pi$
$578$	−2.75047e7	−3.42443
$579$	1.92726e6	0.238915
$580$	0	0
$581$	2.51364e6	0.308932
$582$	−560928.	−0.0686435
$583$	3.26975e6	0.398422
$584$	−966913.	−0.117315
$585$	0	0
$586$	−1.80507e7	−2.17146
$587$	1.10253e7	1.32067	0.660334	−	0.750972i	$-0.270415\pi$
$587$	1.10253e7	1.32067	0.660334	+	0.750972i	$0.270415\pi$
$588$	1.79242e6	0.213794
$589$	2.71294e6	0.322219
$590$	0	0
$591$	−775689.	−0.0913521
$592$	−1.35969e7	−1.59454
$593$	−1.27015e7	−1.48327	−0.741633	−	0.670806i	$-0.765948\pi$
$593$	−1.27015e7	−1.48327	−0.741633	+	0.670806i	$0.765948\pi$
$594$	−1.68034e6	−0.195403
$595$	0	0
$596$	−1.44589e7	−1.66732
$597$	−1.41766e6	−0.162794
$598$	5.79915e6	0.663149
$599$	−2.40180e6	−0.273508	−0.136754	−	0.990605i	$-0.543667\pi$
$599$	−2.40180e6	−0.273508	−0.136754	+	0.990605i	$0.543667\pi$
$600$	0	0
$601$	−6.52655e6	−0.737051	−0.368526	−	0.929618i	$-0.620137\pi$
$601$	−6.52655e6	−0.737051	−0.368526	+	0.929618i	$0.620137\pi$
$602$	1.11491e6	0.125386
$603$	−3.88393e6	−0.434989
$604$	764880.	0.0853102
$605$	0	0
$606$	−2.83100e6	−0.313154
$607$	−1.25664e7	−1.38433	−0.692164	−	0.721740i	$-0.743343\pi$
$607$	−1.25664e7	−1.38433	−0.692164	+	0.721740i	$0.743343\pi$
$608$	−1.64165e7	−1.80103
$609$	170518.	0.0186306
$610$	0	0
$611$	−2.50496e6	−0.271455
$612$	−1.72802e7	−1.86496
$613$	−5.17046e6	−0.555748	−0.277874	−	0.960617i	$-0.589630\pi$
$613$	−5.17046e6	−0.555748	−0.277874	+	0.960617i	$0.589630\pi$
$614$	2.20458e7	2.35996
$615$	0	0
$616$	42717.4	0.00453579
$617$	7.58899e6	0.802548	0.401274	−	0.915958i	$-0.368567\pi$
$617$	7.58899e6	0.802548	0.401274	+	0.915958i	$0.368567\pi$
$618$	−1.07137e6	−0.112841
$619$	−3.72082e6	−0.390312	−0.195156	−	0.980772i	$-0.562521\pi$
$619$	−3.72082e6	−0.390312	−0.195156	+	0.980772i	$0.562521\pi$
$620$	0	0
$621$	−6.49373e6	−0.675717
$622$	1.68563e7	1.74698
$623$	1.03411e6	0.106745
$624$	524111.	0.0538843
$625$	0	0
$626$	−1.17882e7	−1.20230
$627$	862763.	0.0876441
$628$	1.68374e7	1.70364
$629$	3.10527e7	3.12948
$630$	0	0
$631$	1.80988e7	1.80958	0.904788	−	0.425862i	$-0.140029\pi$
$631$	1.80988e7	1.80958	0.904788	+	0.425862i	$0.140029\pi$
$632$	889575.	0.0885910
$633$	345909.	0.0343125
$634$	1.28888e7	1.27347
$635$	0	0
$636$	−2.65527e6	−0.260295
$637$	−2.76779e6	−0.270262
$638$	−2.78660e6	−0.271033
$639$	−6.31511e6	−0.611827
$640$	0	0
$641$	3.95349e6	0.380046	0.190023	−	0.981780i	$-0.439144\pi$
$641$	3.95349e6	0.380046	0.190023	+	0.981780i	$0.439144\pi$
$642$	−1.95838e6	−0.187526
$643$	2.15047e6	0.205119	0.102559	−	0.994727i	$-0.467297\pi$
$643$	2.15047e6	0.205119	0.102559	+	0.994727i	$0.467297\pi$
$644$	2.96876e6	0.282072
$645$	0	0
$646$	3.52768e7	3.32589
$647$	−7.72603e6	−0.725598	−0.362799	−	0.931867i	$-0.618179\pi$
$647$	−7.72603e6	−0.725598	−0.362799	+	0.931867i	$0.618179\pi$
$648$	−788432.	−0.0737610
$649$	193816.	0.0180625
$650$	0	0
$651$	−91628.2	−0.00847377
$652$	4.38763e6	0.404213
$653$	−7.07138e6	−0.648965	−0.324482	−	0.945892i	$-0.605190\pi$
$653$	−7.07138e6	−0.648965	−0.324482	+	0.945892i	$0.605190\pi$
$654$	−2.22123e6	−0.203072
$655$	0	0
$656$	−1.12819e7	−1.02358
$657$	−1.47035e7	−1.32895
$658$	−2.49342e6	−0.224507
$659$	8.83103e6	0.792133	0.396066	−	0.918222i	$-0.370375\pi$
$659$	8.83103e6	0.792133	0.396066	+	0.918222i	$0.370375\pi$
$660$	0	0
$661$	−8.94111e6	−0.795954	−0.397977	−	0.917395i	$-0.630288\pi$
$661$	−8.94111e6	−0.795954	−0.397977	+	0.917395i	$0.630288\pi$
$662$	2.36500e7	2.09743
$663$	−1.19697e6	−0.105755
$664$	1.85492e6	0.163270
$665$	0	0
$666$	2.67323e7	2.33535
$667$	−1.07689e7	−0.937253
$668$	−1.40064e7	−1.21447
$669$	2.79421e6	0.241376
$670$	0	0
$671$	2.19620e6	0.188307
$672$	554458.	0.0473637
$673$	2.00280e7	1.70451	0.852257	−	0.523123i	$-0.175233\pi$
$673$	2.00280e7	1.70451	0.852257	+	0.523123i	$0.175233\pi$
$674$	−1.94534e7	−1.64948
$675$	0	0
$676$	967766.	0.0814524
$677$	1.44506e6	0.121175	0.0605877	−	0.998163i	$-0.480703\pi$
$677$	1.44506e6	0.121175	0.0605877	+	0.998163i	$0.480703\pi$
$678$	−2.41806e6	−0.202019
$679$	443417.	0.0369095
$680$	0	0
$681$	−2.49432e6	−0.206103
$682$	1.49738e6	0.123274
$683$	−3.37739e6	−0.277032	−0.138516	−	0.990360i	$-0.544233\pi$
$683$	−3.37739e6	−0.277032	−0.138516	+	0.990360i	$0.544233\pi$
$684$	1.56186e7	1.27645
$685$	0	0
$686$	−5.58235e6	−0.452904
$687$	1.98668e6	0.160596
$688$	−6.36356e6	−0.512542
$689$	4.10018e6	0.329044
$690$	0	0
$691$	−1.36877e7	−1.09052	−0.545262	−	0.838265i	$-0.683570\pi$
$691$	−1.36877e7	−1.09052	−0.545262	+	0.838265i	$0.683570\pi$
$692$	−99939.8	−0.00793366
$693$	649590.	0.0513814
$694$	−4.29565e6	−0.338555
$695$	0	0
$696$	125833.	0.00984626
$697$	2.57657e7	2.00891
$698$	6.87620e6	0.534207
$699$	4.46417e6	0.345579
$700$	0	0
$701$	2.12015e7	1.62957	0.814784	−	0.579765i	$-0.196856\pi$
$701$	2.12015e7	1.62957	0.814784	+	0.579765i	$0.196856\pi$
$702$	−2.10710e6	−0.161377
$703$	−2.80669e7	−2.14194
$704$	−4.92005e6	−0.374143
$705$	0	0
$706$	2.79963e7	2.11392
$707$	2.23792e6	0.168382
$708$	−157392.	−0.0118005
$709$	4.77971e6	0.357097	0.178549	−	0.983931i	$-0.442860\pi$
$709$	4.77971e6	0.357097	0.178549	+	0.983931i	$0.442860\pi$
$710$	0	0
$711$	1.35275e7	1.00356
$712$	763115.	0.0564144
$713$	5.78668e6	0.426291
$714$	−1.19146e6	−0.0874648
$715$	0	0
$716$	−1.46143e7	−1.06536
$717$	3.72747e6	0.270780
$718$	1.55971e6	0.112910
$719$	−1.56246e7	−1.12716	−0.563580	−	0.826062i	$-0.690576\pi$
$719$	−1.56246e7	−1.12716	−0.563580	+	0.826062i	$0.690576\pi$
$720$	0	0
$721$	846921.	0.0606743
$722$	−1.17866e7	−0.841487
$723$	4.42095e6	0.314536
$724$	2.20772e6	0.156530
$725$	0	0
$726$	−3.74611e6	−0.263778
$727$	−1.10204e7	−0.773327	−0.386664	−	0.922221i	$-0.626373\pi$
$727$	−1.10204e7	−0.773327	−0.386664	+	0.922221i	$0.626373\pi$
$728$	53566.4	0.00374597
$729$	−1.07838e7	−0.751543
$730$	0	0
$731$	1.45332e7	1.00593
$732$	−1.78347e6	−0.123024
$733$	3.38982e6	0.233033	0.116516	−	0.993189i	$-0.462827\pi$
$733$	3.38982e6	0.233033	0.116516	+	0.993189i	$0.462827\pi$
$734$	−1.09016e7	−0.746880
$735$	0	0
$736$	−3.50162e7	−2.38273
$737$	2.25072e6	0.152635
$738$	2.21809e7	1.49913
$739$	2.83527e7	1.90978	0.954891	−	0.296956i	$-0.0959714\pi$
$739$	2.83527e7	1.90978	0.954891	+	0.296956i	$0.0959714\pi$
$740$	0	0
$741$	1.08188e6	0.0723825
$742$	4.08129e6	0.272137
$743$	2.16151e7	1.43643	0.718216	−	0.695820i	$-0.244959\pi$
$743$	2.16151e7	1.43643	0.718216	+	0.695820i	$0.244959\pi$
$744$	−67616.5	−0.00447837
$745$	0	0
$746$	−3.19927e7	−2.10477
$747$	2.82072e7	1.84952
$748$	1.00138e7	0.654401
$749$	1.54812e6	0.100832
$750$	0	0
$751$	1.11361e7	0.720500	0.360250	−	0.932856i	$-0.382691\pi$
$751$	1.11361e7	0.720500	0.360250	+	0.932856i	$0.382691\pi$
$752$	1.42317e7	0.917721
$753$	−4.44347e6	−0.285585
$754$	−3.49431e6	−0.223838
$755$	0	0
$756$	−1.07869e6	−0.0686422
$757$	4.00933e6	0.254291	0.127146	−	0.991884i	$-0.459418\pi$
$757$	4.00933e6	0.254291	0.127146	+	0.991884i	$0.459418\pi$
$758$	6.24255e6	0.394629
$759$	1.84027e6	0.115952
$760$	0	0
$761$	3.18411e7	1.99309	0.996543	−	0.0830812i	$-0.0264761\pi$
$761$	3.18411e7	1.99309	0.996543	+	0.0830812i	$0.0264761\pi$
$762$	−8.66334e6	−0.540503
$763$	1.75590e6	0.109191
$764$	1.37325e7	0.851170
$765$	0	0
$766$	1.08439e7	0.667752
$767$	243040.	0.0149173
$768$	−2.95351e6	−0.180690
$769$	−2.53683e7	−1.54695	−0.773474	−	0.633828i	$-0.781483\pi$
$769$	−2.53683e7	−1.54695	−0.773474	+	0.633828i	$0.781483\pi$
$770$	0	0
$771$	1.92117e6	0.116394
$772$	−2.02182e7	−1.22095
$773$	−2.50569e7	−1.50827	−0.754134	−	0.656720i	$-0.771943\pi$
$773$	−2.50569e7	−1.50827	−0.754134	+	0.656720i	$0.771943\pi$
$774$	1.25112e7	0.750664
$775$	0	0
$776$	327217.	0.0195066
$777$	947947.	0.0563289
$778$	−1.28534e6	−0.0761325
$779$	−2.32883e7	−1.37497
$780$	0	0
$781$	3.65958e6	0.214686
$782$	7.52453e7	4.40010
$783$	3.91284e6	0.228080
$784$	1.57249e7	0.913690
$785$	0	0
$786$	−1.91671e6	−0.110663
$787$	−2.15591e7	−1.24078	−0.620389	−	0.784294i	$-0.713025\pi$
$787$	−2.15591e7	−1.24078	−0.620389	+	0.784294i	$0.713025\pi$
$788$	8.13747e6	0.466846
$789$	−5.18839e6	−0.296715
$790$	0	0
$791$	1.91149e6	0.108625
$792$	479361.	0.0271550
$793$	2.75398e6	0.155517
$794$	−1.71545e7	−0.965663
$795$	0	0
$796$	1.48722e7	0.831941
$797$	−2.48604e6	−0.138631	−0.0693157	−	0.997595i	$-0.522082\pi$
$797$	−2.48604e6	−0.138631	−0.0693157	+	0.997595i	$0.522082\pi$
$798$	1.07690e6	0.0598642
$799$	−3.25024e7	−1.80114
$800$	0	0
$801$	1.16044e7	0.639062
$802$	1.94686e7	1.06881
$803$	8.52063e6	0.466319
$804$	−1.82774e6	−0.0997183
$805$	0	0
$806$	1.87767e6	0.101808
$807$	2.47088e6	0.133557
$808$	1.65146e6	0.0889897
$809$	−2.67078e7	−1.43472	−0.717360	−	0.696703i	$-0.754650\pi$
$809$	−2.67078e7	−1.43472	−0.717360	+	0.696703i	$0.754650\pi$
$810$	0	0
$811$	3.53377e6	0.188663	0.0943313	−	0.995541i	$-0.469929\pi$
$811$	3.53377e6	0.188663	0.0943313	+	0.995541i	$0.469929\pi$
$812$	−1.78885e6	−0.0952100
$813$	−1.54868e6	−0.0821741
$814$	−1.54913e7	−0.819457
$815$	0	0
$816$	6.80046e6	0.357531
$817$	−1.31358e7	−0.688495
$818$	1.72723e7	0.902540
$819$	814567.	0.0424343
$820$	0	0
$821$	−8.58501e6	−0.444511	−0.222256	−	0.974988i	$-0.571342\pi$
$821$	−8.58501e6	−0.444511	−0.222256	+	0.974988i	$0.571342\pi$
$822$	−3.60789e6	−0.186240
$823$	−1.58052e7	−0.813396	−0.406698	−	0.913563i	$-0.633320\pi$
$823$	−1.58052e7	−0.813396	−0.406698	+	0.913563i	$0.633320\pi$
$824$	624981.	0.0320663
$825$	0	0
$826$	241921.	0.0123374
$827$	−1.82089e7	−0.925808	−0.462904	−	0.886408i	$-0.653192\pi$
$827$	−1.82089e7	−0.925808	−0.462904	+	0.886408i	$0.653192\pi$
$828$	3.33145e7	1.68872
$829$	1.13685e7	0.574534	0.287267	−	0.957850i	$-0.407253\pi$
$829$	1.13685e7	0.574534	0.287267	+	0.957850i	$0.407253\pi$
$830$	0	0
$831$	704705.	0.0354001
$832$	−6.16960e6	−0.308993
$833$	−3.59128e7	−1.79323
$834$	9.61846e6	0.478840
$835$	0	0
$836$	−9.05093e6	−0.447896
$837$	−2.10257e6	−0.103738
$838$	2.72123e7	1.33861
$839$	−2.63011e7	−1.28994	−0.644970	−	0.764208i	$-0.723130\pi$
$839$	−2.63011e7	−1.28994	−0.644970	+	0.764208i	$0.723130\pi$
$840$	0	0
$841$	−1.40223e7	−0.683642
$842$	−1.66602e6	−0.0809841
$843$	2.66424e6	0.129123
$844$	−3.62881e6	−0.175351
$845$	0	0
$846$	−2.79804e7	−1.34409
$847$	2.96132e6	0.141833
$848$	−2.32947e7	−1.11242
$849$	4.45448e6	0.212094
$850$	0	0
$851$	−5.98666e7	−2.83374
$852$	−2.97183e6	−0.140257
$853$	−2.59081e7	−1.21916	−0.609582	−	0.792723i	$-0.708663\pi$
$853$	−2.59081e7	−1.21916	−0.609582	+	0.792723i	$0.708663\pi$
$854$	2.74129e6	0.128621
$855$	0	0
$856$	1.14242e6	0.0532896
$857$	2.09566e7	0.974697	0.487348	−	0.873208i	$-0.337964\pi$
$857$	2.09566e7	0.974697	0.487348	+	0.873208i	$0.337964\pi$
$858$	597133.	0.0276919
$859$	3.29521e7	1.52370	0.761852	−	0.647751i	$-0.224290\pi$
$859$	3.29521e7	1.52370	0.761852	+	0.647751i	$0.224290\pi$
$860$	0	0
$861$	786550.	0.0361592
$862$	5.05518e7	2.31723
$863$	2.33805e7	1.06863	0.534314	−	0.845286i	$-0.320570\pi$
$863$	2.33805e7	1.06863	0.534314	+	0.845286i	$0.320570\pi$
$864$	1.27230e7	0.579836
$865$	0	0
$866$	−3.30480e7	−1.49744
$867$	−1.09449e7	−0.494498
$868$	961238.	0.0433044
$869$	−7.83911e6	−0.352142
$870$	0	0
$871$	2.82234e6	0.126056
$872$	1.29575e6	0.0577074
$873$	4.97588e6	0.220971
$874$	−6.80103e7	−3.01159
$875$	0	0
$876$	−6.91935e6	−0.304653
$877$	−6.24910e6	−0.274359	−0.137179	−	0.990546i	$-0.543804\pi$
$877$	−6.24910e6	−0.274359	−0.137179	+	0.990546i	$0.543804\pi$
$878$	−1.42439e7	−0.623581
$879$	−7.18290e6	−0.313565
$880$	0	0
$881$	8.57115e6	0.372048	0.186024	−	0.982545i	$-0.440440\pi$
$881$	8.57115e6	0.372048	0.186024	+	0.982545i	$0.440440\pi$
$882$	−3.09162e7	−1.33818
$883$	2.98999e7	1.29053	0.645265	−	0.763958i	$-0.276747\pi$
$883$	2.98999e7	1.29053	0.645265	+	0.763958i	$0.276747\pi$
$884$	1.25570e7	0.540449
$885$	0	0
$886$	3.31579e7	1.41907
$887$	−3.63685e7	−1.55209	−0.776043	−	0.630680i	$-0.782776\pi$
$887$	−3.63685e7	−1.55209	−0.776043	+	0.630680i	$0.782776\pi$
$888$	699531.	0.0297697
$889$	6.84843e6	0.290627
$890$	0	0
$891$	6.94782e6	0.293194
$892$	−2.93131e7	−1.23353
$893$	2.93772e7	1.23277
$894$	−1.11872e7	−0.468144
$895$	0	0
$896$	−648011.	−0.0269657
$897$	2.30764e6	0.0957608
$898$	−3.51445e7	−1.45434
$899$	−3.48680e6	−0.143889
$900$	0	0
$901$	5.32007e7	2.18326
$902$	−1.28537e7	−0.526033
$903$	443654.	0.0181061
$904$	1.41057e6	0.0574083
$905$	0	0
$906$	591809.	0.0239531
$907$	−2.51577e7	−1.01544	−0.507718	−	0.861523i	$-0.669511\pi$
$907$	−2.51577e7	−1.01544	−0.507718	+	0.861523i	$0.669511\pi$
$908$	2.61670e7	1.05327
$909$	2.51132e7	1.00807
$910$	0	0
$911$	−4.68717e7	−1.87118	−0.935588	−	0.353092i	$-0.885130\pi$
$911$	−4.68717e7	−1.87118	−0.935588	+	0.353092i	$0.885130\pi$
$912$	−6.14659e6	−0.244707
$913$	−1.63460e7	−0.648983
$914$	−3.54787e7	−1.40476
$915$	0	0
$916$	−2.08415e7	−0.820712
$917$	1.51517e6	0.0595031
$918$	−2.73401e7	−1.07076
$919$	−879544.	−0.0343533	−0.0171767	−	0.999852i	$-0.505468\pi$
$919$	−879544.	−0.0343533	−0.0171767	+	0.999852i	$0.505468\pi$
$920$	0	0
$921$	8.77264e6	0.340785
$922$	2.54604e7	0.986367
$923$	4.58901e6	0.177302
$924$	305691.	0.0117788
$925$	0	0
$926$	−5.25341e7	−2.01332
$927$	9.50388e6	0.363247
$928$	2.10992e7	0.804261
$929$	2.05492e7	0.781187	0.390593	−	0.920563i	$-0.372270\pi$
$929$	2.05492e7	0.781187	0.390593	+	0.920563i	$0.372270\pi$
$930$	0	0
$931$	3.24597e7	1.22736
$932$	−4.68320e7	−1.76605
$933$	6.70761e6	0.252269
$934$	−1.88761e7	−0.708018
$935$	0	0
$936$	601105.	0.0224265
$937$	−3.29842e7	−1.22732	−0.613659	−	0.789571i	$-0.710303\pi$
$937$	−3.29842e7	−1.22732	−0.613659	+	0.789571i	$0.710303\pi$
$938$	2.80934e6	0.104255
$939$	−4.69087e6	−0.173616
$940$	0	0
$941$	1.67975e7	0.618400	0.309200	−	0.950997i	$-0.399939\pi$
$941$	1.67975e7	0.618400	0.309200	+	0.950997i	$0.399939\pi$
$942$	1.30276e7	0.478341
$943$	−4.96737e7	−1.81906
$944$	−1.38081e6	−0.0504316
$945$	0	0
$946$	−7.25017e6	−0.263403
$947$	1.07215e7	0.388492	0.194246	−	0.980953i	$-0.437774\pi$
$947$	1.07215e7	0.388492	0.194246	+	0.980953i	$0.437774\pi$
$948$	6.36590e6	0.230059
$949$	1.06846e7	0.385118
$950$	0	0
$951$	5.12881e6	0.183893
$952$	695036.	0.0248551
$953$	1.79930e7	0.641758	0.320879	−	0.947120i	$-0.396022\pi$
$953$	1.79930e7	0.641758	0.320879	+	0.947120i	$0.396022\pi$
$954$	4.57989e7	1.62924
$955$	0	0
$956$	−3.91036e7	−1.38380
$957$	−1.10886e6	−0.0391380
$958$	9.42158e6	0.331673
$959$	2.85206e6	0.100141
$960$	0	0
$961$	−2.67555e7	−0.934555
$962$	−1.94256e7	−0.676764
$963$	1.73725e7	0.603664
$964$	−4.63786e7	−1.60740
$965$	0	0
$966$	2.29702e6	0.0791992
$967$	−4.83225e7	−1.66182	−0.830908	−	0.556409i	$-0.812179\pi$
$967$	−4.83225e7	−1.66182	−0.830908	+	0.556409i	$0.812179\pi$
$968$	2.18529e6	0.0749585
$969$	1.40376e7	0.480269
$970$	0	0
$971$	4.13992e7	1.40911	0.704553	−	0.709652i	$-0.251148\pi$
$971$	4.13992e7	1.40911	0.704553	+	0.709652i	$0.251148\pi$
$972$	−1.82898e7	−0.620931
$973$	−7.60345e6	−0.257471
$974$	−2.95322e7	−0.997468
$975$	0	0
$976$	−1.56464e7	−0.525764
$977$	−2.77983e7	−0.931713	−0.465856	−	0.884860i	$-0.654254\pi$
$977$	−2.77983e7	−0.931713	−0.465856	+	0.884860i	$0.654254\pi$
$978$	3.39483e6	0.113494
$979$	−6.72472e6	−0.224242
$980$	0	0
$981$	1.97041e7	0.653709
$982$	−6.85891e7	−2.26974
$983$	2.08694e7	0.688853	0.344426	−	0.938813i	$-0.388073\pi$
$983$	2.08694e7	0.688853	0.344426	+	0.938813i	$0.388073\pi$
$984$	580430.	0.0191100
$985$	0	0
$986$	−4.53395e7	−1.48520
$987$	−992202.	−0.0324196
$988$	−1.13496e7	−0.369904
$989$	−2.80185e7	−0.910866
$990$	0	0
$991$	5.53538e7	1.79045	0.895227	−	0.445611i	$-0.147014\pi$
$991$	5.53538e7	1.79045	0.895227	+	0.445611i	$0.147014\pi$
$992$	−1.13377e7	−0.365802
$993$	9.41101e6	0.302875
$994$	4.56787e6	0.146638
$995$	0	0
$996$	1.32741e7	0.423990
$997$	−3.35323e6	−0.106838	−0.0534189	−	0.998572i	$-0.517012\pi$
$997$	−3.35323e6	−0.106838	−0.0534189	+	0.998572i	$0.517012\pi$
$998$	5.09973e7	1.62077
$999$	2.17523e7	0.689590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.h.1.2	✓	9
5.2	odd	4		325.6.b.h.274.3		18
5.3	odd	4		325.6.b.h.274.16		18
5.4	even	2		325.6.a.i.1.8	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.2	✓	9	1.1	even	1	trivial
325.6.a.i.1.8	yes	9	5.4	even	2
325.6.b.h.274.3		18	5.2	odd	4
325.6.b.h.274.16		18	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)

\(f(q)\)	\(=\)	\(q-8.11691 q^{2} -3.22994 q^{3} +33.8842 q^{4} +26.2172 q^{6} -20.7248 q^{7} -15.2938 q^{8} -232.567 q^{9} +134.772 q^{11} -109.444 q^{12} +169.000 q^{13} +168.222 q^{14} -960.156 q^{16} +2192.81 q^{17} +1887.73 q^{18} -1981.97 q^{19} +66.9401 q^{21} -1093.93 q^{22} -4227.53 q^{23} +49.3980 q^{24} -1371.76 q^{26} +1536.06 q^{27} -702.244 q^{28} +2547.33 q^{29} -1368.81 q^{31} +8282.90 q^{32} -435.305 q^{33} -17798.9 q^{34} -7880.36 q^{36} +14161.1 q^{37} +16087.5 q^{38} -545.861 q^{39} +11750.1 q^{41} -543.347 q^{42} +6627.63 q^{43} +4566.63 q^{44} +34314.5 q^{46} -14822.2 q^{47} +3101.25 q^{48} -16377.5 q^{49} -7082.67 q^{51} +5726.43 q^{52} +24261.4 q^{53} -12468.0 q^{54} +316.961 q^{56} +6401.66 q^{57} -20676.4 q^{58} +1438.11 q^{59} +16295.7 q^{61} +11110.5 q^{62} +4819.92 q^{63} -36506.5 q^{64} +3533.33 q^{66} +16700.2 q^{67} +74301.7 q^{68} +13654.7 q^{69} +27153.9 q^{71} +3556.83 q^{72} +63222.7 q^{73} -114944. q^{74} -67157.5 q^{76} -2793.12 q^{77} +4430.70 q^{78} -58165.8 q^{79} +51552.5 q^{81} -95374.1 q^{82} -121286. q^{83} +2268.21 q^{84} -53795.9 q^{86} -8227.72 q^{87} -2061.17 q^{88} -49897.1 q^{89} -3502.50 q^{91} -143246. q^{92} +4421.18 q^{93} +120311. q^{94} -26753.3 q^{96} -21395.4 q^{97} +132934. q^{98} -31343.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9 - 1422 * q^11 + 1567 * q^12 + 1521 * q^13 - 342 * q^14 - 1061 * q^16 + 648 * q^17 + 418 * q^18 - 408 * q^19 - 3912 * q^21 + 4345 * q^22 + 1839 * q^23 - 8469 * q^24 - 845 * q^26 - 7649 * q^27 - 2836 * q^28 - 8737 * q^29 + 748 * q^31 - 423 * q^32 - 356 * q^33 - 17789 * q^34 + 512 * q^36 - 15486 * q^37 - 3425 * q^38 - 1859 * q^39 - 28676 * q^41 + 6876 * q^42 + 28665 * q^43 - 30599 * q^44 - 12056 * q^46 - 29452 * q^47 + 64759 * q^48 - 40907 * q^49 - 31006 * q^51 + 15379 * q^52 + 75977 * q^53 - 102761 * q^54 - 23002 * q^56 - 38038 * q^57 + 142384 * q^58 - 88142 * q^59 + 28165 * q^61 - 137308 * q^62 + 41492 * q^63 - 100845 * q^64 + 42577 * q^66 - 94754 * q^67 + 89267 * q^68 - 181747 * q^69 - 70562 * q^71 - 263778 * q^72 + 60602 * q^73 - 135676 * q^74 + 46373 * q^76 - 140292 * q^77 - 14027 * q^78 - 164073 * q^79 - 69935 * q^81 - 72887 * q^82 - 22458 * q^83 - 345656 * q^84 - 294920 * q^86 - 87031 * q^87 + 430607 * q^88 - 252698 * q^89 + 2028 * q^91 - 237824 * q^92 + 56556 * q^93 - 501606 * q^94 - 319181 * q^96 + 137986 * q^97 + 378699 * q^98 - 757776 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more