LMFDB - Embedded newform 3100.3.d.g.1301.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.6
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.g.1301.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.39902i q^{3} +1.77923 q^{7} -2.55337 q^{9} +O(q^{10})$	$q-3.39902i q^{3} +1.77923 q^{7} -2.55337 q^{9} +7.22505i q^{11} +15.0852i q^{13} +10.2180i q^{17} -4.10670 q^{19} -6.04766i q^{21} -22.1968i q^{23} -21.9123i q^{27} -23.4213i q^{29} +(15.2061 + 27.0143i) q^{31} +24.5581 q^{33} +14.2189i q^{37} +51.2749 q^{39} -59.1342 q^{41} +45.7674i q^{43} +0.904686 q^{47} -45.8343 q^{49} +34.7313 q^{51} -3.44549i q^{53} +13.9588i q^{57} -95.8482 q^{59} -28.0564i q^{61} -4.54304 q^{63} -96.7221 q^{67} -75.4473 q^{69} -36.5041 q^{71} +50.7150i q^{73} +12.8551i q^{77} -102.733i q^{79} -97.4606 q^{81} +49.3268i q^{83} -79.6095 q^{87} -151.619i q^{89} +26.8401i q^{91} +(91.8224 - 51.6860i) q^{93} -99.2776 q^{97} -18.4482i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{7} - 72 q^{9}+O(q^{10}) $ 22 * q + 4 * q^7 - 72 * q^9	$ 22 q + 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} - 34 q^{33} - 62 q^{39} + 64 q^{41} - 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} - 4 q^{63} + 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} - 62 q^{87} + 16 q^{93} + 186 q^{97}+O(q^{100}) $ 22 * q + 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 - 34 * q^33 - 62 * q^39 + 64 * q^41 - 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 - 4 * q^63 + 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 - 62 * q^87 + 16 * q^93 + 186 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	3.39902i		−	1.13301i	−0.824059	−	0.566504i	$-0.808296\pi$
$3$		−	3.39902i		−	1.13301i	0.824059	−	0.566504i	$-0.191704\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.77923			0.254176			0.127088	−	0.991891i	$-0.459437\pi$
$7$	1.77923			0.254176			0.127088	+	0.991891i	$0.459437\pi$
$8$		0			0
$9$	−2.55337			−0.283708
$10$		0			0
$11$			7.22505i			0.656823i	0.944535	+	0.328412i	$0.106513\pi$
$11$			7.22505i			0.656823i	−0.944535	+	0.328412i	$0.893487\pi$
$12$		0			0
$13$			15.0852i			1.16040i	0.814475	+	0.580199i	$0.197025\pi$
$13$			15.0852i			1.16040i	−0.814475	+	0.580199i	$0.802975\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			10.2180i			0.601060i	0.953772	+	0.300530i	$0.0971636\pi$
$17$			10.2180i			0.601060i	−0.953772	+	0.300530i	$0.902836\pi$
$18$		0			0
$19$	−4.10670			−0.216142			−0.108071	−	0.994143i	$-0.534467\pi$
$19$	−4.10670			−0.216142			−0.108071	+	0.994143i	$0.534467\pi$
$20$		0			0
$21$		−	6.04766i		−	0.287984i
$22$		0			0
$23$		−	22.1968i		−	0.965077i	−0.875875	−	0.482538i	$-0.839715\pi$
$23$		−	22.1968i		−	0.965077i	0.875875	−	0.482538i	$-0.160285\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	21.9123i		−	0.811565i
$28$		0			0
$29$		−	23.4213i		−	0.807630i	−0.914841	−	0.403815i	$-0.867684\pi$
$29$		−	23.4213i		−	0.807630i	0.914841	−	0.403815i	$-0.132316\pi$
$30$		0			0
$31$	15.2061	+	27.0143i	0.490520	+	0.871430i
$31$	15.2061	+	27.0143i	0.490520	+	0.871430i
$32$		0			0
$33$	24.5581			0.744186
$34$		0			0
$35$		0			0
$36$		0			0
$37$			14.2189i			0.384294i	0.981366	+	0.192147i	$0.0615450\pi$
$37$			14.2189i			0.384294i	−0.981366	+	0.192147i	$0.938455\pi$
$38$		0			0
$39$	51.2749			1.31474
$40$		0			0
$41$	−59.1342			−1.44230			−0.721149	−	0.692780i	$-0.756386\pi$
$41$	−59.1342			−1.44230			−0.721149	+	0.692780i	$0.756386\pi$
$42$		0			0
$43$			45.7674i			1.06436i	0.846632	+	0.532179i	$0.178627\pi$
$43$			45.7674i			1.06436i	−0.846632	+	0.532179i	$0.821373\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.904686			0.0192486			0.00962432	−	0.999954i	$-0.496936\pi$
$47$	0.904686			0.0192486			0.00962432	+	0.999954i	$0.496936\pi$
$48$		0			0
$49$	−45.8343			−0.935394
$50$		0			0
$51$	34.7313			0.681006
$52$		0			0
$53$		−	3.44549i		−	0.0650092i	−0.999472	−	0.0325046i	$-0.989652\pi$
$53$		−	3.44549i		−	0.0650092i	0.999472	−	0.0325046i	$-0.0103484\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			13.9588i			0.244891i
$58$		0			0
$59$	−95.8482			−1.62455			−0.812273	−	0.583278i	$-0.801770\pi$
$59$	−95.8482			−1.62455			−0.812273	+	0.583278i	$0.801770\pi$
$60$		0			0
$61$		−	28.0564i		−	0.459941i	−0.973198	−	0.229971i	$-0.926137\pi$
$61$		−	28.0564i		−	0.459941i	0.973198	−	0.229971i	$-0.0738630\pi$
$62$		0			0
$63$	−4.54304			−0.0721118
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−96.7221			−1.44361			−0.721807	−	0.692095i	$-0.756688\pi$
$67$	−96.7221			−1.44361			−0.721807	+	0.692095i	$0.756688\pi$
$68$		0			0
$69$	−75.4473			−1.09344
$70$		0			0
$71$	−36.5041			−0.514142			−0.257071	−	0.966393i	$-0.582757\pi$
$71$	−36.5041			−0.514142			−0.257071	+	0.966393i	$0.582757\pi$
$72$		0			0
$73$			50.7150i			0.694726i	0.937731	+	0.347363i	$0.112923\pi$
$73$			50.7150i			0.694726i	−0.937731	+	0.347363i	$0.887077\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			12.8551i			0.166949i
$78$		0			0
$79$		−	102.733i		−	1.30041i	−0.759758	−	0.650206i	$-0.774683\pi$
$79$		−	102.733i		−	1.30041i	0.759758	−	0.650206i	$-0.225317\pi$
$80$		0			0
$81$	−97.4606			−1.20322
$82$		0			0
$83$			49.3268i			0.594299i	0.954831	+	0.297150i	$0.0960360\pi$
$83$			49.3268i			0.594299i	−0.954831	+	0.297150i	$0.903964\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−79.6095			−0.915052
$88$		0			0
$89$		−	151.619i		−	1.70358i	−0.523880	−	0.851792i	$-0.675516\pi$
$89$		−	151.619i		−	1.70358i	0.523880	−	0.851792i	$-0.324484\pi$
$90$		0			0
$91$			26.8401i			0.294946i
$92$		0			0
$93$	91.8224	−	51.6860i	0.987337	−	0.555764i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−99.2776			−1.02348			−0.511740	−	0.859140i	$-0.670999\pi$
$97$	−99.2776			−1.02348			−0.511740	+	0.859140i	$0.670999\pi$
$98$		0			0
$99$		−	18.4482i		−	0.186346i
$100$		0			0
$101$	−189.989			−1.88108			−0.940538	−	0.339688i	$-0.889678\pi$
$101$	−189.989			−1.88108			−0.940538	+	0.339688i	$0.889678\pi$
$102$		0			0
$103$	131.541			1.27710			0.638550	−	0.769580i	$-0.279535\pi$
$103$	131.541			1.27710			0.638550	+	0.769580i	$0.279535\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−114.581			−1.07085			−0.535425	−	0.844583i	$-0.679848\pi$
$107$	−114.581			−1.07085			−0.535425	+	0.844583i	$0.679848\pi$
$108$		0			0
$109$	−89.7590			−0.823477			−0.411738	−	0.911302i	$-0.635078\pi$
$109$	−89.7590			−0.823477			−0.411738	+	0.911302i	$0.635078\pi$
$110$		0			0
$111$	48.3303			0.435408
$112$		0			0
$113$	−114.416			−1.01253			−0.506267	−	0.862377i	$-0.668975\pi$
$113$	−114.416			−1.01253			−0.506267	+	0.862377i	$0.668975\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	38.5180i		−	0.329214i
$118$		0			0
$119$			18.1803i			0.152775i
$120$		0			0
$121$	68.7986			0.568583
$122$		0			0
$123$			200.999i			1.63414i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			30.0650i			0.236732i	0.992970	+	0.118366i	$0.0377656\pi$
$127$			30.0650i			0.236732i	−0.992970	+	0.118366i	$0.962234\pi$
$128$		0			0
$129$	155.565			1.20593
$130$		0			0
$131$	110.690			0.844960			0.422480	−	0.906372i	$-0.361160\pi$
$131$	110.690			0.844960			0.422480	+	0.906372i	$0.361160\pi$
$132$		0			0
$133$	−7.30677			−0.0549382
$134$		0			0
$135$		0			0
$136$		0			0
$137$			50.7339i			0.370320i	0.982708	+	0.185160i	$0.0592804\pi$
$137$			50.7339i			0.370320i	−0.982708	+	0.185160i	$0.940720\pi$
$138$		0			0
$139$			73.6616i			0.529939i	0.964257	+	0.264970i	$0.0853619\pi$
$139$			73.6616i			0.529939i	−0.964257	+	0.264970i	$0.914638\pi$
$140$		0			0
$141$		−	3.07505i		−	0.0218089i
$142$		0			0
$143$	−108.991			−0.762176
$144$		0			0
$145$		0			0
$146$		0			0
$147$			155.792i			1.05981i
$148$		0			0
$149$	118.412			0.794714			0.397357	−	0.917664i	$-0.369927\pi$
$149$	118.412			0.794714			0.397357	+	0.917664i	$0.369927\pi$
$150$		0			0
$151$			98.3653i			0.651426i	0.945469	+	0.325713i	$0.105604\pi$
$151$			98.3653i			0.651426i	−0.945469	+	0.325713i	$0.894396\pi$
$152$		0			0
$153$		−	26.0904i		−	0.170526i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	15.1040			0.0962036			0.0481018	−	0.998842i	$-0.484683\pi$
$157$	15.1040			0.0962036			0.0481018	+	0.998842i	$0.484683\pi$
$158$		0			0
$159$	−11.7113			−0.0736559
$160$		0			0
$161$		−	39.4932i		−	0.245300i
$162$		0			0
$163$	120.406			0.738690			0.369345	−	0.929292i	$-0.379582\pi$
$163$	120.406			0.738690			0.369345	+	0.929292i	$0.379582\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			73.5726i			0.440554i	0.975437	+	0.220277i	$0.0706962\pi$
$167$			73.5726i			0.440554i	−0.975437	+	0.220277i	$0.929304\pi$
$168$		0			0
$169$	−58.5626			−0.346525
$170$		0			0
$171$	10.4859			0.0613212
$172$		0			0
$173$	161.701			0.934691			0.467345	−	0.884075i	$-0.345210\pi$
$173$	161.701			0.934691			0.467345	+	0.884075i	$0.345210\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			325.790i			1.84062i
$178$		0			0
$179$		−	287.271i		−	1.60486i	−0.596744	−	0.802432i	$-0.703539\pi$
$179$		−	287.271i		−	1.60486i	0.596744	−	0.802432i	$-0.296461\pi$
$180$		0			0
$181$			220.494i			1.21820i	0.793094	+	0.609100i	$0.208469\pi$
$181$			220.494i			1.21820i	−0.793094	+	0.609100i	$0.791531\pi$
$182$		0			0
$183$	−95.3644			−0.521117
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−73.8258			−0.394790
$188$		0			0
$189$		−	38.9870i		−	0.206281i
$190$		0			0
$191$	−99.5606			−0.521260			−0.260630	−	0.965439i	$-0.583930\pi$
$191$	−99.5606			−0.521260			−0.260630	+	0.965439i	$0.583930\pi$
$192$		0			0
$193$	−377.949			−1.95828			−0.979142	−	0.203175i	$-0.934874\pi$
$193$	−377.949			−1.95828			−0.979142	+	0.203175i	$0.934874\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			221.273i			1.12321i	0.827404	+	0.561607i	$0.189817\pi$
$197$			221.273i			1.12321i	−0.827404	+	0.561607i	$0.810183\pi$
$198$		0			0
$199$		−	111.162i		−	0.558603i	−0.960203	−	0.279302i	$-0.909897\pi$
$199$		−	111.162i		−	0.558603i	0.960203	−	0.279302i	$-0.0901029\pi$
$200$		0			0
$201$			328.761i			1.63563i
$202$		0			0
$203$		−	41.6719i		−	0.205280i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			56.6765i			0.273800i
$208$		0			0
$209$		−	29.6711i		−	0.141967i
$210$		0			0
$211$	−125.547			−0.595009			−0.297505	−	0.954720i	$-0.596154\pi$
$211$	−125.547			−0.595009			−0.297505	+	0.954720i	$0.596154\pi$
$212$		0			0
$213$			124.078i			0.582527i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	27.0553	+	48.0648i	0.124679	+	0.221497i
$218$		0			0
$219$	172.382			0.787131
$220$		0			0
$221$	−154.141			−0.697470
$222$		0			0
$223$		−	103.811i		−	0.465518i	−0.972534	−	0.232759i	$-0.925225\pi$
$223$		−	103.811i		−	0.465518i	0.972534	−	0.232759i	$-0.0747754\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	363.058			1.59937			0.799687	−	0.600417i	$-0.204999\pi$
$227$	363.058			1.59937			0.799687	+	0.600417i	$0.204999\pi$
$228$		0			0
$229$		−	318.829i		−	1.39227i	−0.717913	−	0.696133i	$-0.754902\pi$
$229$		−	318.829i		−	1.39227i	0.717913	−	0.696133i	$-0.245098\pi$
$230$		0			0
$231$	43.6947			0.189154
$232$		0			0
$233$	−57.6489			−0.247420			−0.123710	−	0.992318i	$-0.539479\pi$
$233$	−57.6489			−0.247420			−0.123710	+	0.992318i	$0.539479\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−349.191			−1.47338
$238$		0			0
$239$			189.319i			0.792131i	0.918222	+	0.396066i	$0.129625\pi$
$239$			189.319i			0.792131i	−0.918222	+	0.396066i	$0.870375\pi$
$240$		0			0
$241$		−	290.177i		−	1.20405i	−0.798476	−	0.602027i	$-0.794360\pi$
$241$		−	290.177i		−	1.20405i	0.798476	−	0.602027i	$-0.205640\pi$
$242$		0			0
$243$			134.061i			0.551691i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	61.9503i		−	0.250811i
$248$		0			0
$249$	167.663			0.673346
$250$		0			0
$251$			35.9850i			0.143367i	0.997427	+	0.0716833i	$0.0228371\pi$
$251$			35.9850i			0.143367i	−0.997427	+	0.0716833i	$0.977163\pi$
$252$		0			0
$253$	160.373			0.633885
$254$		0			0
$255$		0			0
$256$		0			0
$257$	140.107			0.545164			0.272582	−	0.962133i	$-0.412122\pi$
$257$	140.107			0.545164			0.272582	+	0.962133i	$0.412122\pi$
$258$		0			0
$259$			25.2987i			0.0976783i
$260$		0			0
$261$			59.8032i			0.229131i
$262$		0			0
$263$			35.3151i			0.134278i	0.997744	+	0.0671390i	$0.0213871\pi$
$263$			35.3151i			0.134278i	−0.997744	+	0.0671390i	$0.978613\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−515.357			−1.93018
$268$		0			0
$269$			233.760i			0.868997i	0.900673	+	0.434498i	$0.143074\pi$
$269$			233.760i			0.868997i	−0.900673	+	0.434498i	$0.856926\pi$
$270$		0			0
$271$			71.8362i			0.265078i	0.991178	+	0.132539i	$0.0423130\pi$
$271$			71.8362i			0.265078i	−0.991178	+	0.132539i	$0.957687\pi$
$272$		0			0
$273$	91.2300			0.334176
$274$		0			0
$275$		0			0
$276$		0			0
$277$			373.787i			1.34941i	0.738086	+	0.674706i	$0.235730\pi$
$277$			373.787i			1.34941i	−0.738086	+	0.674706i	$0.764270\pi$
$278$		0			0
$279$	−38.8269	−	68.9776i	−0.139164	−	0.247231i
$280$		0			0
$281$	−126.774			−0.451153			−0.225577	−	0.974225i	$-0.572427\pi$
$281$	−126.774			−0.451153			−0.225577	+	0.974225i	$0.572427\pi$
$282$		0			0
$283$	−187.418			−0.662254			−0.331127	−	0.943586i	$-0.607429\pi$
$283$	−187.418			−0.662254			−0.331127	+	0.943586i	$0.607429\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−105.214			−0.366598
$288$		0			0
$289$	184.592			0.638726
$290$		0			0
$291$			337.447i			1.15961i
$292$		0			0
$293$	−175.832			−0.600110			−0.300055	−	0.953922i	$-0.597005\pi$
$293$	−175.832			−0.600110			−0.300055	+	0.953922i	$0.597005\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	158.317			0.533055
$298$		0			0
$299$	334.842			1.11987
$300$		0			0
$301$			81.4310i			0.270535i
$302$		0			0
$303$			645.776i			2.13127i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−237.041			−0.772119			−0.386060	−	0.922474i	$-0.626164\pi$
$307$	−237.041			−0.772119			−0.386060	+	0.922474i	$0.626164\pi$
$308$		0			0
$309$		−	447.112i		−	1.44696i
$310$		0			0
$311$	121.194			0.389691			0.194845	−	0.980834i	$-0.437579\pi$
$311$	121.194			0.389691			0.194845	+	0.980834i	$0.437579\pi$
$312$		0			0
$313$		−	384.617i		−	1.22881i	−0.788991	−	0.614405i	$-0.789396\pi$
$313$		−	384.617i		−	1.22881i	0.788991	−	0.614405i	$-0.210604\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−28.4510			−0.0897509			−0.0448755	−	0.998993i	$-0.514289\pi$
$317$	−28.4510			−0.0897509			−0.0448755	+	0.998993i	$0.514289\pi$
$318$		0			0
$319$	169.220			0.530470
$320$		0			0
$321$			389.463i			1.21328i
$322$		0			0
$323$		−	41.9623i		−	0.129914i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			305.093i			0.933006i
$328$		0			0
$329$	1.60965			0.00489255
$330$		0			0
$331$			292.440i			0.883505i	0.897137	+	0.441752i	$0.145643\pi$
$331$			292.440i			0.883505i	−0.897137	+	0.441752i	$0.854357\pi$
$332$		0			0
$333$		−	36.3060i		−	0.109027i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	113.017i		−	0.335362i	−0.985841	−	0.167681i	$-0.946372\pi$
$337$		−	113.017i		−	0.335362i	0.985841	−	0.167681i	$-0.0536278\pi$
$338$		0			0
$339$			388.904i			1.14721i
$340$		0			0
$341$	−195.180	+	109.865i	−0.572375	+	0.322185i
$342$		0			0
$343$	−168.732			−0.491931
$344$		0			0
$345$		0			0
$346$		0			0
$347$			659.919i			1.90178i	0.309522	+	0.950892i	$0.399831\pi$
$347$			659.919i			1.90178i	−0.309522	+	0.950892i	$0.600169\pi$
$348$		0			0
$349$	231.778			0.664120			0.332060	−	0.943258i	$-0.392256\pi$
$349$	231.778			0.664120			0.332060	+	0.943258i	$0.392256\pi$
$350$		0			0
$351$	330.550			0.941739
$352$		0			0
$353$		−	107.649i		−	0.304954i	−0.988307	−	0.152477i	$-0.951275\pi$
$353$		−	107.649i		−	0.304954i	0.988307	−	0.152477i	$-0.0487250\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	61.7952			0.173096
$358$		0			0
$359$	219.833			0.612349			0.306174	−	0.951975i	$-0.400951\pi$
$359$	219.833			0.612349			0.306174	+	0.951975i	$0.400951\pi$
$360$		0			0
$361$	−344.135			−0.953283
$362$		0			0
$363$		−	233.848i		−	0.644210i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			504.235i			1.37394i	0.726686	+	0.686969i	$0.241059\pi$
$367$			504.235i			1.37394i	−0.726686	+	0.686969i	$0.758941\pi$
$368$		0			0
$369$	150.992			0.409191
$370$		0			0
$371$		−	6.13033i		−	0.0165238i
$372$		0			0
$373$	−680.738			−1.82503			−0.912517	−	0.409038i	$-0.865864\pi$
$373$	−680.738			−1.82503			−0.912517	+	0.409038i	$0.865864\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	353.314			0.937173
$378$		0			0
$379$	313.385			0.826872			0.413436	−	0.910533i	$-0.364329\pi$
$379$	313.385			0.826872			0.413436	+	0.910533i	$0.364329\pi$
$380$		0			0
$381$	102.192			0.268219
$382$		0			0
$383$			326.285i			0.851919i	0.904742	+	0.425960i	$0.140063\pi$
$383$			326.285i			0.851919i	−0.904742	+	0.425960i	$0.859937\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	116.861i		−	0.301967i
$388$		0			0
$389$		−	513.584i		−	1.32027i	−0.751148	−	0.660134i	$-0.770500\pi$
$389$		−	513.584i		−	1.32027i	0.751148	−	0.660134i	$-0.229500\pi$
$390$		0			0
$391$	226.807			0.580069
$392$		0			0
$393$		−	376.237i		−	0.957346i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	53.6063			0.135029			0.0675143	−	0.997718i	$-0.478493\pi$
$397$	53.6063			0.135029			0.0675143	+	0.997718i	$0.478493\pi$
$398$		0			0
$399$			24.8359i			0.0622454i
$400$		0			0
$401$		−	242.604i		−	0.604998i	−0.953150	−	0.302499i	$-0.902179\pi$
$401$		−	242.604i		−	0.604998i	0.953150	−	0.302499i	$-0.0978210\pi$
$402$		0			0
$403$	−407.516	+	229.387i	−1.01121	+	0.569199i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−102.732			−0.252413
$408$		0			0
$409$			430.558i			1.05271i	0.850265	+	0.526354i	$0.176441\pi$
$409$			430.558i			1.05271i	−0.850265	+	0.526354i	$0.823559\pi$
$410$		0			0
$411$	172.446			0.419576
$412$		0			0
$413$	−170.536			−0.412921
$414$		0			0
$415$		0			0
$416$		0			0
$417$	250.377			0.600426
$418$		0			0
$419$	476.030			1.13611			0.568055	−	0.822991i	$-0.307696\pi$
$419$	476.030			1.13611			0.568055	+	0.822991i	$0.307696\pi$
$420$		0			0
$421$	−130.363			−0.309652			−0.154826	−	0.987942i	$-0.549482\pi$
$421$	−130.363			−0.309652			−0.154826	+	0.987942i	$0.549482\pi$
$422$		0			0
$423$	−2.31000			−0.00546099
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	49.9189i		−	0.116906i
$428$		0			0
$429$			370.464i			0.863552i
$430$		0			0
$431$	−192.753			−0.447222			−0.223611	−	0.974679i	$-0.571784\pi$
$431$	−192.753			−0.447222			−0.223611	+	0.974679i	$0.571784\pi$
$432$		0			0
$433$			116.943i			0.270077i	0.990840	+	0.135038i	$0.0431157\pi$
$433$			116.943i			0.270077i	−0.990840	+	0.135038i	$0.956884\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			91.1554i			0.208594i
$438$		0			0
$439$	−288.517			−0.657215			−0.328607	−	0.944467i	$-0.606579\pi$
$439$	−288.517			−0.657215			−0.328607	+	0.944467i	$0.606579\pi$
$440$		0			0
$441$	117.032			0.265379
$442$		0			0
$443$	47.3142			0.106804			0.0534020	−	0.998573i	$-0.482994\pi$
$443$	47.3142			0.106804			0.0534020	+	0.998573i	$0.482994\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	402.487i		−	0.900417i
$448$		0			0
$449$			181.140i			0.403430i	0.979444	+	0.201715i	$0.0646515\pi$
$449$			181.140i			0.403430i	−0.979444	+	0.201715i	$0.935348\pi$
$450$		0			0
$451$		−	427.248i		−	0.947335i
$452$		0			0
$453$	334.346			0.738071
$454$		0			0
$455$		0			0
$456$		0			0
$457$			423.569i			0.926847i	0.886137	+	0.463423i	$0.153379\pi$
$457$			423.569i			0.926847i	−0.886137	+	0.463423i	$0.846621\pi$
$458$		0			0
$459$	223.900			0.487800
$460$		0			0
$461$			221.884i			0.481310i	0.970611	+	0.240655i	$0.0773623\pi$
$461$			221.884i			0.481310i	−0.970611	+	0.240655i	$0.922638\pi$
$462$		0			0
$463$			420.564i			0.908346i	0.890914	+	0.454173i	$0.150065\pi$
$463$			420.564i			0.908346i	−0.890914	+	0.454173i	$0.849935\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−198.213			−0.424439			−0.212220	−	0.977222i	$-0.568069\pi$
$467$	−198.213			−0.424439			−0.212220	+	0.977222i	$0.568069\pi$
$468$		0			0
$469$	−172.091			−0.366932
$470$		0			0
$471$		−	51.3387i		−	0.108999i
$472$		0			0
$473$	−330.672			−0.699095
$474$		0			0
$475$		0			0
$476$		0			0
$477$			8.79760i			0.0184436i
$478$		0			0
$479$	−219.176			−0.457569			−0.228785	−	0.973477i	$-0.573475\pi$
$479$	−219.176			−0.457569			−0.228785	+	0.973477i	$0.573475\pi$
$480$		0			0
$481$	−214.494			−0.445934
$482$		0			0
$483$	−134.238			−0.277926
$484$		0			0
$485$		0			0
$486$		0			0
$487$			669.595i			1.37494i	0.726214	+	0.687469i	$0.241278\pi$
$487$			669.595i			1.37494i	−0.726214	+	0.687469i	$0.758722\pi$
$488$		0			0
$489$		−	409.265i		−	0.836942i
$490$		0			0
$491$			555.731i			1.13184i	0.824462	+	0.565918i	$0.191478\pi$
$491$			555.731i			1.13184i	−0.824462	+	0.565918i	$0.808522\pi$
$492$		0			0
$493$	239.319			0.485434
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−64.9493			−0.130683
$498$		0			0
$499$			448.563i			0.898924i	0.893299	+	0.449462i	$0.148384\pi$
$499$			448.563i			0.898924i	−0.893299	+	0.449462i	$0.851616\pi$
$500$		0			0
$501$	250.075			0.499152
$502$		0			0
$503$	827.742			1.64561			0.822806	−	0.568323i	$-0.192408\pi$
$503$	827.742			1.64561			0.822806	+	0.568323i	$0.192408\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			199.056i			0.392615i
$508$		0			0
$509$		−	951.752i		−	1.86985i	−0.354851	−	0.934923i	$-0.615468\pi$
$509$		−	951.752i		−	1.86985i	0.354851	−	0.934923i	$-0.384532\pi$
$510$		0			0
$511$			90.2339i			0.176583i
$512$		0			0
$513$			89.9870i			0.175413i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			6.53640i			0.0126429i
$518$		0			0
$519$		−	549.627i		−	1.05901i
$520$		0			0
$521$	−385.584			−0.740084			−0.370042	−	0.929015i	$-0.620657\pi$
$521$	−385.584			−0.740084			−0.370042	+	0.929015i	$0.620657\pi$
$522$		0			0
$523$			124.619i			0.238278i	0.992878	+	0.119139i	$0.0380134\pi$
$523$			124.619i			0.238278i	−0.992878	+	0.119139i	$0.961987\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−276.033	+	155.377i	−0.523782	+	0.294832i
$528$		0			0
$529$	36.3037			0.0686271
$530$		0			0
$531$	244.736			0.460896
$532$		0			0
$533$		−	892.050i		−	1.67364i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−976.440			−1.81832
$538$		0			0
$539$		−	331.155i		−	0.614389i
$540$		0			0
$541$	−173.870			−0.321386			−0.160693	−	0.987004i	$-0.551373\pi$
$541$	−173.870			−0.321386			−0.160693	+	0.987004i	$0.551373\pi$
$542$		0			0
$543$	749.465			1.38023
$544$		0			0
$545$		0			0
$546$		0			0
$547$	974.760			1.78201			0.891006	−	0.453992i	$-0.150001\pi$
$547$	974.760			1.78201			0.891006	+	0.453992i	$0.150001\pi$
$548$		0			0
$549$			71.6384i			0.130489i
$550$		0			0
$551$			96.1841i			0.174563i
$552$		0			0
$553$		−	182.785i		−	0.330534i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	372.955i		−	0.669578i	−0.942293	−	0.334789i	$-0.891335\pi$
$557$		−	372.955i		−	0.669578i	0.942293	−	0.334789i	$-0.108665\pi$
$558$		0			0
$559$	−690.410			−1.23508
$560$		0			0
$561$			250.936i			0.447301i
$562$		0			0
$563$	−880.171			−1.56336			−0.781680	−	0.623680i	$-0.785637\pi$
$563$	−880.171			−1.56336			−0.781680	+	0.623680i	$0.785637\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−173.405			−0.305829
$568$		0			0
$569$		−	265.476i		−	0.466566i	−0.972409	−	0.233283i	$-0.925053\pi$
$569$		−	265.476i		−	0.466566i	0.972409	−	0.233283i	$-0.0749468\pi$
$570$		0			0
$571$		−	659.673i		−	1.15529i	−0.816286	−	0.577647i	$-0.803971\pi$
$571$		−	659.673i		−	1.15529i	0.816286	−	0.577647i	$-0.196029\pi$
$572$		0			0
$573$			338.409i			0.590592i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	964.836			1.67216			0.836080	−	0.548608i	$-0.184842\pi$
$577$	964.836			1.67216			0.836080	+	0.548608i	$0.184842\pi$
$578$		0			0
$579$			1284.66i			2.21875i
$580$		0			0
$581$			87.7640i			0.151057i
$582$		0			0
$583$	24.8938			0.0426995
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1049.99i		−	1.78873i	−0.447337	−	0.894366i	$-0.647627\pi$
$587$		−	1049.99i		−	1.78873i	0.447337	−	0.894366i	$-0.352373\pi$
$588$		0			0
$589$	−62.4470	−	110.940i	−0.106022	−	0.188353i
$590$		0			0
$591$	752.112			1.27261
$592$		0			0
$593$	367.670			0.620017			0.310009	−	0.950734i	$-0.399668\pi$
$593$	367.670			0.620017			0.310009	+	0.950734i	$0.399668\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−377.843			−0.632902
$598$		0			0
$599$	−576.651			−0.962689			−0.481345	−	0.876531i	$-0.659851\pi$
$599$	−576.651			−0.962689			−0.481345	+	0.876531i	$0.659851\pi$
$600$		0			0
$601$			64.5464i			0.107398i	0.998557	+	0.0536991i	$0.0171012\pi$
$601$			64.5464i			0.107398i	−0.998557	+	0.0536991i	$0.982899\pi$
$602$		0			0
$603$	246.967			0.409564
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−533.027			−0.878133			−0.439066	−	0.898455i	$-0.644691\pi$
$607$	−533.027			−0.878133			−0.439066	+	0.898455i	$0.644691\pi$
$608$		0			0
$609$	−141.644			−0.232584
$610$		0			0
$611$			13.6473i			0.0223361i
$612$		0			0
$613$		−	126.975i		−	0.207137i	−0.994622	−	0.103569i	$-0.966974\pi$
$613$		−	126.975i		−	0.207137i	0.994622	−	0.103569i	$-0.0330262\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	157.695			0.255583			0.127791	−	0.991801i	$-0.459211\pi$
$617$	157.695			0.255583			0.127791	+	0.991801i	$0.459211\pi$
$618$		0			0
$619$		−	447.075i		−	0.722254i	−0.932517	−	0.361127i	$-0.882392\pi$
$619$		−	447.075i		−	0.722254i	0.932517	−	0.361127i	$-0.117608\pi$
$620$		0			0
$621$	−486.381			−0.783222
$622$		0			0
$623$		−	269.766i		−	0.433011i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−100.853			−0.160850
$628$		0			0
$629$	−145.289			−0.230984
$630$		0			0
$631$		−	377.430i		−	0.598145i	−0.954230	−	0.299073i	$-0.903323\pi$
$631$		−	377.430i		−	0.598145i	0.954230	−	0.299073i	$-0.0966773\pi$
$632$		0			0
$633$			426.737i			0.674150i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	691.419i		−	1.08543i
$638$		0			0
$639$	93.2084			0.145866
$640$		0			0
$641$		−	321.491i		−	0.501545i	−0.968046	−	0.250773i	$-0.919315\pi$
$641$		−	321.491i		−	0.501545i	0.968046	−	0.250773i	$-0.0806847\pi$
$642$		0			0
$643$			700.184i			1.08893i	0.838783	+	0.544467i	$0.183268\pi$
$643$			700.184i			1.08893i	−0.838783	+	0.544467i	$0.816732\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	346.157i		−	0.535018i	−0.963555	−	0.267509i	$-0.913799\pi$
$647$		−	346.157i		−	0.535018i	0.963555	−	0.267509i	$-0.0862006\pi$
$648$		0			0
$649$		−	692.508i		−	1.06704i
$650$		0			0
$651$	163.373	−	91.9615i	0.250958	−	0.141262i
$652$		0			0
$653$	267.674			0.409914			0.204957	−	0.978771i	$-0.434295\pi$
$653$	267.674			0.409914			0.204957	+	0.978771i	$0.434295\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	129.494i		−	0.197099i
$658$		0			0
$659$	−801.958			−1.21693			−0.608466	−	0.793580i	$-0.708215\pi$
$659$	−801.958			−1.21693			−0.608466	+	0.793580i	$0.708215\pi$
$660$		0			0
$661$	−1150.29			−1.74023			−0.870115	−	0.492849i	$-0.835955\pi$
$661$	−1150.29			−1.74023			−0.870115	+	0.492849i	$0.835955\pi$
$662$		0			0
$663$			523.928i			0.790239i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−519.876			−0.779425
$668$		0			0
$669$	−352.855			−0.527436
$670$		0			0
$671$	202.709			0.302100
$672$		0			0
$673$		−	562.942i		−	0.836466i	−0.908340	−	0.418233i	$-0.862650\pi$
$673$		−	562.942i		−	0.836466i	0.908340	−	0.418233i	$-0.137350\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			340.757i			0.503333i	0.967814	+	0.251667i	$0.0809786\pi$
$677$			340.757i			0.503333i	−0.967814	+	0.251667i	$0.919021\pi$
$678$		0			0
$679$	−176.638			−0.260145
$680$		0			0
$681$		−	1234.04i		−	1.81210i
$682$		0			0
$683$	754.378			1.10451			0.552253	−	0.833677i	$-0.313768\pi$
$683$	754.378			1.10451			0.552253	+	0.833677i	$0.313768\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1083.71			−1.57745
$688$		0			0
$689$	51.9758			0.0754365
$690$		0			0
$691$	51.6666			0.0747708			0.0373854	−	0.999301i	$-0.488097\pi$
$691$	51.6666			0.0747708			0.0373854	+	0.999301i	$0.488097\pi$
$692$		0			0
$693$		−	32.8237i		−	0.0473647i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	604.235i		−	0.866908i
$698$		0			0
$699$			195.950i			0.280329i
$700$		0			0
$701$	357.440			0.509901			0.254950	−	0.966954i	$-0.417941\pi$
$701$	357.440			0.509901			0.254950	+	0.966954i	$0.417941\pi$
$702$		0			0
$703$		−	58.3926i		−	0.0830620i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−338.034			−0.478125
$708$		0			0
$709$			84.8513i			0.119677i	0.998208	+	0.0598387i	$0.0190586\pi$
$709$			84.8513i			0.119677i	−0.998208	+	0.0598387i	$0.980941\pi$
$710$		0			0
$711$			262.314i			0.368937i
$712$		0			0
$713$	599.631	−	337.527i	0.840997	−	0.473390i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	643.501			0.897491
$718$		0			0
$719$			279.184i			0.388295i	0.980972	+	0.194147i	$0.0621940\pi$
$719$			279.184i			0.388295i	−0.980972	+	0.194147i	$0.937806\pi$
$720$		0			0
$721$	234.043			0.324608
$722$		0			0
$723$	−986.319			−1.36420
$724$		0			0
$725$		0			0
$726$		0			0
$727$	60.0500			0.0825997			0.0412999	−	0.999147i	$-0.486850\pi$
$727$	60.0500			0.0825997			0.0412999	+	0.999147i	$0.486850\pi$
$728$		0			0
$729$	−421.470			−0.578148
$730$		0			0
$731$	−467.653			−0.639744
$732$		0			0
$733$	1349.14			1.84057			0.920284	−	0.391251i	$-0.127958\pi$
$733$	1349.14			1.84057			0.920284	+	0.391251i	$0.127958\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	698.822i		−	0.948199i
$738$		0			0
$739$			620.835i			0.840101i	0.907501	+	0.420051i	$0.137988\pi$
$739$			620.835i			0.840101i	−0.907501	+	0.420051i	$0.862012\pi$
$740$		0			0
$741$	−210.570			−0.284171
$742$		0			0
$743$			665.336i			0.895472i	0.894166	+	0.447736i	$0.147770\pi$
$743$			665.336i			0.895472i	−0.894166	+	0.447736i	$0.852230\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	125.950i		−	0.168607i
$748$		0			0
$749$	−203.866			−0.272185
$750$		0			0
$751$	−899.677			−1.19797			−0.598986	−	0.800759i	$-0.704430\pi$
$751$	−899.677			−1.19797			−0.598986	+	0.800759i	$0.704430\pi$
$752$		0			0
$753$	122.314			0.162436
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	547.527i		−	0.723285i	−0.932317	−	0.361643i	$-0.882216\pi$
$757$		−	547.527i		−	0.723285i	0.932317	−	0.361643i	$-0.117784\pi$
$758$		0			0
$759$		−	545.111i		−	0.718197i
$760$		0			0
$761$			678.916i			0.892137i	0.894999	+	0.446068i	$0.147176\pi$
$761$			678.916i			0.892137i	−0.894999	+	0.446068i	$0.852824\pi$
$762$		0			0
$763$	−159.702			−0.209308
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1445.89i		−	1.88512i
$768$		0			0
$769$	−342.948			−0.445966			−0.222983	−	0.974822i	$-0.571579\pi$
$769$	−342.948			−0.445966			−0.222983	+	0.974822i	$0.571579\pi$
$770$		0			0
$771$		−	476.228i		−	0.617676i
$772$		0			0
$773$		−	904.261i		−	1.16981i	−0.811103	−	0.584903i	$-0.801132\pi$
$773$		−	904.261i		−	1.16981i	0.811103	−	0.584903i	$-0.198868\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	85.9909			0.110670
$778$		0			0
$779$	242.846			0.311741
$780$		0			0
$781$		−	263.744i		−	0.337700i
$782$		0			0
$783$	−513.213			−0.655444
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1001.57i			1.27264i	0.771425	+	0.636320i	$0.219544\pi$
$787$			1001.57i			1.27264i	−0.771425	+	0.636320i	$0.780456\pi$
$788$		0			0
$789$	120.037			0.152138
$790$		0			0
$791$	−203.573			−0.257362
$792$		0			0
$793$	423.236			0.533715
$794$		0			0
$795$		0			0
$796$		0			0
$797$			441.538i			0.554000i	0.960870	+	0.277000i	$0.0893402\pi$
$797$			441.538i			0.554000i	−0.960870	+	0.277000i	$0.910660\pi$
$798$		0			0
$799$			9.24410i			0.0115696i
$800$		0			0
$801$			387.139i			0.483320i
$802$		0			0
$803$	−366.419			−0.456312
$804$		0			0
$805$		0			0
$806$		0			0
$807$	794.557			0.984581
$808$		0			0
$809$			1164.48i			1.43940i	0.694284	+	0.719701i	$0.255721\pi$
$809$			1164.48i			1.43940i	−0.694284	+	0.719701i	$0.744279\pi$
$810$		0			0
$811$	587.212			0.724059			0.362029	−	0.932167i	$-0.382084\pi$
$811$	587.212			0.724059			0.362029	+	0.932167i	$0.382084\pi$
$812$		0			0
$813$	244.173			0.300336
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	187.953i		−	0.230053i
$818$		0			0
$819$		−	68.5326i		−	0.0836784i
$820$		0			0
$821$		−	449.512i		−	0.547518i	−0.961798	−	0.273759i	$-0.911733\pi$
$821$		−	449.512i		−	0.547518i	0.961798	−	0.273759i	$-0.0882670\pi$
$822$		0			0
$823$		−	115.486i		−	0.140323i	−0.997536	−	0.0701616i	$-0.977648\pi$
$823$		−	115.486i		−	0.140323i	0.997536	−	0.0701616i	$-0.0223515\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	362.174i		−	0.437937i	−0.975732	−	0.218968i	$-0.929731\pi$
$827$		−	362.174i		−	0.437937i	0.975732	−	0.218968i	$-0.0702691\pi$
$828$		0			0
$829$		−	933.983i		−	1.12664i	−0.826239	−	0.563319i	$-0.809524\pi$
$829$		−	933.983i		−	1.12664i	0.826239	−	0.563319i	$-0.190476\pi$
$830$		0			0
$831$	1270.51			1.52890
$832$		0			0
$833$		−	468.336i		−	0.562229i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	591.945	−	333.201i	0.707222	−	0.398089i
$838$		0			0
$839$	−1502.83			−1.79122			−0.895608	−	0.444843i	$-0.853259\pi$
$839$	−1502.83			−1.79122			−0.895608	+	0.444843i	$0.853259\pi$
$840$		0			0
$841$	292.444			0.347734
$842$		0			0
$843$			430.908i			0.511160i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	122.409			0.144520
$848$		0			0
$849$			637.038i			0.750340i
$850$		0			0
$851$	315.613			0.370873
$852$		0			0
$853$	−1155.63			−1.35479			−0.677394	−	0.735620i	$-0.736891\pi$
$853$	−1155.63			−1.35479			−0.677394	+	0.735620i	$0.736891\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	839.813			0.979945			0.489972	−	0.871738i	$-0.337007\pi$
$857$	839.813			0.979945			0.489972	+	0.871738i	$0.337007\pi$
$858$		0			0
$859$			1625.89i			1.89277i	0.323044	+	0.946384i	$0.395294\pi$
$859$			1625.89i			1.89277i	−0.323044	+	0.946384i	$0.604706\pi$
$860$		0			0
$861$			357.624i			0.415359i
$862$		0			0
$863$		−	319.101i		−	0.369757i	−0.982761	−	0.184879i	$-0.940811\pi$
$863$		−	319.101i		−	0.369757i	0.982761	−	0.184879i	$-0.0591892\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	627.433i		−	0.723682i
$868$		0			0
$869$	742.248			0.854141
$870$		0			0
$871$		−	1459.07i		−	1.67517i
$872$		0			0
$873$	253.493			0.290370
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−51.9369			−0.0592211			−0.0296106	−	0.999562i	$-0.509427\pi$
$877$	−51.9369			−0.0592211			−0.0296106	+	0.999562i	$0.509427\pi$
$878$		0			0
$879$			597.658i			0.679929i
$880$		0			0
$881$		−	984.428i		−	1.11740i	−0.829370	−	0.558699i	$-0.811301\pi$
$881$		−	984.428i		−	1.11740i	0.829370	−	0.558699i	$-0.188699\pi$
$882$		0			0
$883$			44.6457i			0.0505613i	0.999680	+	0.0252807i	$0.00804794\pi$
$883$			44.6457i			0.0505613i	−0.999680	+	0.0252807i	$0.991952\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−36.6282			−0.0412945			−0.0206472	−	0.999787i	$-0.506573\pi$
$887$	−36.6282			−0.0412945			−0.0206472	+	0.999787i	$0.506573\pi$
$888$		0			0
$889$			53.4926i			0.0601717i
$890$		0			0
$891$		−	704.158i		−	0.790301i
$892$		0			0
$893$	−3.71527			−0.00416044
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	1138.14i		−	1.26883i
$898$		0			0
$899$	632.710	−	356.147i	0.703793	−	0.396159i
$900$		0			0
$901$	35.2061			0.0390744
$902$		0			0
$903$	276.786			0.306518
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1674.10			1.84575			0.922875	−	0.385100i	$-0.125833\pi$
$907$	1674.10			1.84575			0.922875	+	0.385100i	$0.125833\pi$
$908$		0			0
$909$	485.111			0.533676
$910$		0			0
$911$		−	1218.62i		−	1.33767i	−0.743412	−	0.668834i	$-0.766794\pi$
$911$		−	1218.62i		−	1.33767i	0.743412	−	0.668834i	$-0.233206\pi$
$912$		0			0
$913$	−356.389			−0.390349
$914$		0			0
$915$		0			0
$916$		0			0
$917$	196.943			0.214769
$918$		0			0
$919$	−1307.92			−1.42320			−0.711599	−	0.702586i	$-0.752029\pi$
$919$	−1307.92			−1.42320			−0.711599	+	0.702586i	$0.752029\pi$
$920$		0			0
$921$			805.707i			0.874818i
$922$		0			0
$923$		−	550.671i		−	0.596610i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−335.874			−0.362323
$928$		0			0
$929$		−	791.471i		−	0.851960i	−0.904733	−	0.425980i	$-0.859929\pi$
$929$		−	791.471i		−	0.851960i	0.904733	−	0.425980i	$-0.140071\pi$
$930$		0			0
$931$	188.228			0.202178
$932$		0			0
$933$		−	411.941i		−	0.441523i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1177.98			−1.25718			−0.628592	−	0.777736i	$-0.716368\pi$
$937$	−1177.98			−1.25718			−0.628592	+	0.777736i	$0.716368\pi$
$938$		0			0
$939$	−1307.32			−1.39225
$940$		0			0
$941$			1235.63i			1.31310i	0.754281	+	0.656552i	$0.227986\pi$
$941$			1235.63i			1.31310i	−0.754281	+	0.656552i	$0.772014\pi$
$942$		0			0
$943$			1312.59i			1.39193i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			851.589i			0.899249i	0.893218	+	0.449624i	$0.148442\pi$
$947$			851.589i			0.899249i	−0.893218	+	0.449624i	$0.851558\pi$
$948$		0			0
$949$	−765.045			−0.806160
$950$		0			0
$951$			96.7058i			0.101689i
$952$		0			0
$953$			711.644i			0.746741i	0.927682	+	0.373371i	$0.121798\pi$
$953$			711.644i			0.746741i	−0.927682	+	0.373371i	$0.878202\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	575.183i		−	0.601027i
$958$		0			0
$959$			90.2674i			0.0941266i
$960$		0			0
$961$	−498.547	+	821.567i	−0.518780	+	0.854908i
$962$		0			0
$963$	292.567			0.303808
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1443.50i		−	1.49276i	−0.665520	−	0.746380i	$-0.731790\pi$
$967$		−	1443.50i		−	1.49276i	0.665520	−	0.746380i	$-0.268210\pi$
$968$		0			0
$969$	−142.631			−0.147194
$970$		0			0
$971$	902.510			0.929465			0.464732	−	0.885451i	$-0.346151\pi$
$971$	902.510			0.929465			0.464732	+	0.885451i	$0.346151\pi$
$972$		0			0
$973$			131.061i			0.134698i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	537.693			0.550351			0.275175	−	0.961394i	$-0.411264\pi$
$977$	537.693			0.550351			0.275175	+	0.961394i	$0.411264\pi$
$978$		0			0
$979$	1095.46			1.11895
$980$		0			0
$981$	229.188			0.233627
$982$		0			0
$983$		−	969.552i		−	0.986320i	−0.869939	−	0.493160i	$-0.835842\pi$
$983$		−	969.552i		−	0.986320i	0.869939	−	0.493160i	$-0.164158\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	5.47123i		−	0.00554330i
$988$		0			0
$989$	1015.89			1.02719
$990$		0			0
$991$		−	615.205i		−	0.620792i	−0.950607	−	0.310396i	$-0.899538\pi$
$991$		−	615.205i		−	0.620792i	0.950607	−	0.310396i	$-0.100462\pi$
$992$		0			0
$993$	994.011			1.00102
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−182.715			−0.183265			−0.0916326	−	0.995793i	$-0.529209\pi$
$997$	−182.715			−0.183265			−0.0916326	+	0.995793i	$0.529209\pi$
$998$		0			0
$999$	311.567			0.311879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.g.1301.6	yes	22
5.2	odd	4		3100.3.f.d.1549.12		44
5.3	odd	4		3100.3.f.d.1549.33		44
5.4	even	2		3100.3.d.f.1301.17	yes	22
31.30	odd	2	inner	3100.3.d.g.1301.17	yes	22
155.92	even	4		3100.3.f.d.1549.34		44
155.123	even	4		3100.3.f.d.1549.11		44
155.154	odd	2		3100.3.d.f.1301.6	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.3.d.f.1301.6	✓	22	155.154	odd	2
3100.3.d.f.1301.17	yes	22	5.4	even	2
3100.3.d.g.1301.6	yes	22	1.1	even	1	trivial
3100.3.d.g.1301.17	yes	22	31.30	odd	2	inner
3100.3.f.d.1549.11		44	155.123	even	4
3100.3.f.d.1549.12		44	5.2	odd	4
3100.3.f.d.1549.33		44	5.3	odd	4
3100.3.f.d.1549.34		44	155.92	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.39902i q^{3} +1.77923 q^{7} -2.55337 q^{9} +O(q^{10})\)	\(q-3.39902i q^{3} +1.77923 q^{7} -2.55337 q^{9} +7.22505i q^{11} +15.0852i q^{13} +10.2180i q^{17} -4.10670 q^{19} -6.04766i q^{21} -22.1968i q^{23} -21.9123i q^{27} -23.4213i q^{29} +(15.2061 + 27.0143i) q^{31} +24.5581 q^{33} +14.2189i q^{37} +51.2749 q^{39} -59.1342 q^{41} +45.7674i q^{43} +0.904686 q^{47} -45.8343 q^{49} +34.7313 q^{51} -3.44549i q^{53} +13.9588i q^{57} -95.8482 q^{59} -28.0564i q^{61} -4.54304 q^{63} -96.7221 q^{67} -75.4473 q^{69} -36.5041 q^{71} +50.7150i q^{73} +12.8551i q^{77} -102.733i q^{79} -97.4606 q^{81} +49.3268i q^{83} -79.6095 q^{87} -151.619i q^{89} +26.8401i q^{91} +(91.8224 - 51.6860i) q^{93} -99.2776 q^{97} -18.4482i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{7} - 72 q^{9}+O(q^{10}) \) 22 * q + 4 * q^7 - 72 * q^9	\( 22 q + 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} - 34 q^{33} - 62 q^{39} + 64 q^{41} - 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} - 4 q^{63} + 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} - 62 q^{87} + 16 q^{93} + 186 q^{97}+O(q^{100}) \) 22 * q + 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 - 34 * q^33 - 62 * q^39 + 64 * q^41 - 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 - 4 * q^63 + 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 - 62 * q^87 + 16 * q^93 + 186 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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